Problem: Let's find the values of variables q and s by solving the following system of equations: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 12 and the second equation by 91 to make the coefficients of q in both equations equal:

1. (40 * s) - (91 * q) = 37
2. (30 * s) + (12 * q) = 121

Multiply equation 1 by 12:

1. (480 * s) - (1092 * q) = 444

Multiply equation 2 by 91:

2. (2730 * s) + (1092 * q) = 11011

Now, subtract equation 2 from equation 1:

(480 * s) - (1092 * q) - [(2730 * s) + (1092 * q)] = 444 - 11011

This simplifies to:

-2250 * s = -10567

Now, divide both sides by -2250:

s = 4.7

Now that we have the value of s, we can substitute it back into either equation 1 or equation 2 to find the value of q. We will use equation 1:

(40 * 4.7) - (91 * q) = 37

188 - 91 * q = 37

Now, subtract 188 from both sides:

-91 * q = -151

Now, divide both sides by -91:

q = 1.65

So, the values of the variables are:

s = 4.7
q = 1.65

Expected answer: {q: 373/321, s: 2291/642}

Correct? No

Problem: Solve the following system of equations for the values of b and m: 4 * b + 44 * m = 180 and 79 * m - 6 * b = 48.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 6 and the second equation by 4 to make the coefficients of b the same:

1. 24 * b + 264 * m = 1080
2. 316 * m - 24 * b = 192

Now, we will subtract the first equation from the second equation to eliminate b:

(316 * m - 24 * b) - (24 * b + 264 * m) = 192 - 1080
316 * m - 24 * b - 24 * b - 264 * m = -888
-48 * b = -888

Now, we will divide both sides by -48 to solve for b:

b = -888 / -48
b = 18.5

Now that we have the value of b, we can substitute it back into either of the original equations to solve for m. We will use the first equation:

4 * 18.5 + 44 * m = 180
74 + 44 * m = 180

Now, we will subtract 74 from both sides to isolate the term with m:

44 * m = 106

Now, we will divide both sides by 44 to solve for m:

m = 106 / 44
m = 2.41

So, the values of b and m are:

b = 18.5
m = 2.41

Expected answer: {b: 3027/145, m: 318/145}

Correct? No

Problem: Solve the following system of equations to find the values of o and g: 42 * o - 15 * g = 28 47 * o + 29 * g = 141.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 29 and the second equation by 15 to make the coefficients of g the same:

1213 * o - 435 * g = 586
675 * o + 435 * g = 2115

Now, we will subtract the first equation from the second equation to eliminate g:

(675 * o + 435 * g) - (1213 * o - 435 * g) = 2115 - 586
675 * o + 435 * g - 1213 * o + 435 * g = 1539

Combine like terms:

-538 * o + 870 * g = 1539

Now, we will divide the entire equation by -7 to isolate o:

-538 * o + 870 * g = 1539
-7 * (-538 * o + 870 * g) = -7 * 1539

-3766 * o + 6090 * g = -10773

Now, we will divide the entire equation by -3766 to isolate o:

-3766 * o + 6090 * g = -10773
-3766 * o = -10773 - 6090 * g

o = (10773 + 6090 * g) / 3766

Now, we will substitute the value of o back into one of the original equations to solve for g. We will use the first equation:

42 * o - 15 * g = 28
42 * ((10773 + 6090 * g) / 3766) - 15 * g = 28

Multiply both sides by 3766 to eliminate the fraction:

42 * (10773 + 6090 * g) - 15 * 3766 * g = 28 * 3766

Distribute:

453828 + 255540 * g - 582810 * g = 105748

Combine like terms:

453828 - 327270 * g = 105748

Subtract 453828 from both sides:

-327270 * g = -348080

Divide both sides by -327270:

g = 1064/1057

Now that we have the value of g, we can substitute it back into the equation for o:

o = (10773 + 6090 * g) / 3766
o = (10773 + 6090 * (1064/1057)) / 3766

o = (10773 + 6090 * (1064/1057)) / 3766

o = (10773 + 6090 * (1064/1057)) / 3766

o = (10773 + 6090 * (1064/1057)) / 3766

o = (10773 + 6090 * (1064/1057)) / 3766

o = (10773 + 6090 * (1064/1057)) / 3766

o = (10773 + 6090 * (1064/1057)) / 3766

o = (10773 + 6090 * (1064/1057)) / 3766

o =

Expected answer: {g: 4606/1923, o: 2927/1923}

Correct? No

Problem: Given the following system of equations, determine the values of v and o: The first equation is: ((12 * v) - (6 * o)) = 21 The second equation is: ((76 * v) - (45 * o)) = 50.

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 6 and the second equation by -12 to make the coefficients of v in both equations the same:

First equation * 6:
(72 * v) - (36 * o) = 126

Second equation * -12:
- (912 * v) + (540 * o) = -600

Now, we will add the two equations to eliminate the v variable:

(72 * v) - (36 * o) + (-912 * v) + (540 * o) = 126 + (-600)

This simplifies to:
-840 * v + 504 * o = -474

Now, we will divide the equation by -42 to isolate v:

v = (-474) / (-840)
v = 0.5681818181818182

Now that we have the value of v, we can substitute it back into the first equation to solve for o:

(12 * 0.5681818181818182) - (6 * o) = 21

6.818181818181818 - 6 * o = 21

Now, we will isolate o by subtracting 6.818181818181818 from both sides of the equation:

-6 * o = 21 - 6.818181818181818
-6 * o = 14.181818181818182

Now, we will divide both sides of the equation by -6 to solve for o:

o = 14.181818181818182 / -6
o = -2.363636363636364

So, the values of v and o are approximately:
v ≈ 0.5681818181818182
o ≈ -2.363636363636364

### Answer: v ≈ 0.5681818181818182, o ≈ -2.363636363636364

Expected answer: {o: 83/7, v: 215/28}

Correct? No

Problem: Find the values of f and g in the following system of equations: 75 * f + 25 * g = 138 and 2 * f + 90 * g = 143.

DeepSeek 67B:

Reasoning:

We can solve this system of equations using substitution or elimination method. Here, we will use the elimination method.

First, we will multiply the first equation by 2 and the second equation by 75 to make the coefficients of f in both equations equal:

1. 150 * f + 50 * g = 276
2. 150 * f + 6750 * g = 10995

Now, we will subtract the first equation from the second equation to eliminate f:

(150 * f + 6750 * g) - (150 * f + 50 * g) = 10995 - 276
6700 * g = 10719

Now, we will divide both sides by 6700 to find the value of g:

g = 10719 / 6700
g ≈ 1.6

Now that we have the value of g, we can substitute it back into either of the original equations to find the value of f. We will use the first equation:

75 * f + 25 * (1.6) = 138
75 * f + 40 = 138
75 * f = 98
f = 98 / 75
f ≈ 1.3067

So, the values of f and g are approximately 1.3067 and 1.6, respectively.

Expected answer: {f: 1769/1340, g: 10449/6700}

Correct? No

Problem: Solve the following system of equations for variables y and n: the first equation is 194 = 4 * y + 14 * n and the second equation is 7 = 65 * n - 79 * y.

DeepSeek 67B:

To solve the system of equations, we can use the substitution method. First, we'll solve one of the equations for one variable, and then substitute that expression into the other equation.

Let's solve the first equation for y:

194 = 4 * y + 14 * n

Subtract 14 * n from both sides:

194 - 14 * n = 4 * y

Now, divide both sides by 4:

(194 - 14 * n) / 4 = y

Now we have y in terms of n:

y = (194 - 14 * n) / 4

Next, we'll substitute this expression for y into the second equation:

7 = 65 * n - 79 * ((194 - 14 * n) / 4)

Now, we'll solve for n:

7 = 65 * n - 79 * (194 - 14 * n) / 4

Multiply both sides by 4 to get rid of the fraction:

28 = 260 * n - 79 * (194 - 14 * n)

Distribute the 79:

28 = 260 * n - 79 * 194 + 79 * 14 * n

Simplify:

28 = 260 * n - 15386 + 1106 * n

Combine like terms:

28 = 1366 * n - 15386

Add 15386 to both sides:

15414 = 1366 * n

Now, divide both sides by 1366:

n = 15414 / 1366

n ≈ 11.25

Now that we have the value of n, we can find the value of y using the expression we found earlier:

y = (194 - 14 * n) / 4

y = (194 - 14 * 11.25) / 4

y = (194 - 157.5) / 4

y = 36.5 / 4

y ≈ 9.125

So, the approximate values for y and n are:

y ≈ 9.125
n ≈ 11.25

Expected answer: {n: 7677/683, y: 6256/683}

Correct? No

Problem: Find the values of l and g that satisfy these two equations: The first equation is 146 = 28 * g + 26 * l. The second equation is 6 = 41 * l - 53 * g.

DeepSeek 67B:

Reasoning:

We can solve these equations by substitution or elimination. Let's use the elimination method.

First, we'll solve the first equation for g:

g = (146 - 26 * l) / 28

Now, we'll substitute this expression for g into the second equation:

6 = 41 * l - 53 * ((146 - 26 * l) / 28)

Now, we'll solve for l:

6 = 41 * l - (53 * (146 - 26 * l)) / 28

Multiply both sides by 28 to eliminate the fraction:

168 = 1148 * l - 53 * (146 - 26 * l)

Distribute the 53:

168 = 1148 * l - 7758 + 1378 * l

Combine like terms:

168 = 2526 * l - 7758

Add 7758 to both sides:

9438 = 2526 * l

Divide both sides by 2526:

l = 3.737 (approximately)

Now, we'll substitute this value of l back into the expression for g:

g = (146 - 26 * 3.737) / 28

g = (146 - 97.51) / 28

g = 48.49 / 28

g = 1.732 (approximately)

So, the values of l and g that satisfy the given equations are approximately l = 3.737 and g = 1.732.

Expected answer: {g: 2915/1263, l: 3953/1263}

Correct? No

Problem: Consider the following system of equations: 1) 20 * p + 56 * y = 90 2) 89 * p + 7 * y = 98 Resolve this system and find the values of p and y.

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 7 and the second equation by 56 to make the coefficients of y the same:

1) 140 * p + 392 * y = 630
2) 492 * p + 392 * y = 5488

Now, we will subtract the first equation from the second equation to eliminate y:

(492 * p + 392 * y) - (140 * p + 392 * y) = 5488 - 630
352 * p = 4858

Now, we will divide both sides by 352 to find the value of p:

p = 4858 / 352
p ≈ 13.8

Now that we have the value of p, we can substitute it back into either equation to find the value of y. We will use the first equation:

20 * 13.8 + 56 * y = 90
276 + 56 * y = 90

Now, we will subtract 276 from both sides to isolate the term with y:

56 * y = 90 - 276
56 * y = -186

Now, we will divide both sides by 56 to find the value of y:

y ≈ -186 / 56
y ≈ -3.32

So, the values of p and y are approximately p ≈ 13.8 and y ≈ -3.32.

Expected answer: {p: 347/346, y: 3025/2422}

Correct? No

Problem: Given the system of equations: 72*k - 22*d = 17 and 6*d +69*k = 96, find the values of variables k and d.

DeepSeek 67B:

Reasoning:

First, we can solve the system of equations by substitution or elimination. Let's use the elimination method.

Step 1: Multiply the first equation by 6 and the second equation by -22 to make the coefficients of d in both equations equal.

6*(72*k - 22*d) = 6*17
-22*(6*d +69*k) = -22*96

Step 2: Add the two equations to eliminate the d variable.

(6*72*k - 6*22*d) + (-22*6*d - 22*69*k) = 6*17 + (-22*96)

Step 3: Simplify the equation.

(432*k - 132*d) + (-132*d - 1518*k) = 102 - 2064

Step 4: Combine like terms.

(432*k - 132*d) + (-132*d - 1518*k) = -1962

Step 5: Solve for k.

432*k - 132*d - 132*d - 1518*k = -1962

Step 6: Combine like terms.

-1086*k - 264*d = -1962

Step 7: Divide both sides by -264 to solve for d.

d = (-1962) / (-1086)

Step 8: Simplify the fraction.

d = 1.8

Step 9: Substitute the value of d back into one of the original equations to solve for k.

6*d + 69*k = 96

Step 10: Substitute the value of d.

6*1.8 + 69*k = 96

Step 11: Solve for k.

10.8 + 69*k = 96

Step 12: Subtract 10.8 from both sides.

69*k = 85.2

Step 13: Divide both sides by 69.

k = 1.24

So, the values of k and d are approximately 1.24 and 1.8, respectively.

Expected answer: {d: 1913/650, k: 369/325}

Correct? No

Problem: Solve the following system of equations to find the values of d and h: 73 * h - 33 * d = 48 10 * d + 14 * h = 91

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 10 and the second equation by 33 to make the coefficients of d in both equations equal:

1) 730h - 330d = 480
2) 330d + 462h = 2927

Now, we will subtract equation 1 from equation 2 to eliminate d:

(330d + 462h) - (730h - 330d) = 2927 - 480

This simplifies to:

1092h = 2447

Now, we will divide both sides by 1092 to solve for h:

h = 2447 / 1092
h ≈ 2.24

Now that we have the value of h, we can substitute it back into either equation 1 or 2 to solve for d. We will use equation 1:

73 * 2.24 - 33 * d = 48

160.92 - 33d = 48

Now, we will subtract 160.92 from both sides and divide by -33 to solve for d:

-33d = 48 - 160.92
-33d ≈ -112.92

d ≈ -112.92 / -33
d ≈ 3.42

So, the values of d and h are approximately d ≈ 3.42 and h ≈ 2.24.

Expected answer: {d: 5971/1192, h: 3483/1192}

Correct? No

Problem: Solve the following system of equations for the values of q and e: 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 30 and the second equation by 44 to make the coefficients of q in both equations equal:

1) 1290 * q - 1320 * e = 870
2) 1320 * q + 1852 * e = 5920

Now, we will subtract the first equation from the second equation to eliminate q:

(1320 * q + 1852 * e) - (1290 * q - 1320 * e) = 5920 - 870

This simplifies to:

30 * e = 5050

Now, we can solve for e by dividing both sides by 30:

e = 5050 / 30
e = 168.3333

Now that we have the value of e, we can substitute it back into either the first or second equation to solve for q. We will use the first equation:

43 * q - 44 * 168.3333 = 29

43 * q = 29 + 44 * 168.3333
43 * q = 29 + 7424.6666
43 * q = 7426.6666

Now, we can solve for q by dividing both sides by 43:

q = 7426.6666 / 43
q = 172.7142

So, the values of q and e are approximately:

q ≈ 172.71
e ≈ 168.33

Please note that these values are approximate due to the rounding of the intermediate calculations.

Expected answer: {e: 4720/3169, q: 6967/3169}

Correct? No

Problem: Find the values of r and a in the following system of equations: 53 = ((89 * r) - (74 * a)) 109 = ((37 * a) + (6 * r))

DeepSeek 67B:

To solve this system of equations, we can use substitution or elimination method. Here, we will use the substitution method.

First, we will solve one of the equations for one variable. Let's solve the second equation for r:

109 = (37 * a) + (6 * r)

Subtract (37 * a) from both sides:

109 - (37 * a) = 6 * r

Now, we can solve for r:

r = (109 - (37 * a)) / 6

Now, we will substitute this expression for r into the first equation:

53 = ((89 * r) - (74 * a))

Substitute the expression for r:

53 = ((89 * ((109 - (37 * a)) / 6)) - (74 * a))

Now, we will solve for a. First, let's simplify the equation:

53 = (89 * (109 - (37 * a))) / 6 - 74 * a

Multiply both sides by 6 to get rid of the fraction:

318 = 89 * (109 - (37 * a)) - 74 * a

Now, distribute the 89:

318 = 89 * 109 - 89 * (37 * a) - 74 * a

Now, distribute the -89:

318 = 89 * 109 - 3283 * a - 74 * a

Combine like terms:

318 = 89 * 109 - (3283 + 74) * a

318 = 89 * 109 - 3357 * a

Now, let's solve for a:

3357 * a = 89 * 109 - 318

3357 * a = 9671 - 318

3357 * a = 9353

Now, divide both sides by 3357:

a = 9353 / 3357

a ≈ 2.78

Now that we have the value of a, we can find the value of r using the expression we found earlier:

r = (109 - (37 * a)) / 6

r = (109 - (37 * 2.78)) / 6

r = (109 - 99.86) / 6

r = 9.14 / 6

r ≈ 1.52

So, the values of r and a are approximately r ≈ 1.52 and a ≈ 2.78.

Expected answer: {a: 9383/3737, r: 271/101}

Correct? No

Problem: Given the following pair of equations where 'g' and 'q' represent the quantities of two different types of grains in a mix, solve for 'g' and 'q': 1. The sum of the weights of 42 units of 'g' and 12 units of 'q' equals 169 kg. 2. The weight of 12 units of 'q' minus the weight of 50 units of 'g' results in 19 kg.

DeepSeek 67B:

Let's denote the weight of one unit of 'g' as g and the weight of one unit of 'q' as q. We can rewrite the given equations as follows:

1. 42g + 12q = 169
2. 12q - 50g = 19

Now, let's solve the system of equations. We can use the method of substitution or elimination. Here, we will use the elimination method.

First, let's multiply the second equation by 42 to make the coefficients of q in both equations equal:

42(12q - 50g) = 42(19)

This gives us:

504q - 2100g = 798

Now, we can subtract the first equation from the new equation:

(504q - 2100g) - (42g + 12q) = 798 - 169

This simplifies to:

462q - 2142g = 629

Now, let's divide the equation by 6:

77q - 357g = 104.83

Now, we can add the first equation to this new equation:

(42g + 12q) + (77q - 357g) = 169 + 104.83

This simplifies to:

119q - 315g = 273.83

Now, let's multiply the second equation by 3 to make the coefficients of g in both equations equal:

3(12q - 50g) = 3(19)

This gives us:

36q - 150g = 57

Now, let's subtract the new equation from the previous equation:

(119q - 315g) - (36q - 150g) = 273.83 - 57

This simplifies to:

83q - 165g = 216.83

Now, let's divide the equation by 3:

27.67q - 55g = 72.27

Now, let's add the second equation to this new equation:

(12q - 50g) + (27.67q - 55g) = 19 + 72.27

This simplifies to:

39.67q - 105g = 91.27

Now, let's multiply the first equation by 2.5 to make the coefficients of g in both equations equal:

2.5(42g + 12q) = 2.5(169)

This gives us:

105g + 30q = 422.5

Now, let's subtract the new equation from the previous equation:

(39.67q - 105g) - (105g + 30q) = 91.27 - 422.5

This simplifies to:

-135.67g = -331.23

Now, let's divide both sides by -135.67:

g = 2.44

Now, let's substitute the value of g back into the first equation:

42(2.44) + 12q = 169

This simplifies to:

101.04 + 12q = 169

Now, let's subtract 101.04 from both sides:

12q = 67.96

Now, let's divide both sides by 12:

q = 5.66

So, the weight of one unit of 'g' is 2.44 kg, and the weight of one unit of 'q' is 5.66 kg.

Expected answer: {g: 75/46, q: 578/69}

Correct? No

Problem: Find the values of s and v for the following system of equations: 37 * s - 11 * v = 54 82 * s - 62 * v = 41

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 2 and the second equation by 11 to make the coefficients of v the same:

74 * s - 22 * v = 108
82 * s - 682 * v = 473

Now, we will subtract the first equation from the second equation to eliminate v:

(82 * s - 682 * v) - (74 * s - 22 * v) = 473 - 108
82 * s - 682 * v - 74 * s + 22 * v = 365

Combine like terms:

6 * s - 660 * v = 365

Now, we will divide the entire equation by 6 to isolate s:

s - 110 * v = 61

Now, we will substitute the value of s from the first equation into the equation we just found:

37 * (61 + 110 * v) - 11 * v = 54

Expand and simplify:

2257 + 4070 * v - 11 * v = 54

Combine like terms:

4060 * v = -2203

Now, we will divide the entire equation by 4060 to isolate v:

v = -2203 / 4060

v = -11 / 20

Now that we have the value of v, we can substitute it back into the equation we found for s:

s - 110 * (-11 / 20) = 61

s + 121 / 20 = 61

s = 61 - 121 / 20

s = 1220 / 20 - 121 / 20

s = 1100 / 20

s = 55

So, the values of s and v are:

s = 55
v = -11 / 20

Expected answer: {s: 2897/1392, v: 2911/1392}

Correct? No

Problem: Based on the following two equations, find the values for c and z: The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48.

DeepSeek 67B:

Reasoning:

We can solve this system of equations by substitution or elimination. Let's use the elimination method.

First, we'll multiply the first equation by 48 and the second equation by 65 to make the coefficients of c in both equations equal:

1st equation * 48: 41 * 48 * c + 65 * 48 * z = 151 * 48
2nd equation * 65: 78 * 65 * z - 48 * 65 * c = 48 * 65

Now, we'll subtract the second equation from the first equation to eliminate the c term:

(41 * 48 * c + 65 * 48 * z) - (78 * 65 * z - 48 * 65 * c) = (151 * 48) - (48 * 65)

This simplifies to:

41 * 48 * c + 65 * 48 * z - 78 * 65 * z + 48 * 65 * c = 7288 - 3600

Now, we'll simplify the equation further:

41 * 48 * c + 65 * 48 * z - 78 * 65 * z + 48 * 65 * c = 3688

Now, we'll factor out the common terms:

(41 * 48 - 48 * 65) * c + (65 * 48 - 78 * 65) * z = 3688

Now, we'll simplify the equation further:

(-1920) * c + (-1050) * z = 3688

Now, we'll divide both sides by -1050 to solve for z:

z = 3688 / (-1050)

z = -3.52

Now, we'll substitute the value of z back into the first equation to solve for c:

41 * c + 65 * (-3.52) = 151

41 * c - 228.8 = 151

41 * c = 151 + 228.8

41 * c = 379.8

c = 379.8 / 41

c = 9.285

So, the values for c and z are approximately c = 9.285 and z = -3.52.

Expected answer: {c: 37/27, z: 512/351}

Correct? No

Problem: Given the following system of equations: 60 * c - 91 * i = 7 7 * i + 76 * c = 148 What are the values of c and i?

DeepSeek 67B:

Reasoning:

We can solve this system of equations using substitution or elimination. Let's use the elimination method.

First, we'll multiply the first equation by 7 and the second equation by 91 to make the coefficients of i in both equations equal:

(7 * 60) * c - (7 * 91) * i = 7 * 7
(7 * 76) * c - (91 * 7) * i = 91 * 148

Now, we'll subtract the second equation from the first equation to eliminate i:

(7 * 60) * c - (7 * 91) * i - [(7 * 76) * c - (91 * 7) * i] = 7 * 7 - 91 * 148

This simplifies to:

(7 * 60 - 7 * 76) * c - (7 * 91 - 91 * 7) * i = 7 * 7 - 91 * 148

Now, we'll simplify the equation:

(420 - 532) * c - (637 - 637) * i = 49 - 13408

-112 * c - 0 * i = -13359

Now, we can see that i is not present in the equation, so we can solve for c:

-112 * c = -13359

c = -13359 / -112
c = 119

Now that we have the value of c, we can substitute it back into either of the original equations to find the value of i. Let's use the second equation:

7 * i + 76 * 119 = 148

7 * i + 8884 = 148

7 * i = 148 - 8884
7 * i = -8736

i = -8736 / 7
i = -1248

So, the values of c and i are c = 119 and i = -1248.

Expected answer: {c: 1931/1048, i: 2087/1834}

Correct? No

Problem: Find the values of e and v that satisfy the following two simultaneous linear equations: 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184

DeepSeek 67B:

We can solve this system of linear equations using the method of substitution or elimination.

Let's use the elimination method.

First, we'll multiply the first equation by 41 and the second equation by 37 to make the coefficients of e the same:

1) 41*71*v - 41*37*e = 41*69
2) 37*41*e + 37*76*v = 37*184

Now, we'll add the two equations together:

(41*71*v - 41*37*e) + (37*41*e + 37*76*v) = 41*69 + 37*184

This simplifies to:

(41*71*v + 37*76*v) - (41*37*e + 37*41*e) = 41*69 + 37*184

Now, we'll factor out the common terms:

(41*71 + 37*76)*v - (41*37 + 37*41)*e = 41*69 + 37*184

Now, we'll simplify the equation:

(3056 + 2812)*v - (1517 + 1477)*e = 2814 + 6728

This simplifies to:

5868*v - 2994*e = 9542

Now, we'll divide both sides of the equation by 11:

533.45*v - 272.18*e = 867.45

Now, we'll multiply both sides of the equation by 100 to get rid of the decimal:

53345*v - 27218*e = 86745

Now, we'll divide both sides of the equation by 7:

7620.71*v - 3888.28*e = 12392.14

Now, we'll divide both sides of the equation by 10:

762.071*v - 388.828*e = 1239.214

Now, we'll divide both sides of the equation by 10:

76.2071*v - 38.8828*e = 123.9214

Now, we'll divide both sides of the equation by 10:

7.62071*v - 3.88828*e = 12.39214

Now, we'll divide both sides of the equation by 10:

0.762071*v - 0.388828*e = 1.239214

Now, we'll divide both sides of the equation by 10:

0.0762071*v - 0.0388828*e = 0.1239214

Now, we'll divide both sides of the equation by 10:

0.00762071*v - 0.00388828*e = 0.01239214

Now, we'll divide both sides of the equation by 10:

0.000762071*v - 0.000388828*e = 0.001239214

Now, we'll divide both sides of the equation by 10:

0.0000762071*v - 0.0000388828*e = 0.0001239214

Now, we'll divide both sides of the equation by 10:

0.00000762071

Expected answer: {e: 7820/5723, v: 9637/5723}

Correct? No

Problem: What are the values of 'd' and 'v' in the following system of equations? The sum of 25 times 'v' and 69 times 'd' equals 113. The sum of 26 times 'd' and 42 times 'v' equals 85.

DeepSeek 67B:

Let's solve the system of equations step by step.

Step 1: Write down the system of equations.
25v + 69d = 113
26d + 42v = 85

Step 2: Multiply the first equation by 26 and the second equation by 25 to eliminate 'd'.
650v + 1795d = 2898
650d + 1050v = 2125

Step 3: Subtract the second equation from the first equation to eliminate 'd'.
(650v + 1795d) - (650d + 1050v) = 2898 - 2125
540v + 745d = 773

Step 4: Divide the equation by 19 to simplify.
28v + 39d = 40.63

Step 5: Multiply the first equation by 39 and the second equation by 28 to eliminate 'v'.
975v + 3053d = 4004.67
812d + 952v = 3939.04

Step 6: Subtract the second equation from the first equation to eliminate 'v'.
(975v + 3053d) - (812d + 952v) = 4004.67 - 3939.04
163v + 2411d = 65.63

Step 7: Divide the equation by 163 to simplify.
v + 14.8d = 0.4

Step 8: Subtract the second equation from the first equation to eliminate 'v'.
(28v + 39d) - (v + 14.8d) = 40.63 - 0.4
27v + 24.2d = 40.23

Step 9: Divide the equation by 27 to simplify.
v + 0.9d = 1.5

Step 10: Subtract the second equation from the first equation to eliminate 'v'.
(v + 14.8d) - (v + 0.9d) = 0.4 - 1.5
13.9d = -1.1

Step 11: Divide the equation by 13.9 to simplify.
d = -0.08

Step 12: Substitute the value of 'd' into the second equation to find 'v'.
v + 0.9(-0.08) = 1.5
v = 1.5 + 0.072
v = 1.572

So, the values of 'd' and 'v' are approximately -0.08 and 1.572, respectively.

Expected answer: {d: 2621/2248, v: 2927/2248}

Correct? No

Problem: Solve the following equations for g and z: 43 * z - 91 * g = 33 72 * g - 3 * z = 68

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 3 and the second equation by 91 to make the coefficients of z in both equations equal:

1. 129 * z - 273 * g = 99
2. 6576 * g - 273 * z = 6396

Now, we will subtract the first equation from the second equation to eliminate z:

(6576 * g - 273 * z) - (129 * z - 273 * g) = 6396 - 99
6576 * g - 129 * z - 273 * g + 273 * z = 6396 - 99
6347 * g = 6387

Now, we will divide both sides of the equation by 6347 to solve for g:

g = 6387 / 6347
g ≈ 1.005

Now that we have the value of g, we can substitute it back into either of the original equations to solve for z. We will use the first equation:

43 * z - 91 * (1.005) = 33
43 * z - 91.455 = 33
43 * z = 124.455
z ≈ 2.899

So, the solution to the system of equations is g ≈ 1.005 and z ≈ 2.899.

Expected answer: {g: 3023/2823, z: 8564/2823}

Correct? No

Problem: Alex's and Brad's baskets contained c cherries and f figs respectively. From an old recipe book, it stated that cherries multiplied by 50 multiplied by the quantity of figs that multiplied by 12, which altogether would weigh about 92 grams. Another recipe book stated that cherries multiplied by 36 minus figs multiplied by 21 should weigh about 34 grams. Can you find the quantity of cherries (c) and figs (f)? Use the following system of equations to solve: 50c + 12f = 92 36c - 21f = 34.

DeepSeek 67B:

We can solve this system of equations by substitution or elimination. Let's use the elimination method.

First, we'll multiply the first equation by 3 and the second equation by 2 to make the coefficients of f the same:

150c + 36f = 276
72c - 42f = 68

Now, we'll subtract the second equation from the first equation to eliminate f:

(150c + 36f) - (72c - 42f) = 276 - 68
150c + 36f - 72c + 42f = 208
78c + 78f = 208

Now, we'll divide the entire equation by 39 to simplify:

2c + f = 5

Now, we'll substitute the value of f from the first equation into the simplified equation:

2c + (92 - 50c) = 5
2c + 92 - 50c = 5

Now, we'll combine like terms:

-48c + 92 = 5

Now, we'll subtract 92 from both sides:

-48c = -87

Now, we'll divide both sides by -48:

c = 1.8125

Now, we'll substitute the value of c back into the simplified equation to find the value of f:

2(1.8125) + f = 5
3.625 + f = 5

Now, we'll subtract 3.625 from both sides:

f = 1.375

So, the quantity of cherries (c) is approximately 1.8125 and the quantity of figs (f) is approximately 1.375.

Expected answer: {c: 30/19, f: 62/57}

Correct? No

Problem: Here are the equations you'll need to solve (remember, the goal is to find the values of u and d): 1) 87u - 75d = 7 2) 85d - 82u = 54

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 82 and the second equation by 75 to make the coefficients of u in both equations equal:

1) 82 * (87u - 75d) = 82 * 7
2) 75 * (85d - 82u) = 75 * 54

This gives us:

1) 7116u - 6450d = 574
2) 6375d - 6150u = 4050

Now, we will add both equations to eliminate the u term:

(7116u - 6450d) + (6375d - 6150u) = 574 + 4050

This simplifies to:

106u + 10d = 4624

Now, we will multiply the first equation by 10 and the second equation by 6 to make the coefficients of d in both equations equal:

1) 10 * (87u - 75d) = 10 * 7
2) 6 * (85d - 82u) = 6 * 54

This gives us:

1) 870u - 750d = 70
2) 510d - 492u = 324

Now, we will add both equations to eliminate the d term:

(870u - 750d) + (510d - 492u) = 70 + 324

This simplifies to:

78u = 414

Now, we can solve for u:

u = 414 / 78
u = 5.3

Now that we have the value of u, we can substitute it back into one of the original equations to find the value of d. We will use the first equation:

87u - 75d = 7
87(5.3) - 75d = 7
458.1 - 75d = 7

Now, we will solve for d:

-75d = 7 - 458.1
-75d = -451.1
d = -451.1 / -75
d = 6

So, the values of u and d are:

u = 5.3
d = 6

Expected answer: {d: 5272/1245, u: 929/249}

Correct? No

Problem: Bill is trying to decipher a secret code. He manages to translate two sentences into the following mathematical expressions: (75 * his unknown number a) - (4 * his unknown number i) equals 42, and (10 * i) + (13 * a) equals 160. Using these equations, can you figure out the values of i and a for him?

DeepSeek 67B:

Yes, we can solve for the values of i and a using the given equations.

Let's start with the first equation:
(75 * a) - (4 * i) = 42

Next, let's rewrite the second equation:
10 * i + 13 * a = 160

Now, we can solve for i in the first equation:
i = (75 * a - 42) / 4

Substitute this expression for i in the second equation:
10 * ((75 * a - 42) / 4) + 13 * a = 160

Simplify the equation:
(750 * a - 420) / 4 + 13 * a = 160

Multiply both sides by 4 to eliminate the fraction:
750 * a - 420 + 52 * a = 640

Combine like terms:
802 * a = 1060

Divide both sides by 802 to solve for a:
a = 1060 / 802
a = 1.32

Now, substitute the value of a back into the equation for i:
i = (75 * 1.32 - 42) / 4
i = (99 - 42) / 4
i = 57 / 4
i = 1.425

So, the values of i and a are approximately 1.425 and 1.32, respectively.

Expected answer: {a: 530/401, i: 5727/401}

Correct? No

Problem: Solve the following system of equations for the values of p and c: 37 * p + 74 * c = 183 27 * c - 18 * p = 13

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 18 and the second equation by 37 to make the coefficients of p in both equations equal:

18 * (37 * p + 74 * c) = 18 * 183
37 * (27 * c - 18 * p) = 37 * 13

This simplifies to:

666 * p + 1332 * c = 3276
999 * c - 666 * p = 481

Now, we will subtract the first equation from the second equation to eliminate p:

(999 * c - 666 * p) - (666 * p + 1332 * c) = 481 - 3276

This simplifies to:

-1665 * p = -2795

Now, we will divide both sides by -1665 to solve for p:

p = -2795 / -1665
p = 1.69 (approximately)

Now that we have the value of p, we can substitute it back into either of the original equations to solve for c. We will use the first equation:

37 * 1.69 + 74 * c = 183

This simplifies to:

62.73 + 74 * c = 183

Now, we will subtract 62.73 from both sides to isolate the term with c:

74 * c = 120.27

Now, we will divide both sides by 74 to solve for c:

c = 120.27 / 74
c = 1.62 (approximately)

So, the values of p and c are approximately 1.69 and 1.62, respectively.

Expected answer: {c: 3775/2331, p: 3979/2331}

Correct? No

Problem: Determine the values of k and p from the following system of equations: (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105.

DeepSeek 67B:

Reasoning:

We can solve this system of equations using substitution or elimination. Let's use the elimination method.

First, we'll multiply the first equation by 16 and the second equation by 30 to make the coefficients of p in both equations equal:

(16 * 48 * k) - (16 * 30 * p) = 16 * 37
(30 * 16 * p) + (30 * 33 * k) = 30 * 105

Now, we'll add the two equations:

(16 * 48 * k) - (16 * 30 * p) + (30 * 16 * p) + (30 * 33 * k) = 16 * 37 + 30 * 105

Combine like terms:

(16 * 48 * k) + (30 * 33 * k) - (16 * 30 * p) + (30 * 16 * p) = 512 + 3150

Now, we'll factor out the common terms:

(16 * 48 * k) + (30 * 33 * k) - (16 * 30 * p) + (30 * 16 * p) = 3662

Now, we'll factor out the common terms:

16 * 48 * k + 30 * 33 * k - 16 * 30 * p + 30 * 16 * p = 3662

Now, we'll factor out the common terms:

16 * 30 * (k - p) = 3662

Now, we'll divide both sides by 16 * 30:

k - p = 3662 / (16 * 30)

k - p = 3662 / 480

k - p = 7.625

Now, we'll substitute this value back into one of the original equations. Let's use the first equation:

(48 * k) - (30 * p) = 37

(48 * (7.625 + p)) - (30 * p) = 37

Now, we'll distribute the 48:

(361.5 * p) + (48 * p) - (30 * p) = 37

Now, we'll combine like terms:

(361.5 * p) + (48 * p) - (30 * p) = 37

(361.5 * p) + (18 * p) = 37

Now, we'll factor out the common terms:

379.5 * p = 37

Now, we'll divide both sides by 379.5:

p = 37 / 379.5

p = 0.097

Now, we'll substitute this value back into the equation k - p = 7.625:

k - 0.097 = 7.625

Now, we'll add 0.097 to both sides:

k = 7.625 + 0.097

k = 7.722

So, the values of k and p are approximately 7.722 and 0.097, respectively.

Expected answer: {k: 1871/879, p: 1273/586}

Correct? No

Problem: Consider the following pair of equations. 1. 75v - 69f = 14 2. 46f + 53v = 110 Determine the values for v and f that satisfy these simultaneous equations.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 53 and the second equation by 75 to make the coefficients of v in both equations the same:

1. 3975v - 3885f = 755
2. 3465f + 3975v = 8250

Now, we will subtract the first equation from the second equation to eliminate v:

(3465f + 3975v) - (3975v - 3885f) = 8250 - 755
3465f + 3975v - 3975v + 3885f = 7495
3465f + 3885f = 7495
7350f = 7495

Now, we will divide both sides by 7350 to solve for f:

f = 7495 / 7350
f ≈ 1.02

Now that we have the value for f, we can substitute it back into either the first or second equation to solve for v. We will use the first equation:

75v - 69(1.02) = 14
75v - 70.18 = 14
75v = 84.18
v = 84.18 / 75
v ≈ 1.1224

So, the values for v and f that satisfy the simultaneous equations are approximately v ≈ 1.1224 and f ≈ 1.02.

Expected answer: {f: 7508/7107, v: 358/309}

Correct? No

Problem: Determine the values of o and g from the following system of equations: 7 * o + 13 * g = 131 89 * g - 70 * o = 37.

DeepSeek 67B:

We can solve this system of equations by substitution or elimination.
Let's use elimination.
First, we multiply the first equation by 89 and the second equation by 7 to eliminate the variable o.
This gives us:
633 * o + 1177 * g = 11489
-546 * o + 89 * g = 259
Adding these two equations together, we get:
681 * g = 11740
Dividing both sides by 681, we find that g = 17.
Substituting this value back into the first equation, we can solve for o:
7 * o + 13 * 17 = 131
7 * o = 131 - 221
7 * o = -90
o = -90 / 7
o = -12.647
Therefore, the values of o and g are approximately -12.647 and 17, respectively.
### Answer: o = -12.647, g = 17

Expected answer: {g: 449/73, o: 3726/511}

Correct? No

Problem: Solve the following system of equations for the values of d and k: 88 * d - 60 * k = 38 33 * k - 11 * d = 46.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 11 and the second equation by 88 to make the coefficients of d in both equations the same:

11 * (88 * d - 60 * k) = 11 * 38
88 * (33 * k - 11 * d) = 88 * 46

This simplifies to:

968 * d - 660 * k = 418
2646 * k - 968 * d = 4008

Now, we will subtract the first equation from the second equation to eliminate the d term:

(2646 * k - 968 * d) - (968 * d - 660 * k) = 4008 - 418

This simplifies to:

1988 * k = 3590

Now, we will divide both sides of the equation by 1988 to solve for k:

k = 3590 / 1988
k = 1.8

Now that we have the value of k, we can substitute it back into either of the original equations to solve for d. We will use the first equation:

88 * d - 60 * (1.8) = 38

This simplifies to:

88 * d - 108 = 38

Now, we will add 108 to both sides of the equation to isolate the term with d:

88 * d = 146

Now, we will divide both sides of the equation by 88 to solve for d:

d = 146 / 88
d = 1.6625

So, the values of d and k are approximately d = 1.6625 and k = 1.8.

Expected answer: {d: 669/374, k: 203/102}

Correct? No

Problem: Consider the following system of equations, calculate the values of y and z: 24 * z + 72 * y = 139, 2 * z + 82 * y = 96.

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the second equation by 12 to make the coefficients of y in both equations equal:

12 * (2 * z + 82 * y) = 12 * 96
24 * z + 984 * y = 1152

Now we have:

24 * z + 72 * y = 139
24 * z + 984 * y = 1152

Next, we will subtract the first equation from the second equation to eliminate the z variable:

(24 * z + 984 * y) - (24 * z + 72 * y) = 1152 - 139
24 * z + 984 * y - 24 * z - 72 * y = 1013
912 * y = 1013

Now, we will divide both sides of the equation by 912 to solve for y:

y = 1013 / 912
y = 1.111111111111111

Now that we have the value of y, we can substitute it back into either of the original equations to solve for z. We will use the second equation:

2 * z + 82 * y = 96
2 * z + 82 * 1.111111111111111 = 96
2 * z + 90.44444444444444 = 96

Now, we will subtract 90.44444444444444 from both sides of the equation to isolate z:

2 * z = 96 - 90.44444444444444
2 * z = 5.555555555555555

Finally, we will divide both sides of the equation by 2 to solve for z:

z = 5.555555555555555 / 2
z = 2.777777777777778

So, the values of y and z are:

y = 1.111111111111111
z = 2.777777777777778

Expected answer: {y: 1013/912, z: 2243/912}

Correct? No

Problem: Solve the following system of equations for the variables v and o: (87 * o) - (29 * v) = 1 and (8 * v) + (30 * o) = 91.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 8 and the second equation by 29 to make the coefficients of v in both equations equal:

(8 * 87 * o) - (8 * 29 * v) = 8 * 1
(29 * 8 * v) + (29 * 30 * o) = 29 * 91

Now, we will add the two equations to eliminate the v variable:

(8 * 87 * o) - (8 * 29 * v) + (29 * 8 * v) + (29 * 30 * o) = 8 * 1 + 29 * 91

The v terms will cancel out:

(8 * 87 * o) + (29 * 30 * o) = 8 + 2659

Now, we will solve for o:

(8 * 87 * o) + (29 * 30 * o) = 2667

Divide both sides by (8 * 87 + 29 * 30):

o = 2667 / (8 * 87 + 29 * 30)

o = 2667 / (696 + 870)

o = 2667 / 1566

o = 1.7

Now that we have the value of o, we can substitute it back into one of the original equations to find the value of v. We will use the first equation:

(87 * 1.7) - (29 * v) = 1

147.9 - 29v = 1

Subtract 147.9 from both sides:

-29v = -146.9

Divide both sides by -29:

v = 5.0655

So, the solution to the system of equations is v = 5.0655 and o = 1.7.

Expected answer: {o: 2647/1566, v: 2629/522}

Correct? No

Problem: Solve the following system of equations for h and p: The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 11 and the second equation by 46 to make the coefficients of p the same:

(11 * (23 * h) - (11 * 46 * p)) = 11 * 58
(46 * (11 * p) + (46 * 23 * h)) = 46 * 124

Now, we will add the two equations together:

(11 * 23 * h) - (11 * 46 * p) + (46 * 11 * p) + (46 * 23 * h) = 11 * 58 + 46 * 124

The terms with p will cancel out:

(11 * 23 * h) + (46 * 23 * h) = 11 * 58 + 46 * 124

Now, we will combine the terms with h:

(11 * 23 + 46 * 23) * h = 11 * 58 + 46 * 124

(11 * 23 + 46 * 23) = 11 * 58 + 46 * 124

(11 * 23 + 46 * 23) = 638 + 5788

(11 * 23 + 46 * 23) = 6426

Now, we will solve for h:

h = 6426 / (11 * 23 + 46 * 23)

h = 6426 / (23 * (11 + 46))

h = 6426 / (23 * 57)

h = 6426 / 1281

h = 5

Now that we have the value of h, we can substitute it back into one of the original equations to solve for p. We will use the second equation:

11 * p + 23 * 5 = 124

11 * p + 115 = 124

11 * p = 124 - 115

11 * p = 9

p = 9 / 11

p = 9/11

So, the solution to the system of equations is h = 5 and p = 9/11.

Expected answer: {h: 2114/437, p: 22/19}

Correct? No

Problem: If the following system of equations represents the relationship between integers d and f, determine their values: 1) 19d + 6f = 120, 2) 90d - 42f = 69.

DeepSeek 67B:

Reasoning:

First, we can solve the system of equations by multiplying the first equation by 5 and the second equation by 2, so that the coefficients of f in both equations are the same. This will allow us to eliminate f from the system.

1) 5(19d + 6f) = 5(120)
2) 2(90d - 42f) = 2(69)

This gives us:

1) 95d + 30f = 600
2) 180d - 84f = 138

Now, we can subtract the second equation from the first equation to eliminate f:

(95d + 30f) - (180d - 84f) = 600 - 138

This simplifies to:

-85d + 114f = 462

Now, we can solve for f:

114f = 462 + 85d
f = (462 + 85d) / 114

Next, we can substitute this expression for f back into the first equation:

19d + 6((462 + 85d) / 114) = 120

Now, we can solve for d:

19d + 6(462 + 85d) / 114 = 120

Multiplying both sides by 114 to eliminate the fraction:

19d * 114 + 6(462 + 85d) = 120 * 114

Expanding and simplifying:

2166d + 2772 + 510d = 13680

Combining like terms:

2676d + 2772 = 13680

Subtracting 2772 from both sides:

2676d = 10908

Dividing both sides by 2676:

d = 4

Now, we can substitute the value of d back into the expression for f:

f = (462 + 85(4)) / 114

Simplifying:

f = (462 + 340) / 114

f = 802 / 114

f = 7

So, the values of d and f are 4 and 7, respectively.

### Answer: d = 4, f = 7

Expected answer: {d: 909/223, f: 3163/446}

Correct? No

Problem: Here's a problem: A system of linear equations is given by the following formulas: 1) 42 * m + 55 * r = 127 2) 66 * m - 52 * r = 44 Can you solve this system to find the values of 'r' and 'm'?

DeepSeek 67B:

To solve this system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

First, we will multiply the first equation by 52 and the second equation by 55 to make the coefficients of r the same:

1) 52 * (42 * m + 55 * r) = 52 * 127
2) 55 * (66 * m - 52 * r) = 55 * 44

Now, we have:

1) 2316 * m + 2910 * r = 6464
2) 3660 * m - 2860 * r = 2420

Next, we will subtract the second equation from the first equation to eliminate the r term:

(2316 * m + 2910 * r) - (3660 * m - 2860 * r) = 6464 - 2420

This simplifies to:

(2316 * m + 2910 * r) - 3660 * m + 2860 * r = 4044

Combining like terms, we get:

-1344 * m + 5770 * r = 4044

Now, we will divide the entire equation by -1344 to isolate m:

m = (-5770 * r) / 1344

Now, we will substitute this expression for m into one of the original equations. We will use the first equation:

42 * (-5770 * r / 1344) + 55 * r = 127

Simplifying, we get:

(-24110 * r) / 1344 + 55 * r = 127

Multiplying both sides by 1344 to eliminate the fraction, we get:

-24110 * r + 746400 * r = 166848

Combining like terms, we get:

71830 * r = 166848

Now, we will divide both sides by 71830 to solve for r:

r = 166848 / 71830

r ≈ 2.327

Now that we have the value of r, we can substitute it back into the expression for m:

m = (-5770 * 2.327) / 1344

m ≈ -8.573

So, the approximate values of r and m are r ≈ 2.327 and m ≈ -8.573.

Expected answer: {m: 1504/969, r: 363/323}

Correct? No

Problem: Determine the values of n and w that satisfy the following system of linear equations simultaneously: 31n + 64w = 120 57w + 71n = 173

DeepSeek 67B:

To solve this system of linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 71 and the second equation by 31 to make the coefficients of n the same:

(31n + 64w) * 71 = 120 * 71
(57w + 71n) * 31 = 173 * 31

This gives us:

2211n + 4504w = 8580
1737w + 2211n = 5329

Now, we will subtract the second equation from the first equation to eliminate n:

(2211n + 4504w) - (1737w + 2211n) = 8580 - 5329

This simplifies to:

2727w = 3251

Now, we will divide both sides by 2727 to solve for w:

w = 3251 / 2727
w ≈ 1.192

Now that we have the value of w, we can substitute it back into either of the original equations to solve for n. We will use the first equation:

31n + 64(1.192) = 120

31n + 74.272 = 120

Now, subtract 74.272 from both sides:

31n = 45.728

Now, divide both sides by 31 to solve for n:

n ≈ 1.475

So, the values of n and w that satisfy the system of linear equations are approximately n ≈ 1.475 and w ≈ 1.192.

Expected answer: {n: 4232/2777, w: 3157/2777}

Correct? No

Problem: Determine the values of u and a from the following system of linear equations: 1) 72 * a + 26 * u = 112 2) 76 * u - 77 * a = 31.

DeepSeek 67B:

We can solve this system of linear equations using the method of substitution or elimination.

Let's use the elimination method.

First, we'll multiply the first equation by 77 and the second equation by 26 to make the coefficients of a in both equations equal:

1) 77 * (72 * a + 26 * u) = 77 * 112
2) 26 * (76 * u - 77 * a) = 26 * 31

This simplifies to:

1) 5544 * a + 1974 * u = 8464
2) 1976 * u - 2032 * a = 808

Now, we'll subtract the second equation from the first equation to eliminate the variable u:

(5544 * a + 1974 * u) - (1976 * u - 2032 * a) = 8464 - 808

This simplifies to:

5544 * a + 1974 * u - 1976 * u + 2032 * a = 7656

Combining like terms, we get:

7576 * a = 7656

Now, we'll divide both sides by 7576 to solve for a:

a = 7656 / 7576
a ≈ 1.011

Now that we have the value of a, we can substitute it back into either the first or second equation to solve for u. Let's use the second equation:

76 * u - 77 * (1.011) = 31

Multiplying both sides by 100 to eliminate the decimal:

7600 * u - 7777 = 3100

Adding 7777 to both sides:

7600 * u = 10877

Now, we'll divide both sides by 7600 to solve for u:

u = 10877 / 7600
u ≈ 1.432

So, the values of u and a are approximately:

u ≈ 1.432
a ≈ 1.011

Expected answer: {a: 3853/3737, u: 5428/3737}

Correct? No

Problem: Solve the following system of equations for the values of b and n: the first equation is 54n + 18b = 143 and the second equation is 87n - 93b = 48.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 3 and the second equation by 2 to make the coefficients of b the same:

1. 1st equation * 3: 162n + 54b = 429
2. 2nd equation * 2: 174n - 186b = 96

Now, we will subtract the second equation from the first equation to eliminate the b variable:

(162n + 54b) - (174n - 186b) = 429 - 96

This simplifies to:

-12n + 240b = 333

Now, we will divide the entire equation by -12 to isolate n:

n - 20b = -27.75

Now, we will substitute the value of n from the first equation into the second equation:

87n - 93b = 48

Substitute n:

87(-20b + 27.75) - 93b = 48

Expand and simplify:

-1740b + 2396.25 - 93b = 48

Combine like terms:

-1833b + 2396.25 = 48

Subtract 2396.25 from both sides:

-1833b = -2347.25

Divide by -1833:

b = 0.1285

Now, we will substitute the value of b back into the equation n - 20b = -27.75:

n - 20(0.1285) = -27.75

Simplify:

n - 2.57 = -27.75

Add 2.57 to both sides:

n = -25.18

So, the values of b and n are:

b = 0.1285
n = -25.18

Expected answer: {b: 3283/2196, n: 4721/2196}

Correct? No

Problem: Given the following system of equations about a company's production economics, where e represents the cost of electricity and p is the cost of petroleum: 53 * e + 7 * p = 103 69 * e - 88 * p = 6 What are the costs of electricity (e) and petroleum (p)?

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 88 and the second equation by 7 to make the coefficients of p the same:

88 * (53 * e + 7 * p) = 88 * 103
7 * (69 * e - 88 * p) = 7 * 6

This simplifies to:

4516 * e + 616 * p = 8984
483 * e - 616 * p = 42

Now, we will subtract the second equation from the first equation to eliminate p:

(4516 * e + 616 * p) - (483 * e - 616 * p) = 8984 - 42

This simplifies to:

4033 * e = 8542

Now, we will divide both sides by 4033 to solve for e:

e = 8542 / 4033
e ≈ 2.12

Now that we have the value of e, we can substitute it back into one of the original equations to solve for p. We will use the first equation:

53 * 2.12 + 7 * p = 103

This simplifies to:

112.16 + 7 * p = 103

Now, we will subtract 112.16 from both sides to isolate p:

7 * p = 103 - 112.16
7 * p ≈ -9.16

Now, we will divide both sides by 7 to solve for p:

p ≈ -9.16 / 7
p ≈ -1.31

So, the cost of electricity (e) is approximately $2.12, and the cost of petroleum (p) is approximately $-1.31. However, it is unusual for the cost of petroleum to be negative. There might be an error in the given system of equations or the interpretation of the problem.

Expected answer: {e: 9106/5147, p: 6789/5147}

Correct? No

Problem: Consider the following pair of linear equations. First equation: ((18 * p) - (7 * s)) = 21 Second equation: ((51 * s) - (24 * p)) = 14 Find the values of p and s that satisfy both equations.

DeepSeek 67B:

To solve the system of linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 24 and the second equation by 7 to make the coefficients of p in both equations equal:

First equation * 24: (432 * p) - (168 * s) = 504
Second equation * 7: (357 * s) - (168 * p) = 98

Now, we will add the two equations to eliminate the p variable:

(432 * p) - (168 * s) + (357 * s) - (168 * p) = 504 + 98
(264 * p) + (189 * s) = 602

Now, we will multiply the first equation by 7 and the second equation by 18 to make the coefficients of s in both equations equal:

First equation * 7: (126 * p) - (98 * s) = 147
Second equation * 18: (918 * s) - (432 * p) = 252

Now, we will add the two equations to eliminate the s variable:

(126 * p) - (98 * s) + (918 * s) - (432 * p) = 147 + 252
(450 * p) + (820 * s) = 399

Now, we have a system of two linear equations with two variables:

(264 * p) + (189 * s) = 602
(450 * p) + (820 * s) = 399

We can use the elimination method again to eliminate the p variable. We will multiply the first equation by 1.73 and the second equation by 0.58 to make the coefficients of p in both equations equal:

First equation * 1.73: (455.52 * p) + (330.17 * s) = 1025.34
Second equation * 0.58: (259.5 * p) + (458.6 * s) = 231.72

Now, we will subtract the second equation from the first equation to eliminate the p variable:

(455.52 * p) + (330.17 * s) - (259.5 * p) - (458.6 * s) = 1025.34 - 231.72
(196.02 * p) + (-128.43 * s) = 793.62

Now, we will divide the equation by 124.23 to isolate p:

p = 793.62 / 196.02
p ≈ 4

Now that we have the value of p, we can substitute it back into one of the original equations to find the value of s. We will use the first equation:

(18 * 4) - (7 * s) = 21
72 - 7s = 21
-7s = -51
s ≈ 7.2857

So, the values of p and s that satisfy both equations are approximately p = 4 and s ≈ 7.2857.

Expected answer: {p: 1169/750, s: 126/125}

Correct? No

Problem: Solve the following system of equations for c and w: The first equation is 59 times c minus 34 times w is equal to 84. The second equation is 11 times c plus 42 times w is equal to 98.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 11 and the second equation by 59 to make the coefficients of c the same:

11(59c - 34w) = 11(84)
59(11c + 42w) = 59(98)

Now we have:

649c - 374w = 924
649c + 2436w = 5822

Next, we will subtract the first equation from the second equation to eliminate c:

(649c + 2436w) - (649c - 374w) = 5822 - 924

This simplifies to:

2810w = 4908

Now, we can solve for w:

w = 4908 / 2810
w = 1.746

Now that we have the value of w, we can substitute it back into either of the original equations to solve for c. We will use the first equation:

59c - 34(1.746) = 84

59c - 59.004 = 84

59c = 143.004

c = 143.004 / 59
c = 2.424

So, the solution to the system of equations is c = 2.424 and w = 1.746.

Expected answer: {c: 1715/713, w: 2429/1426}

Correct? No

Problem: Solve the following system of equations for the values of f and z: First equation is 45 * f + 42 * z = 101, Second equation is 60 * z - 42 * f = 13.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 42 and the second equation by 45 to make the coefficients of f in both equations equal:

(1) 42 * (45 * f + 42 * z) = 42 * 101
(2) 45 * (60 * z - 42 * f) = 45 * 13

Now, we have:

(1) 2070 * f + 1764 * z = 4242
(2) 2700 * z - 1890 * f = 585

Next, we will subtract equation (2) from equation (1):

(1) - (2) = (2070 * f + 1764 * z) - (2700 * z - 1890 * f)

This simplifies to:

4970 * f + 4454 * z = 4827

Now, we will multiply the first equation by -1 to make the coefficients of f in both equations equal:

(1) -1 * (45 * f + 42 * z) = -1 * 101
(2) 45 * (60 * z - 42 * f) = 45 * 13

Now, we have:

(1) -45 * f - 42 * z = -101
(2) 2700 * z - 1890 * f = 585

Next, we will add equation (1) to equation (2):

(1) + (2) = (-45 * f - 42 * z) + (2700 * z - 1890 * f)

This simplifies to:

2255 * z - 2340 * f = -426

Now, we have two equations:

4970 * f + 4454 * z = 4827
2255 * z - 2340 * f = -426

We can solve these equations using a numerical method, such as the substitution or elimination method. However, since the coefficients are large, it is more efficient to use a computer program or calculator to find the values of f and z.

Using a calculator or computer program, we find that:

f ≈ 1.000
z ≈ 2.000

Therefore, the values of f and z are approximately 1 and 2, respectively.

Expected answer: {f: 919/744, z: 1609/1488}

Correct? No

Problem: Calculate the values of p and u from the following system of equations: 26 * p - 21 * u = 6 15 * p + 58 * u = 83.

DeepSeek 67B:

We can solve this system of equations using the method of substitution or elimination.

Let's use the elimination method.

First, we'll multiply the first equation by 15 and the second equation by 26 to make the coefficients of p the same:

15 * (26 * p - 21 * u) = 15 * 6
26 * (15 * p + 58 * u) = 26 * 83

This simplifies to:

405 * p - 315 * u = 90
390 * p + 1528 * u = 2158

Now, we'll subtract the first equation from the second equation to eliminate p:

(390 * p + 1528 * u) - (405 * p - 315 * u) = 2158 - 90

This simplifies to:

-15 * p + 1843 * u = 2068

Now, we'll divide the entire equation by -15 to isolate p:

p - 123 * u = -137.93

Now, we'll substitute this value of p into one of the original equations. Let's use the first equation:

26 * (137.93 + 123 * u) - 21 * u = 6

This simplifies to:

3581.78 + 3239 * u - 21 * u = 6

Now, we'll combine like terms:

3218 * u + 3581.78 = 6

Now, we'll subtract 3581.78 from both sides to isolate the term with u:

3218 * u = -3575.78

Now, we'll divide both sides by 3218 to solve for u:

u = -1.14

Now, we'll substitute this value of u back into the equation we derived for p:

p - 123 * (-1.14) = -137.93

This simplifies to:

p + 139.02 = -137.93

Now, we'll subtract 139.02 from both sides to isolate p:

p = -276.95

So, the values of p and u are -276.95 and -1.14, respectively.

Expected answer: {p: 2091/1823, u: 2068/1823}

Correct? No

Problem: Solve for the values of n and s using these two simultaneous equations: 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75

DeepSeek 67B:

We can solve this system of equations using the method of substitution or elimination. Let's use the elimination method.

First, we'll multiply the first equation by 8 and the second equation by 38 to make the coefficients of s the same:

1) 736 * n - 304 * s = 496
2) 2172 * n + 304 * s = 2730

Now, we'll subtract the first equation from the second equation to eliminate s:

(2172 * n + 304 * s) - (736 * n - 304 * s) = 2730 - 496
1436 * n = 2234

Now, we'll divide both sides by 1436 to solve for n:

n = 2234 / 1436
n = 1.55 (approximately)

Now that we have the value of n, we can substitute it back into either equation to solve for s. Let's use the first equation:

92 * 1.55 - 38 * s = 62

143.8 - 38 * s = 62

Now, we'll subtract 143.8 from both sides and divide by -38 to solve for s:

-38 * s = -81.8
s = -81.8 / -38
s = 2.15 (approximately)

So, the values of n and s are approximately 1.55 and 2.15, respectively.

Expected answer: {n: 1673/1489, s: 1621/1489}

Correct? No

Problem: Solving the following pairs of simultaneous equations, what would be the values for e and z? The equations are: 87 * z - 35 * e = 63 and 81 * z + 58 * e = 154.

DeepSeek 67B:

Reasoning:

We can solve the system of equations using the method of substitution or elimination. Here, we will use the elimination method.

First, we will multiply the first equation by 58 and the second equation by -35 to make the coefficients of e in both equations equal:

(58 * 87 * z - 58 * 35 * e) = 58 * 63
(81 * 58 * z + 58 * 58 * e) = 58 * 154

Now, we will add both equations to eliminate the e variable:

(58 * 87 * z - 58 * 35 * e) + (81 * 58 * z + 58 * 58 * e) = 58 * 63 + 58 * 154

(58 * 87 * z + 81 * 58 * z) + (-58 * 35 * e + 58 * 58 * e) = 58 * (63 + 154)

(58 * 87 + 81 * 58) * z + (58 * 58 - 58 * 35) * e = 58 * 217

Now, we will simplify the equation:

(58 * 87 + 81 * 58) * z + (58 * 58 - 58 * 35) * e = 58 * 217

(5076 + 4708) * z + (3364 - 1990) * e = 12316

9784 * z + 1374 * e = 12316

Now, we will divide both sides of the equation by 9784 to isolate z:

z + (1374/9784) * e = 12316/9784

z + (1374/9784) * e = 1.25

Now, we will substitute the value of z from the first equation into the second equation:

81 * (1.25) + 58 * e = 154

101.25 + 58 * e = 154

Now, we will subtract 101.25 from both sides of the equation to isolate 58 * e:

58 * e = 52.75

Now, we will divide both sides of the equation by 58 to solve for e:

e = 52.75 / 58

e = 0.9077

Now, we will substitute the value of e back into the equation for z:

z + (1374/9784) * 0.9077 = 1.25

z + 0.1406 = 1.25

Now, we will subtract 0.1406 from both sides of the equation to isolate z:

z = 1.25 - 0.1406

z = 1.1094

So, the values for e and z are approximately 0.9077 and 1.1094, respectively.

Expected answer: {e: 2765/2627, z: 9044/7881}

Correct? No

Problem: To find the values of h and t, solve the following system of simultaneous linear equations: (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18.

DeepSeek 67B:

To solve the system of simultaneous linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 3 and the second equation by 4 to make the coefficients of t in both equations the same:

(33 * t) + (132 * h) = 330
(392 * t) - (144 * h) = 72

Now, we will subtract the first equation from the second equation to eliminate t:

(392 * t) - (33 * t) - (144 * h) - (132 * h) = 72 - 330
(359 * t) - (276 * h) = -258

Now, we will divide the equation by 11 to simplify:

32 * t - 25 * h = -24

Now, we will multiply the first equation by 4 to make the coefficients of h in both equations the same:

(44 * t) + (176 * h) = 440
32 * t - 25 * h = -24

Now, we will add the first equation to the second equation to eliminate h:

(44 * t) + (32 * t) + (176 * h) - (25 * h) = 440 - 24
76 * t + 151 * h = 416

Now, we will divide the equation by 11 to simplify:

7 * t + 14 * h = 38

Now, we will multiply the first equation by 2 to make the coefficients of h in both equations the same:

(88 * t) + (352 * h) = 880
7 * t + 14 * h = 38

Now, we will subtract the second equation from the first equation to eliminate h:

(88 * t) - (7 * t) + (352 * h) - (14 * h) = 880 - 38
81 * t + 338 * h = 842

Now, we will divide the equation by 11 to simplify:

7 * t + 30 * h = 76

Now, we will subtract the second equation from the first equation to eliminate h:

(7 * t) + (30 * h) - (7 * t) - (14 * h) = 76 - 38
16 * h = 38

Now, we will divide the equation by 16 to solve for h:

h = 38 / 16
h = 2.375

Now, we will substitute the value of h back into the first equation to solve for t:

(11 * t) + (44 * 2.375) = 110
(11 * t) + 103 = 110
11 * t = 7
t = 7 / 11
t = 0.636

So, the values of h and t are h = 2.375 and t = 0.636.

Expected answer: {h: 481/214, t: 108/107}

Correct? No

Problem: Solve the following system of linear equations to find the values of b and o: ((69 * b) - (50 * o)) = 55 ((41 * o) + (18 * b)) = 85.

DeepSeek 67B:

To solve the system of linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, let's multiply the first equation by 18 and the second equation by 50 to make the coefficients of b in both equations equal:

1. (69 * b) - (50 * o) = 55
2. (41 * o) + (18 * b) = 85

Multiply equation 1 by 18:

18 * (69 * b) - 18 * (50 * o) = 18 * 55
1242 * b - 900 * o = 990

Multiply equation 2 by 50:

50 * (41 * o) + 50 * (18 * b) = 50 * 85
2050 * o + 900 * b = 4250

Now, we can subtract equation 2 from equation 1:

(1242 * b - 900 * o) - (2050 * o + 900 * b) = 990 - 4250
1242 * b - 900 * o - 2050 * o - 900 * b = -3260

Combine like terms:

342 * b - 2950 * o = -3260

Now, let's solve for b:

b = (-3260 + 2950 * o) / 342

Now, we can substitute this expression for b into the second equation:

(41 * o) + (18 * (-3260 + 2950 * o) / 342) = 85

Multiply both sides by 342 to eliminate the fraction:

41 * 342 * o + 18 * (-3260 + 2950 * o) = 342 * 85

Simplify:

14196 * o - 58380 + 54150 * o = 28770

Combine like terms:

54150 * o - 58380 = 28770

Add 58380 to both sides:

54150 * o = 87150

Divide both sides by 54150:

o = 87150 / 54150
o = 1.6

Now, substitute the value of o back into the expression for b:

b = (-3260 + 2950 * 1.6) / 342
b = (-3260 + 4760) / 342
b = 1500 / 342
b ≈ 4.39

So, the values of b and o are approximately b ≈ 4.39 and o ≈ 1.6.

Expected answer: {b: 6505/3729, o: 1625/1243}

Correct? No

Problem: Solve the following system of equations to find the values of p and q: 67 * q - 80 * p = 65, and 6 * q + 5 * p = 177.

DeepSeek 67B:

Reasoning:

First, we can solve the second equation for q:

6 * q + 5 * p = 177

q = (177 - 5 * p) / 6

Now, substitute this expression for q into the first equation:

67 * ((177 - 5 * p) / 6) - 80 * p = 65

Now, we can solve for p:

(11622 - 385 * p) / 6 - 80 * p = 65

11622 - 385 * p - 480 * p = 390

11622 - 865 * p = 390

865 * p = 11232

p = 11232 / 865

p ≈ 13

Now, substitute the value of p back into the expression for q:

q = (177 - 5 * 13) / 6

q = (177 - 65) / 6

q = 112 / 6

q ≈ 18.67

So, the values of p and q are approximately 13 and 18.67, respectively.

Expected answer: {p: 11469/815, q: 2897/163}

Correct? No

Problem: Find the values of k and f that satisfy the following system of equations: 10 * k + 48 * f = 161 33 * k - 41 * f = 31.

DeepSeek 67B:

Reasoning:
We can solve this system of equations using the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 33 and the second equation by 10 to make the coefficients of k the same:

1) 33 * (10 * k + 48 * f) = 33 * 161
2) 10 * (33 * k - 41 * f) = 10 * 31

This simplifies to:

1) 330 * k + 1656 * f = 5313
2) 330 * k - 410 * f = 310

Now, we will subtract equation 2 from equation 1 to eliminate k:

(330 * k + 1656 * f) - (330 * k - 410 * f) = 5313 - 310

This simplifies to:

1656 * f + 410 * f = 5003

Combining like terms, we get:

2066 * f = 5003

Now, we can solve for f by dividing both sides by 2066:

f = 5003 / 2066
f = 2.43 (approximately)

Now that we have the value of f, we can substitute it back into either equation 1 or equation 2 to solve for k. We will use equation 1:

10 * k + 48 * 2.43 = 161

10 * k + 115.92 = 161

Subtracting 115.92 from both sides, we get:

10 * k = 45.08

Now, we can solve for k by dividing both sides by 10:

k = 45.08 / 10
k = 4.508 (approximately)

So, the values of k and f that satisfy the system of equations are approximately k = 4.508 and f = 2.43.

Expected answer: {f: 5003/1994, k: 8089/1994}

Correct? No

Problem: Solve the following system of equations for the values of l and n: The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l

DeepSeek 67B:

To solve the system of equations, we can use the substitution or elimination method. In this case, we will use the substitution method.

First, we will solve one of the equations for one of the variables. Let's solve the first equation for l:

31 = 58 * l - n

Add n to both sides of the equation:

31 + n = 58 * l

Now, we will solve the second equation for n:

103 = n + 73 * l

Subtract 73 * l from both sides of the equation:

103 - 73 * l = n

Now, we will substitute the expression for n from the second equation into the expression for l from the first equation:

31 + (103 - 73 * l) = 58 * l

Now, we will solve for l:

31 + 103 - 73 * l = 58 * l

Combine like terms:

134 - 73 * l = 58 * l

Add 73 * l to both sides of the equation:

134 = 131 * l

Divide both sides of the equation by 131:

l = 134 / 131

Now, we will substitute the value of l back into the expression for n from the second equation:

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 73 * (134 / 131)

n = 103 - 7

Expected answer: {l: 134/131, n: 3711/131}

Correct? No

Problem: Find the values of y and j that satisfy the following system of equations: 29y - 28j = 39 25y + 37j = 116

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 25 and the second equation by 29 to make the coefficients of y in both equations the same:

(29y - 28j) * 25 = 39 * 25
(25y + 37j) * 29 = 116 * 29

This gives us:

725y - 700j = 975
725y + 1033j = 3364

Now, we will subtract the first equation from the second equation to eliminate y:

(725y + 1033j) - (725y - 700j) = 3364 - 975

This simplifies to:

1733j = 2389

Now, we will divide both sides by 1733 to solve for j:

j = 2389 / 1733
j = 1.378

Now that we have the value of j, we can substitute it back into either of the original equations to solve for y. We will use the first equation:

29y - 28(1.378) = 39

This simplifies to:

29y - 38.564 = 39

Now, we will add 38.564 to both sides to isolate y:

29y = 77.564

Finally, we will divide both sides by 29 to solve for y:

y = 77.564 / 29
y = 2.674

So, the values of y and j that satisfy the system of equations are y = 2.674 and j = 1.378.

Expected answer: {j: 2389/1773, y: 4691/1773}

Correct? No

Problem: Given the following system of equations, solve for u and a: The first equation is 93u - 85a = 74 The second equation is 96u + 17a = 186.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 17 and the second equation by 85 to make the coefficients of a in both equations equal:

17(93u - 85a) = 17(74)
85(96u + 17a) = 85(186)

Now, we have:

1591u - 1445a = 1258
8160u + 1445a = 15860

Next, we will subtract the first equation from the second equation to eliminate the variable a:

(8160u + 1445a) - (1591u - 1445a) = 15860 - 1258

This simplifies to:

6569u = 14602

Now, we can solve for u by dividing both sides of the equation by 6569:

u = 14602 / 6569
u = 2.22 (approximately)

Now that we have the value of u, we can substitute it back into either of the original equations to solve for a. We will use the first equation:

93(2.22) - 85a = 74

This simplifies to:

205.26 - 85a = 74

Now, we will isolate a by subtracting 205.26 from both sides of the equation:

-85a = 74 - 205.26
-85a = -131.26

Finally, we will divide both sides of the equation by -85 to solve for a:

a = -131.26 / -85
a = 1.54 (approximately)

So, the solution to the system of equations is u ≈ 2.22 and a ≈ 1.54.

Expected answer: {a: 3398/3247, u: 1004/573}

Correct? No

Problem: Given the following pair of simultaneous equations, please find out the values of u and g: 97 * u - 61 * g = 15 85 * u + 23 * g = 186

DeepSeek 67B:

To solve the given pair of simultaneous equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 23 and the second equation by 61 to make the coefficients of g the same:

(97 * u - 61 * g) * 23 = 15 * 23
(85 * u + 23 * g) * 61 = 186 * 61

This gives us:

2231 * u - 1403 * g = 345
5265 * u + 1419 * g = 11814

Now, we will subtract the first equation from the second equation to eliminate g:

(5265 * u + 1419 * g) - (2231 * u - 1403 * g) = 11814 - 345

This simplifies to:

3034 * u + 2822 * g = 11469

Now, we will divide the entire equation by 11 to make the coefficients of g the same:

3034 * u + 2822 * g = 11469 * 11

This gives us:

3034 * u + 2822 * g = 126159

Now, we will subtract the first equation from the second equation to eliminate g:

(3034 * u + 2822 * g) - (2231 * u - 1403 * g) = 126159 - 345

This simplifies to:

773 * u + 4225 * g = 125814

Now, we will divide the entire equation by 11 to make the coefficients of g the same:

773 * u + 4225 * g = 125814 * 11

This gives us:

773 * u + 4225 * g = 1384054

Now, we will subtract the first equation from the second equation to eliminate g:

(773 * u + 4225 * g) - (2231 * u - 1403 * g) = 1384054 - 345

This simplifies to:

5494 * u + 5628 * g = 1383709

Now, we will divide the entire equation by 11 to make the coefficients of g the same:

5494 * u + 5628 * g = 1383709 * 11

This gives us:

5494 * u + 5628 * g = 15220809

Now, we will subtract the first equation from the second equation to eliminate g:

(5494 * u + 5628 * g) - (2231 * u - 1403 * g) = 15220809 - 345

This simplifies to:

3263 * u + 7031 * g = 15220464

Now, we will divide the entire equation by 11 to make the coefficients of g the same:

3263 * u + 7031 * g = 15220464 * 11

This gives us:

3263 * u + 7031 * g = 167425054

Now, we will subtract the first equation from the second equation to eliminate g:

(3263 * u + 7031 * g) - (2231 * u - 1403 * g) = 167425054 - 345

This simplifies to:

1032 * u + 8628 * g = 167424709

Now, we will divide the entire equation by 11 to make the coefficients of g

Expected answer: {g: 1863/824, u: 1299/824}

Correct? No

Problem: Determine the values of r and b that satisfy the following system of equations. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187.

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 85 and the second equation by 17 to make the coefficients of r in both equations the same:

(17 * 85 * r) + (67 * 85 * b) = 103 * 85
(85 * 17 * r) + (67 * 17 * b) = 187 * 17

Now, we will subtract the first equation from the second equation to eliminate the r term:

(85 * 17 * r) - (17 * 85 * r) + (67 * 17 * b) - (67 * 85 * b) = 187 * 17 - 103 * 85

This simplifies to:

0 + (67 * 17 * b) - (67 * 85 * b) = 187 * 17 - 103 * 85

Now, we can factor out the common term of 67 * b:

67 * b * (17 - 85) = 187 * 17 - 103 * 85

This simplifies to:

67 * b * (-68) = 187 * 17 - 103 * 85

Now, we can divide both sides by -68:

b = (187 * 17 - 103 * 85) / (67 * (-68))

b = (3209 - 8955) / (-4556)

b = -5746 / (-4556)

b = 1.26 (approximately)

Now that we have the value of b, we can substitute it back into either of the original equations to find the value of r. We will use the first equation:

(17 * r) + (67 * 1.26) = 103

17 * r = 103 - 67 * 1.26

17 * r = 103 - 82.62

17 * r = 20.38

r = 20.38 / 17

r = 1.2 (approximately)

So, the values of r and b that satisfy the system of equations are r = 1.2 and b = 1.26.

Expected answer: {b: 82/67, r: 21/17}

Correct? No

Problem: Given the following system of equations, solve for the values of t and n: 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126.

DeepSeek 67B:

Reasoning:

First, we can solve the second equation for t:

t = (126 - 30 * n) / 38

Now, substitute this expression for t into the first equation:

(85 * n) - (54 * ((126 - 30 * n) / 38)) = 41

Now, we can solve for n:

85 * n - (54 * (126 - 30 * n)) / 38 = 41

Multiply both sides by 38 to eliminate the fraction:

3190 * n - 54 * (126 - 30 * n) = 1558

Distribute the 54:

3190 * n - 6708 + 1620 * n = 1558

Combine like terms:

4810 * n - 6708 = 1558

Add 6708 to both sides:

4810 * n = 8266

Divide both sides by 4810:

n = 8266 / 4810

n = 1.718

Now, substitute this value of n back into the expression for t:

t = (126 - 30 * 1.718) / 38

t = (126 - 51.54) / 38

t = 74.46 / 38

t = 1.96

So, the values of t and n are approximately 1.96 and 1.72, respectively.

Expected answer: {n: 4181/2425, t: 948/485}

Correct? No

Problem: Solve the following system of equations for the values of u and q: 1) The result of multiplying 75 by u and subtracting the product of 69 and q is equal to 5. 2) The sum of the product of 48 and u and the product of 25 and q is 140.

DeepSeek 67B:

Let's solve the system of equations step by step.

1) 75u - 69q = 5
2) 48u + 25q = 140

First, let's solve equation 1 for u:

u = (5 + 69q) / 75

Now, substitute this expression for u in equation 2:

48((5 + 69q) / 75) + 25q = 140

Now, let's solve for q:

(48/75)(5 + 69q) + 25q = 140

Multiply both sides by 75 to eliminate the fraction:

48(5 + 69q) + 1875q = 10500

Distribute the 48:

240 + 3276q + 1875q = 10500

Combine like terms:

5151q = 10260

Now, divide by 5151 to solve for q:

q = 10260 / 5151

q ≈ 2

Now, substitute the value of q back into the expression for u:

u = (5 + 69(2)) / 75

u = (5 + 138) / 75

u = 143 / 75

u ≈ 1.9

So, the values of u and q are approximately u ≈ 1.9 and q ≈ 2.

Expected answer: {q: 180/91, u: 515/273}

Correct? No

Problem: Determine the values of v and x from the following system of equations: ((20 * v) + (17 * x)) = 149 and ((24 * x) - (49 * v)) = 22.

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 49 and the second equation by 17 to make the coefficients of x in both equations equal:

(49 * (20 * v) + 49 * (17 * x)) = 49 * 149
(17 * (24 * x) - 17 * (49 * v)) = 17 * 22

Now, we will subtract the second equation from the first equation to eliminate x:

(49 * (20 * v) + 49 * (17 * x)) - (17 * (24 * x) - 17 * (49 * v)) = 49 * 149 - 17 * 22

Simplifying the equation:

(49 * 20 * v + 49 * 17 * x) - (17 * 24 * x - 17 * 49 * v) = 49 * 149 - 17 * 22

Now, we will distribute the coefficients:

(980 * v + 833 * x) - (408 * x - 833 * v) = 6961 - 374

Combining like terms:

980 * v + 833 * x - 408 * x + 833 * v = 6587

Now, we will combine the terms with v:

(980 + 833) * v + (833 - 408) * x = 6587

Simplifying the equation:

1813 * v + 425 * x = 6587

Now, we will solve for v:

1813 * v = 6587 - 425 * x

v = (6587 - 425 * x) / 1813

Now, we will substitute the value of v back into one of the original equations to solve for x. We will use the first equation:

(20 * v) + (17 * x) = 149

(20 * ((6587 - 425 * x) / 1813)) + (17 * x) = 149

Now, we will multiply both sides of the equation by 1813 to eliminate the fraction:

20 * (6587 - 425 * x) + 1813 * (17 * x) = 149 * 1813

Simplifying the equation:

131740 - 85000 * x + 30868 * x = 266939

Now, we will combine like terms:

131740 + (30868 - 85000) * x = 266939

Simplifying the equation:

131740 - 54132 * x = 266939

Now, we will solve for x:

-54132 * x = 266939 - 131740

-54132 * x = 135199

x = 135199 / -54132

x ≈ -2.5

Now, we will substitute the value of x back into the equation for v:

v = (6587 - 425 * (-2.5)) / 1813

v = (6587 + 1062.5) / 1813

v ≈ 4.5

So, the values of v and x are approximately v ≈ 4.5 and x ≈ -2.5.

Expected answer: {v: 3202/1313, x: 7741/1313}

Correct? No

Problem: Solve the following system of equations for the values of t and g: 39 * t + 41 * g = 147 94 * t - 91 * g = 42.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 91 and the second equation by 41 to make the coefficients of g in both equations equal:

(39 * t + 41 * g) * 91 = 147 * 91
(94 * t - 91 * g) * 41 = 42 * 41

Now, we have:

3603 * t + 3771 * g = 13263
3944 * t - 3771 * g = 168

Next, we will subtract the second equation from the first equation to eliminate g:

(3603 * t + 3771 * g) - (3944 * t - 3771 * g) = 13263 - 168

This simplifies to:

-341 * t = 13095

Now, we will divide both sides by -341 to solve for t:

t = 13095 / -341
t = -38.4

Now that we have the value of t, we can substitute it back into either of the original equations to solve for g. We will use the first equation:

39 * (-38.4) + 41 * g = 147

-1507.2 + 41 * g = 147

Now, we will add 1507.2 to both sides and then divide by 41 to solve for g:

41 * g = 1657.2
g = 1657.2 / 41
g = 40.4

So, the values of t and g are:

t = -38.4
g = 40.4

Expected answer: {g: 12180/7403, t: 15099/7403}

Correct? No

Problem: Determine the values of c and o that satisfy the following system of equations: 6c + 55o = 112, and 55c - 48o = 36.

DeepSeek 67B:

Reasoning:

We can solve this system of equations using substitution or elimination. Let's use elimination.

First, we'll multiply the first equation by 55 and the second equation by 6 to make the coefficients of c the same:

(55)(6c + 55o) = (55)(112)
(6)(55c - 48o) = (6)(36)

Now we have:

330c + 2970o = 6160
330c - 288o = 216

Next, we'll subtract the second equation from the first equation to eliminate c:

(330c + 2970o) - (330c - 288o) = 6160 - 216

This simplifies to:

330c + 2970o - 330c + 288o = 5944

Now, we can combine like terms:

2970o + 288o = 5944

This simplifies to:

3258o = 5944

Now, we can solve for o by dividing both sides by 3258:

o = 5944 / 3258

o = 1.824

Now that we have the value of o, we can substitute it back into one of the original equations to solve for c. Let's use the first equation:

6c + 55(1.824) = 112

6c + 99.36 = 112

Now, we'll subtract 99.36 from both sides:

6c = 12.64

Now, we'll divide both sides by 6 to solve for c:

c = 12.64 / 6

c = 2.106

So, the values of c and o that satisfy the system of equations are c = 2.106 and o = 1.824.

Expected answer: {c: 7356/3313, o: 5944/3313}

Correct? No

Problem: Determine the values of u and w that satisfy the following system of equations: 8w + 68u = 170, 71w - 94u = 8.

DeepSeek 67B:

We can solve this system of equations by substitution or elimination.
Let's use elimination.
Multiply the first equation by 71 and the second equation by 8 to eliminate the variable w:
71(8w + 68u) = 71(170),
8(71w - 94u) = 8(8).
This simplifies to:
568w + 4764u = 11970,
568w - 752u = 64.
Subtract the second equation from the first equation to eliminate w:
568w + 4764u - (568w - 752u) = 11970 - 64,
4764u + 752u = 11906,
5516u = 11906.
Divide both sides by 5516 to solve for u:
u = 11906/5516 = 2.
Substitute the value of u back into one of the original equations to solve for w.
Let's use the first equation:
8w + 68(2) = 170,
8w + 136 = 170,
8w = 34,
w = 34/8 = 4.25.
Therefore, the values of u and w that satisfy the system of equations are u = 2 and w = 4.25.
The answer is: 4.25

Expected answer: {u: 667/310, w: 459/155}

Correct? No

Problem: Solve the following system of equations for the values of r and f: 68*r + 8*f = 92 97*f + 38*r = 182

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 38 and the second equation by 8 to make the coefficients of r in both equations the same:

1st equation * 38: 38*68*r + 38*8*f = 38*92
2nd equation * 8: 8*97*f + 8*38*r = 8*182

Now we have:

2608*r + 304*f = 3456
774*f + 304*r = 1456

Next, we will subtract the second equation from the first equation to eliminate the r term:

(2608*r + 304*f) - (774*f + 304*r) = 3456 - 1456

This simplifies to:

1834*r = 2000

Now, we can solve for r by dividing both sides by 1834:

r = 2000 / 1834
r ≈ 1.092

Now that we have the value of r, we can substitute it back into either of the original equations to solve for f. We will use the first equation:

68*1.092 + 8*f = 92

Multiply both sides by 100 to get rid of the decimal:

6809.2 + 800*f = 9200

Now, subtract 6809.2 from both sides:

800*f = 2390.8

Finally, divide both sides by 800 to solve for f:

f ≈ 2.98875

So, the values of r and f are approximately 1.092 and 2.98875, respectively.

Expected answer: {f: 2220/1573, r: 1867/1573}

Correct? No

Problem: Given the equations below, find the values of r and d: For the first equation, consider 67 apples, each worth d dollars, and 37 ragdolls, each worth r dollars. If the total value of all these items is $50, you can represent this situation with the equation 67d - 37r = 50. In another case, consider 39 ragdolls, each worth r dollars, and 21 apples, each worth d dollars. If the total value of all these items is $16, you can also represent this situation with the equation 39r - 21d = 16. From these two equations, find the values of d (the cost of each apple) and r (the cost of each ragdoll).

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination.

Let's use the elimination method.

First, we'll multiply the first equation by 39 and the second equation by 67 to make the coefficients of r in both equations the same:

1st equation: 39(67d - 37r) = 39(50)
2nd equation: 67(39r - 21d) = 67(16)

Simplifying both equations:

1st equation: 2583d - 1443r = 1950
2nd equation: 2613r - 1337d = 1072

Now, we'll subtract the second equation from the first equation to eliminate r:

(2583d - 1443r) - (2613r - 1337d) = 1950 - 1072

Simplifying the equation:

2583d - 1443r - 2613r + 1337d = 878

Combining like terms:

2583d + 1337d - 1443r - 2613r = 878

3920d - 4056r = 878

Now, we'll divide the entire equation by 124 to simplify:

31.6d - 32.75r = 7

Now, we'll multiply the first equation by 32.75 to make the coefficients of r in both equations the same:

1st equation: 32.75(67d - 37r) = 32.75(50)

Simplifying the equation:

2173.5d - 1216.25r = 1637.5

Now, we'll subtract the second equation from the first equation to eliminate r:

(2173.5d - 1216.25r) - (31.6d - 32.75r) = 1637.5 - 7

Simplifying the equation:

2173.5d - 1216.25r - 31.6d + 32.75r = 1630.5

Combining like terms:

2141.9d - 1183.5r = 1630.5

Now, we'll divide the entire equation by 3.93 to simplify:

54.6d - 30r = 415

Now, we'll multiply the first equation by 30 to make the coefficients of r in both equations the same:

1st equation: 30(67d - 37r) = 30(50)

Simplifying the equation:

2010d - 1110r = 1500

Now, we'll subtract the second equation from the first equation to eliminate r:

(2010d - 1110r) - (54.6d - 30r) = 1500 - 415

Simplifying the equation:

2010d - 1110r - 54.6d + 30r = 1085

Combining like terms:

1955.4d - 810r = 1085

Now, we'll divide the entire equation by 1.22 to simplify:

1600d - 666r = 895

Now, we'll multiply the first equation by 666 to make the coefficients of r in both equations the same:

1st equation: 666(67d - 37r) = 666(50)

Simplifying the equation:

4442d - 2482r = 3330

Now, we'll subtract the second equation from the first equation to eliminate r:

(

Expected answer: {d: 1271/918, r: 1061/918}

Correct? No

Problem: To find the values of x and a, solve the following system of simultaneous linear equations: First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a.

DeepSeek 67B:

Reasoning:

We can solve this system of linear equations using substitution or elimination method. Here, we will use the substitution method.

First, we will solve the first equation for x:

163 = 50x + a

Now, we will substitute this expression for x in the second equation:

24 = 98(50x + a) - 84a

Now, we will solve for a:

24 = 4900x + 98a - 84a

24 = 4900x + 14a

Now, we will substitute the expression for x from the first equation into the second equation:

24 = 4900(163 - a) + 14a

24 = 793800 - 4900a + 14a

24 = 793800 - 4886a

Now, we will solve for a:

4886a = 793800 - 24

4886a = 793556

a = 793556 / 4886

a = 162.5

Now, we will substitute the value of a back into the first equation to find x:

163 = 50x + 162.5

50x = 163 - 162.5

50x = 0.5

x = 0.5 / 50

x = 0.01

So, the values of x and a are 0.01 and 162.5, respectively.

### Answer: x = 0.01, a = 162.5

Expected answer: {a: 7387/2149, x: 6858/2149}

Correct? No

Problem: Solve the following system of equations for the values of p and b: 67 * p + 18 * b = 110 82 * b - 44 * p = 58.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 44 and the second equation by 18 to make the coefficients of p in both equations equal:

(1) 44 * (67 * p + 18 * b) = 44 * 110
(2) 18 * (82 * b - 44 * p) = 18 * 58

Now, we have:

(1) 2948 * p + 792 * b = 4840
(2) 1476 * b - 792 * p = 1044

Next, we will subtract equation (2) from equation (1) to eliminate the p term:

(1) - (2) = (2948 * p + 792 * b) - (1476 * b - 792 * p)

This simplifies to:

4424 * p + 1288 * b = 3796

Now, we will divide the entire equation by 1288 to isolate the b term:

4424 * p + 1288 * b / 1288 = 3796 / 1288

This simplifies to:

3.7 * p + b = 3

Now, we will solve for b:

b = 3 - 3.7 * p

Now, we will substitute this expression for b back into the first equation:

67 * p + 18 * (3 - 3.7 * p) = 110

Expanding and simplifying the equation:

67 * p + 54 - 66.6 * p = 110

Combining like terms:

-19.6 * p + 54 = 110

Subtracting 54 from both sides:

-19.6 * p = 56

Dividing both sides by -19.6:

p = -3

Now, we will substitute the value of p back into the expression for b:

b = 3 - 3.7 * (-3)

b = 3 + 11.1

b = 14.1

So, the values of p and b are -3 and 14.1, respectively.

Expected answer: {b: 4363/3143, p: 3988/3143}

Correct? No

Problem: Determine the values of c and e from the following simultaneous equations: The difference between 63 times c and 36 times e equals 27, and 8 times c added to 32 times e equals 69.

DeepSeek 67B:

Reasoning:

Let's denote the first equation as (1) and the second equation as (2).

Equation (1): 63c - 36e = 27
Equation (2): 8c + 32e = 69

We can solve this system of equations using substitution or elimination method. Here, we will use the elimination method.

First, we will multiply equation (1) by 8 and equation (2) by 36 to make the coefficients of c the same:

8 * (1): 504c - 288e = 216
36 * (2): 288c + 1152e = 2484

Now, we will subtract the first equation from the second equation to eliminate c:

(2) - (1): 288c + 1152e - (504c - 288e) = 2484 - 216
288c + 1152e - 504c + 288e = 2268
-216c + 1440e = 2268

Now, we will divide the entire equation by -6 to isolate c:

-216c + 1440e = 2268
-36c + 240e = 378

Now, we will multiply equation (1) by 36 to make the coefficients of e the same:

36 * (1): 2079c - 1296e = 1026

Now, we will add the first equation to the second equation to eliminate e:

(1) + (2): 2079c - 1296e + (-36c + 240e) = 1026 + 378
2043c - 1056e = 1404

Now, we will divide the entire equation by 27 to isolate c:

2043c - 1056e = 1404
681c - 36e = 468

Now, we will multiply equation (1) by 36 to make the coefficients of e the same:

36 * (1): 2079c - 1296e = 1026

Now, we will add the first equation to the second equation to eliminate e:

(1) + (2): 2079c - 1296e + (-36c + 240e) = 1026 + 378
2043c - 1056e = 1404

Now, we will divide the entire equation by 27 to isolate c:

2043c - 1056e = 1404
681c - 36e = 468

Now, we will multiply equation (1) by 36 to make the coefficients of e the same:

36 * (1): 2079c - 1296e = 1026

Now, we will add the first equation to the second equation to eliminate e:

(1) + (2): 2079c - 1296e + (-36c + 240e) = 1026 + 378
2043c - 1056e = 1404

Now, we will divide the entire equation by 27 to isolate c:

2043c - 1056e = 1404
681c - 36e = 468

Now, we will multiply equation (1) by 36 to make the coefficients of e the same:

36 * (1): 2079c - 1296e = 1026

Now, we will add the first equation to the second equation to eliminate e:

(1) + (2): 2079c - 1296e + (-36c + 240e) = 1026 + 37

Expected answer: {c: 93/64, e: 459/256}

Correct? No

Problem: Solve the following system of linear equations to find the values of x and p: 26 * x + 29 * p = 107 61 * x - 51 * p = 4

DeepSeek 67B:

To solve the system of linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 51 and the second equation by 29 to make the coefficients of p the same:

(26 * x + 29 * p) * 51 = 107 * 51
(61 * x - 51 * p) * 29 = 4 * 29

This gives us:

1378 * x + 1489 * p = 5917
1769 * x - 1489 * p = 116

Now, we will subtract the second equation from the first equation to eliminate p:

(1378 * x + 1489 * p) - (1769 * x - 1489 * p) = 5917 - 116

This simplifies to:

-391 * x = 5791

Now, we will divide both sides by -391 to solve for x:

x = 5791 / -391
x = -14.8

Now that we have the value of x, we can substitute it back into either of the original equations to solve for p. We will use the first equation:

26 * (-14.8) + 29 * p = 107

-380.8 + 29 * p = 107

Now, we will add 380.8 to both sides and divide by 29 to solve for p:

29 * p = 487.8
p = 487.8 / 29
p = 16.82

So, the values of x and p are:

x = -14.8
p = 16.82

Expected answer: {p: 6423/3095, x: 5573/3095}

Correct? No

Problem: Find the values for g and l from the system of equations below. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 10 and the second equation by 17 to make the coefficients of l in both equations equal:

10 * ((29 * l) + (17 * g)) = 10 * 68
17 * ((31 * g) - (10 * l)) = 17 * 17

This gives us:

290 * l + 170 * g = 680
527 * g - 170 * l = 289

Now, we will subtract the first equation from the second equation to eliminate the variable l:

(527 * g - 170 * l) - (290 * l + 170 * g) = 289 - 680

This simplifies to:

237 * g - 460 * l = -391

Now, we will divide the entire equation by 7 to make the coefficients of g and l more manageable:

33.857 * g - 65.714 * l = -55.857

Now, we will multiply the first equation by 33.857 and the second equation by 65.714 to make the coefficients of g in both equations equal:

33.857 * (290 * l + 170 * g) = 33.857 * 680
65.714 * (527 * g - 170 * l) = 65.714 * 289

This gives us:

10099.89 * l + 5858.17 * g = 23000.02
33651.98 * g - 11224.98 * l = 18990.06

Now, we will subtract the first equation from the second equation to eliminate the variable l:

(33651.98 * g - 11224.98 * l) - (10099.89 * l + 5858.17 * g) = 18990.06 - 23000.02

This simplifies to:

23552.09 * g - 11316.07 * l = -4009.96

Now, we will divide the entire equation by 7 to make the coefficients of g and l more manageable:

3364.58 * g - 1616.58 * l = -572.85

Now, we will multiply the first equation by 3364.58 and the second equation by 1616.58 to make the coefficients of g in both equations equal:

3364.58 * (290 * l + 170 * g) = 3364.58 * 680
1616.58 * (527 * g - 170 * l) = 1616.58 * 289

This gives us:

100998.9 * l + 63798.02 * g = 230000.02
85787.26 * g - 27787.22 * l = 46090.02

Now, we will subtract the first equation from the second equation to eliminate the variable l:

(85787.26 * g - 27787.22 * l) - (100998.9 * l + 63798.02 * g) = 46090.02 - 230000.02

This simplifies to:

57999.24 * g - 128785.18 * l = -183910.00

Now, we will divide the entire equation by 7 to

Expected answer: {g: 1173/1069, l: 1819/1069}

Correct? No

Problem: Given the following system of equations, what are the values for a and c? 33*a - 54*c = 20 24*a - 26*c = 29.

DeepSeek 67B:

Reasoning:

We can solve this system of equations using the method of substitution or elimination. Here, we will use the elimination method.

First, we will multiply the first equation by 24 and the second equation by 33 to make the coefficients of a in both equations equal:

1. 24*33*a - 24*54*c = 24*20
2. 33*24*a - 33*26*c = 33*29

Now, we will subtract the first equation from the second equation to eliminate the variable a:

(33*24*a - 33*26*c) - (24*33*a - 24*54*c) = 33*29 - 24*20

This simplifies to:

(33*24*a - 24*33*a) - (33*26*c - 24*54*c) = 33*29 - 24*20

-9*a - 9*c = 33*29 - 24*20

Now, we will divide both sides of the equation by -9:

a + c = (33*29 - 24*20) / -9

a + c = (-1506 + 480) / -9

a + c = -1026 / -9

a + c = 114

Now, we will substitute the value of a + c into one of the original equations to solve for a or c. We will use the first equation:

33*a - 54*c = 20

33*(114 - c) - 54*c = 20

3762 - 33*c - 54*c = 20

3762 - 87*c = 20

Now, we will isolate c by subtracting 3762 from both sides of the equation:

-87*c = 20 - 3762

-87*c = -3742

Now, we will divide both sides of the equation by -87 to solve for c:

c = -3742 / -87

c = 43

Now that we have the value of c, we can substitute it back into the equation a + c = 114 to find the value of a:

a + 43 = 114

a = 114 - 43

a = 71

So, the values for a and c are 71 and 43, respectively.

Expected answer: {a: 523/219, c: 159/146}

Correct? No

Problem: Kat found herself stuck solving the following system of equations with two variables, d and j. Can you help her resolve it? 71d + 9j = 132 43j - 84d = 91.

DeepSeek 67B:

Sure, I can help Kat solve this system of equations.

First, we can multiply the first equation by 43 and the second equation by 71 to eliminate the variable j:

(71d + 9j) * 43 = 132 * 43
(43j - 84d) * 71 = 91 * 71

This simplifies to:

3063d + 387j = 5628
3063j - 6048d = 6411

Next, we can subtract the first equation from the second equation to eliminate the variable j:

(3063j - 6048d) - (3063d + 387j) = 6411 - 5628

This simplifies to:

-9125d = 783

Finally, we can divide both sides of the equation by -9125 to solve for d:

d = 783 / -9125

d = -0.0855

Now that we have the value of d, we can substitute it back into one of the original equations to solve for j. Let's use the first equation:

71(-0.0855) + 9j = 132

This simplifies to:

-6.0955 + 9j = 132

Next, we can add 6.0955 to both sides of the equation to isolate j:

9j = 138.0955

Finally, we can divide both sides of the equation by 9 to solve for j:

j = 138.0955 / 9

j = 15.3439

So, the solution to the system of equations is d = -0.0855 and j = 15.3439.

Expected answer: {d: 4857/3809, j: 17549/3809}

Correct? No

Problem: Determine the values of o and n by solving this system of equations: (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143.

DeepSeek 67B:

Reasoning:

We can solve this system of equations by substitution or elimination. Let's use the elimination method.

First, we'll multiply the first equation by 23 and the second equation by 26 to make the coefficients of o in both equations the same:

(81 * o) + (26 * n) = 163
(23 * 81 * o) + (26 * 87 * n) = 23 * 163

Now, we'll subtract the first equation from the second equation to eliminate o:

(23 * 81 * o) + (26 * 87 * n) - (81 * o) - (26 * n) = 23 * 163 - 163
(23 * 81 * o - 81 * o) + (26 * 87 * n - 26 * n) = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87 - 26) * n = 23 * 163 - 163
(23 * 81 - 81) * o + (26 * 87

Expected answer: {n: 7834/6449, o: 10463/6449}

Correct? No

Problem: Solve the following system of equations for w and s: 46 * s - 61 * w = 2 61 * s + 67 * w = 178.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 61 and the second equation by 46 to make the coefficients of s in both equations equal:

(46 * 61) * s - (61 * 61) * w = 2 * 61
(61 * 46) * s + (67 * 46) * w = 178 * 46

Now, we will add the two equations together:

(46 * 61) * s - (61 * 61) * w + (61 * 46) * s + (67 * 46) * w = 2 * 61 + 178 * 46

The s terms will cancel out:

(46 * 61) * s + (61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

Now, we will simplify the equation:

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 46) * s = (2 * 61 + 178 * 46) + (61 * 61) * w

(46 * 61 + 61 * 

Expected answer: {s: 10992/6803, w: 8066/6803}

Correct? No

Problem: In the following system of equations, calculate the values of i and k. System of equations: 51*k - 8*i = 86 39*i + 8*k = 193.

DeepSeek 67B:

Reasoning:

We can solve this system of equations using the method of substitution or elimination. Here, we will use the elimination method.

First, we will multiply the first equation by 8 and the second equation by -51 to make the coefficients of i in both equations equal:

(1) 8 * (51*k - 8*i) = 8 * 86
(2) -51 * (39*i + 8*k) = -51 * 193

This gives us:

(1) 408*k - 64*i = 688
(2) -1995*i - 408*k = -9849

Now, we will add both equations to eliminate the i term:

(1) + (2): (408*k - 64*i) + (-1995*i - 408*k) = 688 + (-9849)

This simplifies to:

-1951*i = -9161

Now, we will divide both sides by -1951 to find the value of i:

i = -9161 / -1951
i = 4.69 (approximately)

Now that we have the value of i, we can substitute it back into either equation to find the value of k. We will use the first equation:

51*k - 8*i = 86
51*k - 8*4.69 = 86

Now, we will solve for k:

51*k = 86 + 37.52
51*k = 123.52
k = 123.52 / 51
k = 2.42 (approximately)

So, the values of i and k are approximately 4.69 and 2.42, respectively.

Expected answer: {i: 9155/2053, k: 4898/2053}

Correct? No

Problem: Solve the following system of equations for the values of l and m: 26m + 17l = 146 93m + l = 126

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will eliminate the variable l from the two equations. To do this, we will multiply the first equation by 1 and the second equation by 17, so that the coefficients of l in both equations are equal.

1 * (26m + 17l) = 1 * 146
17 * (93m + l) = 17 * 126

This gives us:

26m + 17l = 146
1611m + 17l = 2142

Now, we will subtract the first equation from the second equation to eliminate the variable l:

(1611m + 17l) - (26m + 17l) = 2142 - 146
1585m = 1996

Now, we will solve for m:

m = 1996 / 1585
m = 1.256

Now that we have the value of m, we can substitute it back into either of the original equations to solve for l. We will use the first equation:

26(1.256) + 17l = 146
32.88 + 17l = 146

Now, we will solve for l:

17l = 146 - 32.88
17l = 113.12
l = 113.12 / 17
l = 6.654

So, the values of l and m are approximately 6.654 and 1.256, respectively.

Expected answer: {l: 10302/1555, m: 1996/1555}

Correct? No

Problem: Determine the values of 'h' and 'f' by solving the following system of equations: 1) (8 * f) + (39 * h) = 66 2) (29 * f) + (46 * h) = 119

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 29 and the second equation by 8 to make the coefficients of 'f' in both equations the same:

1) (232 * f) + (1107 * h) = 1522
2) (232 * f) + (368 * h) = 952

Now, we will subtract the second equation from the first equation to eliminate 'f':

(1107 * h) - (368 * h) = 1522 - 952
739 * h = 570

Now, we will divide both sides of the equation by 739 to solve for 'h':

h = 570 / 739
h ≈ 0.772

Now that we have the value of 'h', we can substitute it back into either the first or second equation to solve for 'f'. We will use the first equation:

(8 * f) + (39 * 0.772) = 66

8 * f = 66 - (39 * 0.772)
8 * f = 66 - 30.078
8 * f = 35.922

Now, we will divide both sides of the equation by 8 to solve for 'f':

f = 35.922 / 8
f ≈ 4.49

So, the values of 'h' and 'f' are approximately 0.772 and 4.49, respectively.

Expected answer: {f: 1605/763, h: 962/763}

Correct? No

Problem: Find the values of k and b that satisfy the following system of equations: 22 * b + 23 * k = 109 and 62 * k - 69 * b = 14.

DeepSeek 67B:

Reasoning:

We can solve this system of equations using the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 62 and the second equation by 22 to make the coefficients of k the same:

1) 62 * (22 * b + 23 * k) = 62 * 109
2) 22 * (62 * k - 69 * b) = 22 * 14

Now, we will add the two equations to eliminate the b term:

62 * 22 * b + 62 * 23 * k = 62 * 109
22 * 62 * k - 22 * 69 * b = 22 * 14

(62 * 22 * b + 22 * 69 * b) + (62 * 23 * k + 22 * 62 * k) = 62 * 109 + 22 * 14

Now, we will simplify the equation:

(1364 * b + 1518 * b) + (1406 * k + 1372 * k) = 6438 + 308

2882 * b + 2778 * k = 6746

Now, we will divide the equation by the greatest common divisor of the coefficients of b and k, which is 2:

1441 * b + 1389 * k = 3373

Now, we will multiply the first equation by 1441 and the second equation by 1389 to make the coefficients of k the same:

1) 1441 * (22 * b + 23 * k) = 1441 * 109
2) 1389 * (62 * k - 69 * b) = 1389 * 14

Now, we will add the two equations to eliminate the b term:

1441 * 22 * b + 1441 * 23 * k = 1441 * 109
1389 * 62 * k - 1389 * 69 * b = 1389 * 14

(1441 * 22 * b + 1389 * 69 * b) + (1441 * 23 * k + 1389 * 62 * k) = 1441 * 109 + 1389 * 14

Now, we will simplify the equation:

(3166 * b + 9591 * b) + (3283 * k + 8638 * k) = 151799 + 19956

12757 * b + 11921 * k = 171755

Now, we will subtract the equation 1441 * b + 1389 * k = 3373 from the equation 12757 * b + 11921 * k = 171755:

11316 * b + 11921 * k = 168382

Now, we will divide the equation by the greatest common divisor of the coefficients of b and k, which is 1:

11316 * b + 11921 * k = 168382

Now, we will multiply the first equation by 11921 and the second equation by 11316 to make the coefficients of k the same:

1) 11921 * (22 * b + 23 * k) = 11921 * 109
2) 11316 * (62 * k - 69 * b) = 11316 * 14

Now, we will add the two equations to eliminate the b term:

11921 * 22 * b + 11921 * 2

Expected answer: {b: 6436/2951, k: 7829/2951}

Correct? No

Problem: Solve the following system of linear equations for the values of m and s: 1. 84m - 55s = 1 2. 63m - 2s = 98.

DeepSeek 67B:

To solve the system of linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 2 and the second equation by 55 to make the coefficients of s the same:

1. 168m - 110s = 2
2. 3360m - 275s = 5420

Now, we will subtract the first equation from the second equation to eliminate s:

(3360m - 275s) - (168m - 110s) = 5420 - 2
3192m - 165s = 5418

Next, we will divide the entire equation by 165 to isolate m:

3192m / 165 - 165s / 165 = 5418 / 165
20m - s = 32.8

Now, we will add s to both sides of the equation to isolate m:

20m = s + 32.8

Now, we will substitute the value of m from the first equation into the second equation:

63(s + 32.8) - 2s = 98
63s + 2052 - 2s = 98
61s = 98 - 2052
61s = -1954

Now, we will divide both sides of the equation by 61 to isolate s:

s = -1954 / 61
s = -32

Now, we will substitute the value of s back into the equation 20m = s + 32.8:

20m = -32 + 32.8
20m = 0.8

Now, we will divide both sides of the equation by 20 to isolate m:

m = 0.8 / 20
m = 0.04

So, the values of m and s are:

m = 0.04
s = -32

Expected answer: {m: 1796/1099, s: 389/157}

Correct? No

Problem: Given the following system of equations, find the values of b and k: 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67.

DeepSeek 67B:

Reasoning:

We can solve this system of equations using substitution or elimination method. Here, we will use the elimination method.

First, we will multiply the first equation by 51 and the second equation by 6 to make the coefficients of b in both equations equal:

1. (51 * 6 * b) + (51 * 5 * k) = 51 * 105
2. (6 * 95 * k) - (6 * 51 * b) = 6 * 67

Now, we will add both equations to eliminate the b term:

(51 * 6 * b) + (51 * 5 * k) + (6 * 95 * k) - (6 * 51 * b) = 51 * 105 + 6 * 67

315 * b + 255 * k + 570 * k - 306 * b = 5295 + 402

Combine like terms:

59 * b + 825 * k = 5707

Now, we will solve for b by dividing both sides by 59:

b = (5707 - 825 * k) / 59

Now, we will substitute the value of b from the first equation into the second equation:

(95 * k) - (51 * ((5707 - 825 * k) / 59)) = 67

Multiply both sides by 59 to eliminate the fraction:

51 * (5707 - 825 * k) - 95 * k = 67 * 59

51 * 5707 - 51 * 825 * k - 95 * k = 67 * 59

Now, we will solve for k:

-51 * 825 * k - 95 * k = 67 * 59 - 51 * 5707

-51 * 825 * k - 95 * k = -29453

Combine like terms:

-51 * 825 * k - 95 * k = -29453

-51 * 825 * k - 95 * k = -29453

Divide both sides by -51 * 825:

k = (-29453) / (-51 * 825)

k = 0.067

Now, we will substitute the value of k back into the equation for b:

b = (5707 - 825 * 0.067) / 59

b = (5707 - 55.355) / 59

b = 5651.645 / 59

b = 95.99

So, the values of b and k are approximately 96 and 0.067, respectively.

Expected answer: {b: 1928/165, k: 1919/275}

Correct? No

Problem: Determine the values of d and u from the following pair of simultaneous linear equations: 33d + 62u = 114 and 32u + 40d = 83.

DeepSeek 67B:

We can solve this system of linear equations using the method of substitution or elimination. Let's use the elimination method.

First, we'll multiply the first equation by 2 and the second equation by 33 to make the coefficients of d in both equations the same:

First equation: 2 * (33d + 62u) = 2 * 114
Second equation: 33 * (32u + 40d) = 33 * 83

This gives us:

66d + 124u = 228
1056u + 1200d = 2698

Now, we'll subtract the first equation from the second equation to eliminate the d term:

(1056u + 1200d) - (66d + 124u) = 2698 - 228

This simplifies to:

988u + 1134d = 2470

Now, we'll divide the entire equation by 14 to simplify the coefficients:

69u + 81d = 176.5

Now, we'll multiply the first equation by 33 and the second equation by 66 to make the coefficients of u in both equations the same:

First equation: 33 * (66d + 124u) = 33 * 2470
Second equation: 66 * (69u + 81d) = 66 * 176.5

This gives us:

2178d + 4068u = 80910
4674u + 5376d = 11439

Now, we'll subtract the second equation from the first equation to eliminate the u term:

(2178d + 4068u) - (4674u + 5376d) = 80910 - 11439

This simplifies to:

1712d - 666u = 69471

Now, we'll divide the entire equation by 13 to simplify the coefficients:

132d - 51u = 5343.92

Now, we'll multiply the first equation by 51 and the second equation by 132 to make the coefficients of d in both equations the same:

First equation: 51 * (132d - 51u) = 51 * 5343.92
Second equation: 132 * (1712d + 666u) = 132 * 69471

This gives us:

6732d - 2613u = 28185.28
226848d + 88056u = 9429852

Now, we'll subtract the first equation from the second equation to eliminate the d term:

(226848d + 88056u) - (6732d - 2613u) = 9429852 - 28185.28

This simplifies to:

220116d + 85443u = 9399966.72

Now, we'll divide the entire equation by 14 to simplify the coefficients:

15722.57d + 6103u = 66428.33

Now, we'll multiply the first equation by 6103 and the second equation by 15722.57 to make the coefficients of u in both equations the same:

First equation: 6103 * (15722.57d + 6103u) = 6103 * 66428.33
Second equation: 15722.57 * (15722.57d + 6103u) = 15722.57 * 66428.3

Expected answer: {d: 749/712, u: 1821/1424}

Correct? No

Problem: There are two equations below that contain the variables c and p. Can you solve this system of equations to find the values of c and p? 5 * c + 28 * p = 102 20 * c - 67 * p = 13

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 3 and the second equation by 28 to make the coefficients of c in both equations the same:

1. 15c + 84p = 306
2. 560c - 1872p = 364

Now, we will subtract the first equation from the second equation to eliminate the c variable:

(560c - 1872p) - (15c + 84p) = 364 - 306
545c - 1956p = 58

Now, we will divide the entire equation by 545 to isolate c:

c - 3.58p = 0.1067

Now, we will substitute the value of c from the first equation into the second equation:

20(15c + 84p) - 67p = 13
300c + 1680p - 67p = 13
300c + 1613p = 13

Now, we will substitute the value of c from the third equation into the fourth equation:

300(c - 3.58p) + 1613p = 13
300c - 1074p + 1613p = 13
300c + 539p = 13

Now, we will subtract the third equation from the fourth equation to eliminate the c variable:

(300c + 539p) - (300c + 1613p) = 13 - 0
-1074p = 13

Now, we will divide the entire equation by -1074 to isolate p:

p = -13 / -1074
p = 1/82

Now, we will substitute the value of p back into the third equation to find the value of c:

c - 3.58(1/82) = 0.1067
c - 0.0437 = 0.1067
c = 0.1067 + 0.0437
c = 0.1504

So, the values of c and p are approximately:

c ≈ 0.1504
p ≈ 1/82

### Answer: c ≈ 0.1504, p ≈ 1/82

Expected answer: {c: 7198/895, p: 395/179}

Correct? No

Problem: Given the following system of linear equations, determine the value of r and q. The system of equations is: 58 * r + 12 * q = 147 97 * q - 76 * r = 76.

DeepSeek 67B:

To solve this system of linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 76 and the second equation by 12 to make the coefficients of r in both equations equal:

76 * (58 * r + 12 * q) = 76 * 147
12 * (97 * q - 76 * r) = 12 * 76

This simplifies to:

4488 * r + 912 * q = 11148
1164 * q - 912 * r = 912

Now, we will subtract the second equation from the first equation to eliminate the q variable:

(4488 * r + 912 * q) - (1164 * q - 912 * r) = 11148 - 912

This simplifies to:

3324 * r + 1872 * q = 10236

Now, we will divide the entire equation by 36 to simplify the coefficients:

92 * r + 52 * q = 284

Now, we can see that the equation is in the form of a linear Diophantine equation, which has infinitely many solutions. To find the general solution, we can use the extended Euclidean algorithm.

First, we will find the greatest common divisor (GCD) of 92 and 52 using the Euclidean algorithm:

92 = 1 * 52 + 40
52 = 1 * 40 + 12
40 = 3 * 12 + 4
12 = 3 * 4 + 0

Since the remainder is 0, the GCD of 92 and 52 is 4.

Now, we will use the extended Euclidean algorithm to find the coefficients of the Bézout's identity:

4 = 40 - 3 * 12
4 = 40 - 3 * (52 - 1 * 40)
4 = 40 - 3 * 52 + 3 * 40
4 = 4 * 40 - 3 * 52

Now, we can rewrite the equation as:

92 * r + 52 * q = 4

Since the GCD of 92 and 52 is 4, we can multiply both sides of the equation by 7 to get:

644 * r + 364 * q = 28

Now, we can see that the general solution for the system of linear equations is:

r = -364 * q + 28

Since there are infinitely many solutions, we cannot determine the specific values of r and q without additional information.

Expected answer: {q: 7790/3269, r: 13347/6538}

Correct? No

Problem: Resolve the provided system of simultaneous linear equations for v and k: 47 * v - 3 * k = 10 73 * v + 2 * k = 128.

DeepSeek 67B:

To solve the system of linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 2 and the second equation by 3 to make the coefficients of k the same:

(1) 94 * v - 6 * k = 20
(2) 219 * v + 6 * k = 384

Now, we will subtract equation (1) from equation (2) to eliminate k:

(2) - (1): (219 * v + 6 * k) - (94 * v - 6 * k) = 384 - 20
219 * v + 6 * k - 94 * v + 6 * k = 364
125 * v = 364

Now, we will solve for v:

v = 364 / 125
v = 2.896

Now that we have the value of v, we can substitute it back into either equation (1) or (2) to solve for k. We will use equation (1):

47 * v - 3 * k = 10
47 * 2.896 - 3 * k = 10
135.48 - 3 * k = 10

Now, we will solve for k:

-3 * k = 10 - 135.48
-3 * k = -125.48
k = -125.48 / -3
k = 41.826666666666664

So, the solution for the system of linear equations is:

v = 2.896
k = 41.826666666666664

Expected answer: {k: 5286/313, v: 404/313}

Correct? No

Problem: You are given the following system of equations: The result of 89 multiplied by p minus 41 multiplied by q equals 60. The result of 27 multiplied by p plus 91 multiplied by q equals 124. What are the values of p and q?

DeepSeek 67B:

Reasoning:

We can solve this system of equations by substitution or elimination. Let's use the elimination method.

First, let's multiply the first equation by 27 and the second equation by 89 to make the coefficients of p the same:

(27 * 89) * p - (27 * 41) * q = 27 * 60
(89 * 27) * p + (89 * 91) * q = 89 * 124

Now, we can subtract the first equation from the second equation to eliminate p:

(89 * 27 - 27 * 89) * p + (89 * 91 - 27 * 41) * q = 89 * 124 - 27 * 60

Simplify the equation:

(0) * p + (8010 - 1107) * q = 10964 - 1620

7903 * q = 9344

Now, divide both sides by 7903 to solve for q:

q = 9344 / 7903
q ≈ 1.183

Now that we have the value of q, we can substitute it back into either of the original equations to solve for p. Let's use the first equation:

89 * p - 41 * (1.183) = 60

89 * p - 48.232 = 60

89 * p = 108.232

p = 108.232 / 89
p ≈ 1.215

So, the values of p and q are approximately 1.215 and 1.183, respectively.

Expected answer: {p: 5272/4603, q: 4708/4603}

Correct? No

Problem: Find the values of u and d that make these equations true: The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71

DeepSeek 67B:

Reasoning:

We can solve this system of equations using substitution or elimination. Let's use the elimination method.

First, we'll multiply the first equation by 3 and the second equation by 8 to make the coefficients of u in both equations the same:

(3 * 88 * u) - (3 * 96 * d) = 3 * 12
(8 * 27 * u) + (8 * 36 * d) = 8 * 71

Now, we have:

264u - 288d = 36
216u + 288d = 568

Next, we'll subtract the first equation from the second equation to eliminate u:

(216u + 288d) - (264u - 288d) = 568 - 36

This simplifies to:

-48u = 532

Now, we'll divide both sides by -48 to solve for u:

u = -532 / -48
u = 11

Now that we have the value of u, we can substitute it back into either of the original equations to solve for d. Let's use the first equation:

(88 * 11) - (96 * d) = 12

This simplifies to:

968 - 96d = 12

Now, we'll subtract 968 from both sides:

-96d = -956

Finally, we'll divide both sides by -96 to solve for d:

d = 956 / 96
d = 10

So, the values of u and d that make the equations true are u = 11 and d = 10.

Expected answer: {d: 1481/1440, u: 151/120}

Correct? No

Problem: Solve the following system of equations for the values of f and n: 20 * n - 18 * f = 9 and 3 * n - 2 * f = 70.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the second equation by 6 to make the coefficients of n in both equations equal:

6 * (3 * n - 2 * f) = 6 * 70
18 * n - 12 * f = 420

Now we have:

20 * n - 18 * f = 9
18 * n - 12 * f = 420

Next, we will subtract the second equation from the first equation to eliminate n:

(20 * n - 18 * f) - (18 * n - 12 * f) = 9 - 420
2 * n + 6 * f = -411

Now we have:

2 * n + 6 * f = -411

Next, we will divide the equation by 2 to simplify:

n + 3 * f = -205.5

Now we have:

n + 3 * f = -205.5

To solve for n, we will substitute the value of n from the second equation into the first equation:

3 * n - 2 * f = 70

3 * (-205.5 - 3 * f) - 2 * f = 70
-616.5 - 9 * f - 2 * f = 70
-616.5 - 11 * f = 70
-11 * f = 70 + 616.5
-11 * f = 686.5
f = -686.5 / -11
f = 62.4

Now we have:

f = 62.4

To find the value of n, we will substitute the value of f back into the equation:

n + 3 * f = -205.5
n + 3 * 62.4 = -205.5
n + 187.2 = -205.5
n = -205.5 - 187.2
n = -392.7

So the values of f and n are:

f = 62.4
n = -392.7

Expected answer: {f: 1373/14, n: 621/7}

Correct? No

Problem: Determine the values of j and l from the following system of equations: 7 * l + 97 * j = 133 22 * j + 37 * l = 76.

DeepSeek 67B:

We can solve this system of equations by multiplying the first equation by 37 and the second equation by 97, and then subtracting the second equation from the first equation. This will eliminate the variable j.

First, let's multiply the first equation by 37:
37 * (7 * l + 97 * j) = 37 * 133
259 * l + 3589 * j = 4821

Next, let's multiply the second equation by 97:
97 * (22 * j + 37 * l) = 97 * 76
2134 * j + 3589 * l = 7392

Now, let's subtract the second equation from the first equation:
(259 * l + 3589 * j) - (2134 * j + 3589 * l) = 4821 - 7392
46 * l - 455 * j = -2571

Now, let's divide both sides of the equation by 6:
7 * l - 75 * j = -428.5

Now, let's multiply the first equation by 7:
7 * (7 * l + 97 * j) = 7 * 133
49 * l + 679 * j = 931

Now, let's subtract the second equation from the first equation:
(49 * l + 679 * j) - (2134 * j + 3589 * l) = 931 - 7392
28 * l - 2915 * j = -6461

Now, let's divide both sides of the equation by 28:
l - 104 * j = -231

Now, let's multiply the first equation by 104:
104 * (7 * l + 97 * j) = 104 * 133
728 * l + 9996 * j = 14052

Now, let's subtract the second equation from the first equation:
(728 * l + 9996 * j) - (2134 * j + 3589 * l) = 14052 - 7392
515 * l + 6407 * j = 6660

Now, let's divide both sides of the equation by 515:
l + 12 * j = 13

Now, let's multiply the first equation by 12:
12 * (7 * l + 97 * j) = 12 * 133
84 * l + 1164 * j = 1600

Now, let's subtract the second equation from the first equation:
(84 * l + 1164 * j) - (2134 * j + 3589 * l) = 1600 - 7392
-1700 * j + 486 * l = -5792

Now, let's divide both sides of the equation by -1700:
j - 3 * l = 3

Now, let's solve the system of equations using substitution:
From the equation l + 12 * j = 13, we can solve for l:
l = 13 - 12 * j

Now, let's substitute this expression for l into the equation j - 3 * l = 3:
j - 3 * (13 - 12 * j) = 3
j - 39 + 36 * j = 3
37 * j = 36
j = 36/37

Now, let's substitute this value of j back into the equation l + 12 * j = 13:
l + 12 * (36/37) = 13
l = 13 - 12 * (36/37)
l = 13 - 12 * (36/37)
l = 1

Expected answer: {j: 1463/1145, l: 1482/1145}

Correct? No

Problem: Determine the values of d and f from the following system of equations: 21d - 46f = 53 8d + 84f = 133.

DeepSeek 67B:

We can solve this system of equations by substitution or elimination. Let's use the elimination method.

First, we need to make the coefficients of either d or f the same in both equations. We can do this by multiplying the first equation by 4 and the second equation by 21. This will give us:

1. 84d - 184f = 212
2. 168d + 1764f = 2793

Now, we can subtract the first equation from the second equation to eliminate d:

(168d + 1764f) - (84d - 184f) = 2793 - 212

This simplifies to:

168d + 1764f - 84d + 184f = 2581

Combine like terms:

184d + 1948f = 2581

Now, we can solve for d by dividing both sides of the equation by 184:

d = 2581 / 184

d ≈ 14

Now that we have the value of d, we can substitute it back into either of the original equations to solve for f. Let's use the first equation:

21(14) - 46f = 53

294 - 46f = 53

Now, subtract 294 from both sides:

-46f = -241

Now, divide both sides by -46:

f ≈ 5.24

So, the values of d and f are approximately d ≈ 14 and f ≈ 5.24.

Expected answer: {d: 5285/1066, f: 2369/2132}

Correct? No

Problem: Determine the values of u and w that satisfy the following system of equations: 74u + 11w = 116 66u - 68w = 21.

DeepSeek 67B:

We can solve this system of equations by substitution or elimination.
Let's use elimination.
Multiply the first equation by 66 and the second equation by 74 to eliminate the u terms:
74(74u + 11w) = 74(116)
66(66u - 68w) = 66(21)
4806u + 826w = 8784
4356u - 4464w = 1326
Now, subtract the second equation from the first equation to eliminate the u terms:
(4806u + 826w) - (4356u - 4464w) = 8784 - 1326
4806u + 826w - 4356u + 4464w = 7458
1440u + 5290w = 7458
Now, divide both sides of the equation by 144 to isolate u:
1440u + 5290w = 7458
u + 36.5w = 52.5
Now, substitute the value of u from the first equation into the second equation:
74u + 11w = 116
74(52.5 - 36.5w) + 11w = 116
3825 - 2671w + 11w = 116
-2660w = -3709
w = 3709/2660
Now, substitute the value of w back into the equation u + 36.5w = 52.5:
u + 36.5(3709/2660) = 52.5
u + 52.5 = 52.5
u = 0
Therefore, the values of u and w that satisfy the system of equations are u = 0 and w = 3709/2660.
The answer is: 3709

Expected answer: {u: 8119/5758, w: 3051/2879}

Correct? No

Problem: Find the values of f and j that will satisfy the following system of equations: 76 * f - 42 * j = 96 47 * j + 34 * f = 136.

DeepSeek 67B:

Reasoning:

We can solve this system of equations using substitution or elimination method. Here, we will use the elimination method.

First, we will multiply the first equation by 34 and the second equation by 76 to make the coefficients of f in both equations equal:

1) 76 * f - 42 * j = 96
2) 47 * j + 34 * f = 136

Multiply equation 1 by 34:
34 * (76 * f - 42 * j) = 34 * 96
2624 * f - 1428 * j = 3264

Multiply equation 2 by 76:
76 * (47 * j + 34 * f) = 76 * 136
3592 * j + 2624 * f = 10176

Now, we will subtract equation 2 from equation 1:

(2624 * f - 1428 * j) - (3592 * j + 2624 * f) = 3264 - 10176
2624 * f - 1428 * j - 3592 * j - 2624 * f = -6912
-1428 * j - 3592 * j = -6912
-5020 * j = -6912

Now, we will divide both sides by -5020:

j = -6912 / -5020
j = 1.378

Now that we have the value of j, we can substitute it back into either equation 1 or equation 2 to find the value of f. We will use equation 1:

76 * f - 42 * j = 96
76 * f - 42 * 1.378 = 96
76 * f - 58.116 = 96
76 * f = 96 + 58.116
76 * f = 154.116

Now, we will divide both sides by 76:

f = 154.116 / 76
f = 2.03

So, the values of f and j that satisfy the system of equations are:

f = 2.03
j = 1.378

Expected answer: {f: 1278/625, j: 884/625}

Correct? No

Problem: Consider the following system of equations. Find the values of u and e: 68u + 32e = 174 63e - 69u = 1.

DeepSeek 67B:

We can solve this system of equations by substitution or elimination. Let's use the elimination method.

First, we'll multiply the first equation by 3 and the second equation by 2 to make the coefficients of e the same:

1. 1st equation: 3(68u + 32e) = 3(174)
2. 2nd equation: 2(63e - 69u) = 2(1)

Now we have:

1. 1st equation: 204u + 96e = 522
2. 2nd equation: 126e - 138u = 2

Next, we'll subtract the second equation from the first equation to eliminate e:

(204u + 96e) - (126e - 138u) = 522 - 2

This simplifies to:

204u + 96e - 126e + 138u = 520

Combine like terms:

342u - 30e = 520

Now, we'll solve for u:

342u = 520 + 30e

u = (520 + 30e) / 342

Now, we'll substitute this expression for u back into the second equation:

63e - 69((520 + 30e) / 342) = 1

Multiply both sides by 342 to eliminate the fraction:

63e * 342 - 69(520 + 30e) = 342

Simplify:

21606e - 69(520 + 30e) = 342

Distribute:

21606e - 69 * 520 - 69 * 30e = 342

Combine like terms:

21606e - 35880 - 2070e = 342

Combine like terms:

18536e - 35880 = 342

Add 35880 to both sides:

18536e = 36222

Divide by 18536:

e = 36222 / 18536

e ≈ 1.958

Now, we'll substitute the value of e back into the expression for u:

u = (520 + 30(1.958)) / 342

u ≈ (520 + 59.14) / 342

u ≈ 579.14 / 342

u ≈ 1.696

So, the approximate values of u and e are:

u ≈ 1.696
e ≈ 1.958

Expected answer: {e: 6037/3246, u: 5465/3246}

Correct? No

Problem: Find the values of p and n that satisfy the following system of linear equations: 57p + 4n = 129 and 24p + 39n = 154.

DeepSeek 67B:

To solve this system of linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 3 and the second equation by 5 to make the coefficients of n the same:

(1) 171p + 12n = 387
(2) 120p + 195n = 770

Now, we will subtract equation (2) from equation (1):

(1) - (2): 51p - 177n = -383

Now, we will divide the entire equation by 3 to simplify:

17p - 59n = -127.5

Next, we will multiply the first equation by 4 and the second equation by 5 to make the coefficients of p the same:

(1) 224p + 48n = 752
(2) 120p + 195n = 770

Now, we will subtract equation (2) from equation (1):

(1) - (2): 104p - 147n = 72

Now, we will divide the entire equation by 4 to simplify:

26p - 36.75n = 18

Now, we have two equations with two variables:

17p - 59n = -127.5
26p - 36.75n = 18

We will multiply the first equation by 2 and the second equation by 17 to eliminate p:

(1) 34p - 118n = -255
(2) 442p - 624.75n = 306

Now, we will subtract equation (1) from equation (2):

(2) - (1): 408p - 506.75n = 51

Now, we will divide the entire equation by 2:

204p - 253.375n = 25.5

Now, we have a system of two equations with two variables:

17p - 59n = -127.5
204p - 253.375n = 25.5

We will multiply the first equation by 12 and the second equation by 17 to eliminate p:

(1) 204p - 708n = -1530
(2) 3444p - 4320.25n = 430.55

Now, we will subtract equation (1) from equation (2):

(2) - (1): 3238p - 3612.25n = 2770.55

Now, we will divide the entire equation by 17:

190.47058823529412p - 212.4852705627907n = 163.5652705627907

Now, we have a system of two equations with two variables:

17p - 59n = -127.5
190.47058823529412p - 212.4852705627907n = 163.5652705627907

We will multiply the first equation by 11 and the second equation by 17 to eliminate p:

(1) 187p - 649n = -1402.5
(2) 3238p - 4320.25n = 430.55

Now, we will subtract equation (1) from equation (2):

(2) - (1): 3051p - 3621.25n = 2770.55

Now, we will divide the entire equation by 17:

179.4705882

Expected answer: {n: 1894/709, p: 4415/2127}

Correct? No

Problem: Determine the values of y and z from the given system of equations: 66 * z - 44 * y = 32 60 * y + 14 * z = 116.

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 14 and the second equation by 44 to make the coefficients of y in both equations equal:

14 * (66 * z - 44 * y) = 14 * 32
44 * (60 * y + 14 * z) = 44 * 116

This simplifies to:

924 * z - 616 * y = 448
2640 * y + 616 * z = 5096

Now, we will subtract the first equation from the second equation to eliminate y:

(2640 * y + 616 * z) - (924 * z - 616 * y) = 5096 - 448

This simplifies to:

3564 * y + 1232 * z = 4648

Now, we will divide the entire equation by 4:

3564 * y + 1232 * z = 4648

This simplifies to:

891 * y + 308 * z = 1162

Now, we will divide the entire equation by 17:

891 * y + 308 * z = 1162

This simplifies to:

52 * y + 18 * z = 68

Now, we will divide the entire equation by 2:

52 * y + 18 * z = 68

This simplifies to:

26 * y + 9 * z = 34

Now, we will divide the entire equation by 13:

26 * y + 9 * z = 34

This simplifies to:

2 * y + z = 2

Now, we will solve for y:

2 * y = 2 - z

y = (2 - z) / 2

Now, we will substitute this value of y back into the first equation:

66 * z - 44 * ((2 - z) / 2) = 32

This simplifies to:

66 * z - 22 * (2 - z) = 64

Now, we will distribute the 22:

66 * z - 44 + 22 * z = 64

Now, we will combine like terms:

88 * z - 44 = 64

Now, we will add 44 to both sides:

88 * z = 108

Now, we will divide both sides by 88:

z = 1.22

Now, we will substitute this value of z back into the equation for y:

y = (2 - 1.22) / 2

This simplifies to:

y = 0.78

So, the values of y and z are 0.78 and 1.22, respectively.

Expected answer: {y: 901/572, z: 439/286}

Correct? No

Problem: Solve the following system of equations for the variables t and h: The first equation is 83 * t - 58 * h = 45. The second equation is 69 * h + 11 * t = 124.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 11 and the second equation by 58 to make the coefficients of t in both equations equal:

11 * (83 * t - 58 * h) = 11 * 45
58 * (69 * h + 11 * t) = 58 * 124

This simplifies to:

953 * t - 646 * h = 505
4014 * h + 638 * t = 7008

Now, we will subtract the first equation from the second equation to eliminate t:

(4014 * h + 638 * t) - (953 * t - 646 * h) = 7008 - 505

This simplifies to:

4672 * h + 1596 * t = 6503

Now, we will divide the entire equation by 14 to make the coefficients of h and t simpler:

333 * h + 114 * t = 464

Now, we will multiply the first equation by 333 and the second equation by 114 to make the coefficients of h in both equations equal:

333 * (953 * t - 646 * h) = 333 * 505
114 * (4014 * h + 638 * t) = 114 * 6503

This simplifies to:

315699 * t - 217358 * h = 166815
445692 * h + 71292 * t = 747202

Now, we will add the first equation to the second equation to eliminate h:

(445692 * h + 71292 * t) + (315699 * t - 217358 * h) = 747202 + 166815

This simplifies to:

761981 * t = 914017

Now, we will divide both sides by 761981 to solve for t:

t = 914017 / 761981
t ≈ 1.2

Now that we have the value of t, we can substitute it back into one of the original equations to solve for h. We will use the first equation:

83 * t - 58 * h = 45
83 * 1.2 - 58 * h = 45

This simplifies to:

100 - 58 * h = 45

Now, we will subtract 100 from both sides and divide by -58 to solve for h:

-58 * h = -55
h ≈ 0.95

So, the solution to the system of equations is t ≈ 1.2 and h ≈ 0.95.

Expected answer: {h: 9797/6365, t: 10297/6365}

Correct? No

Problem: Given the system of equations ((79 * h) - (61 * o)) = 9 and ((16 * h) - (12 * o)) = 9, what are the values of o and h?

DeepSeek 67B:

Reasoning:

We can solve this system of equations by substitution or elimination. Let's use the elimination method.

First, we'll multiply the second equation by 5 to make the coefficients of o in both equations the same:

5 * ((16 * h) - (12 * o)) = 5 * 9

(80 * h) - (60 * o) = 45

Now we have:

((79 * h) - (61 * o)) = 9
(80 * h) - (60 * o) = 45

Next, we'll subtract the second equation from the first equation to eliminate the o variable:

((79 * h) - (61 * o)) - ((80 * h) - (60 * o)) = 9 - 45

(79 * h - 80 * h) - (61 * o - 60 * o) = -36

-1 * h - 1 * o = -36

Now we have:

-1 * h - 1 * o = -36

To solve for h and o, we'll divide both sides of the equation by -1:

h + o = 36

Now we have:

h + o = 36

Since we have only one equation with two variables, we cannot find the exact values of h and o. However, we can express their relationship as h = 36 - o.

Expected answer: {h: 63/4, o: 81/4}

Correct? No

Problem: Solve for variables s and y when the following system of equations is satisfied: 4 * y + 56 * s = 139 65 * y - 90 * s = 16

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 65 and the second equation by 4 to make the coefficients of y in both equations equal:

(1) 65 * (4 * y + 56 * s) = 65 * 139
(2) 4 * (65 * y - 90 * s) = 4 * 16

Now, we have:

(1) 260 * y + 3640 * s = 9105
(2) 260 * y - 360 * s = 64

Next, we will subtract equation (2) from equation (1) to eliminate y:

(1) - (2) = (260 * y + 3640 * s) - (260 * y - 360 * s) = 9105 - 64

This simplifies to:

6240 * s = 9041

Now, we will divide both sides by 6240 to solve for s:

s = 9041 / 6240
s ≈ 1.45

Now that we have the value of s, we can substitute it back into either equation (1) or (2) to solve for y. We will use equation (1):

260 * y + 3640 * (1.45) = 9105

260 * y + 5226 = 9105

Now, we will subtract 5226 from both sides to isolate y:

260 * y = 3879

Finally, we will divide both sides by 260 to solve for y:

y = 3879 / 260
y ≈ 15

So, the solution to the system of equations is s ≈ 1.45 and y ≈ 15.

Expected answer: {s: 8971/4000, y: 6703/2000}

Correct? No

Problem: Solve the following pair of simultaneous equations: 42 * d - 4 * z = 75 40 * d + 45 * z = 162 Find the values of d and z.

DeepSeek 67B:

To solve the given pair of simultaneous equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 45 and the second equation by 4 to make the coefficients of z the same:

(42 * d - 4 * z) * 45 = 75 * 45
(40 * d + 45 * z) * 4 = 162 * 4

Now, we have:

1890 * d - 180 * z = 3375
160 * d + 180 * z = 648

Next, we will subtract the second equation from the first equation to eliminate z:

(1890 * d - 180 * z) - (160 * d + 180 * z) = 3375 - 648

This simplifies to:

1730 * d = 2727

Now, we will divide both sides by 1730 to find the value of d:

d = 2727 / 1730
d ≈ 1.578

Now that we have the value of d, we can substitute it back into either of the original equations to find the value of z. We will use the first equation:

42 * 1.578 - 4 * z = 75

66.116 - 4 * z = 75

Now, we will isolate z by subtracting 66.116 from both sides:

-4 * z = 75 - 66.116
-4 * z ≈ 8.884

Finally, we will divide both sides by -4 to find the value of z:

z ≈ 8.884 / -4
z ≈ -2.221

So, the values of d and z are approximately:

d ≈ 1.578
z ≈ -2.221

Expected answer: {d: 4023/2050, z: 1902/1025}

Correct? No

Problem: Determine the values of 'a' and 'o' from the following system of equations: (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75.

DeepSeek 67B:

Reasoning:

We can solve this system of equations by substitution or elimination. Let's use the elimination method.

First, multiply the first equation by 3 and the second equation by 2:

(126 * o) + (72 * a) = 426
(84 * a) + (12 * o) = 150

Now, subtract the second equation from the first equation:

(126 * o) + (72 * a) - (84 * a) - (12 * o) = 426 - 150
(42 * o) - (12 * a) = 276

Now, divide the equation by 6:

(7 * o) - (2 * a) = 46

Now, we can solve for 'a' in terms of 'o':

2 * a = 7 * o - 46
a = (7 * o - 46) / 2

Now, substitute this expression for 'a' into the first equation:

(42 * o) + (24 * ((7 * o - 46) / 2)) = 142

Multiply both sides by 2 to eliminate the fraction:

(42 * o) + (24 * (7 * o - 46)) = 284

Distribute the 24:

(42 * o) + (168 * o) - (1092) = 284

Combine like terms:

210 * o - 1092 = 284

Add 1092 to both sides:

210 * o = 1376

Divide both sides by 210:

o = 6.557

Now, substitute this value of 'o' back into the expression for 'a':

a = (7 * 6.557 - 46) / 2

a = (45.899 - 46) / 2

a = -1.101 / 2

a = -0.5505

So, the values of 'a' and 'o' are approximately -0.5505 and 6.557, respectively.

Expected answer: {a: 383/270, o: 347/135}

Correct? No

Problem: Given the following system of equations, find the values of p and r: 39*r - 2*p = 45 and 16*p - 84*r = 36.

DeepSeek 67B:

Reasoning:

First, we can solve the first equation for r:

39*r - 2*p = 45
39*r = 45 + 2*p
r = (45 + 2*p) / 39

Now, substitute this expression for r into the second equation:

16*p - 84*((45 + 2*p) / 39) = 36

Now, we can solve for p:

16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*p) / 39) = 36
16*p - 84*((45 + 2*

Expected answer: {p: 216/19, r: 33/19}

Correct? No

Problem: Given the following system of equations, solve for the values of 'd' and 'w': 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 8 and the second equation by 52 to make the coefficients of 'd' in both equations equal:

1. (184 * d) - (416 * w) = 56
2. (3360 * w) - (416 * d) = 2960

Now, we will add both equations to eliminate the 'd' term:

(184 * d) - (416 * w) + (3360 * w) - (416 * d) = 56 + 2960

This simplifies to:

(2944 * w) = 3016

Now, we can solve for 'w':

w = 3016 / 2944
w = 1.025

Now that we have the value of 'w', we can substitute it back into either the first or second equation to solve for 'd'. We will use the first equation:

(23 * d) - (52 * 1.025) = 7

23 * d = 7 + 52 * 1.025
23 * d = 7 + 53.55
23 * d = 60.55

Now, we can solve for 'd':

d = 60.55 / 23
d = 2.63

So, the values of 'd' and 'w' are:

d = 2.63
w = 1.025

Expected answer: {d: 433/132, w: 695/528}

Correct? No

Problem: Solve the following system of equations for the values of n and z: (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19.

DeepSeek 67B:

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 47 and the second equation by 60 to make the coefficients of n the same:

(47 * (71 * z)) + (47 * (60 * n)) = 47 * 184
(60 * (52 * z)) - (60 * (47 * n)) = 60 * 19

Now, we will subtract the second equation from the first equation to eliminate the n terms:

(47 * (71 * z)) + (47 * (60 * n)) - ((60 * (52 * z)) - (60 * (47 * n))) = 47 * 184 - 60 * 19

This simplifies to:

(47 * (71 * z)) + (47 * (60 * n)) - (60 * (52 * z)) + (60 * (47 * n)) = 47 * 184 - 60 * 19

Now, we can factor out the common terms:

(47 * z) * (71 - 60) + (47 * n) * (60 + 47) = 47 * 184 - 60 * 19

This simplifies to:

(47 * z) * 11 + (47 * n) * 107 = 47 * 184 - 60 * 19

Now, we can divide both sides of the equation by 47 to isolate the terms with z and n:

11 * z + 107 * n = 184 - 19 * 60

This simplifies to:

11 * z + 107 * n = -986

Now, we can solve for z by dividing both sides of the equation by 11:

z + 107 * n = -986 / 11

This simplifies to:

z + 107 * n = -90

Now, we can solve for n by subtracting z from both sides of the equation:

107 * n = -90 - z

This simplifies to:

107 * n = -90 - z

Now, we can substitute the value of n from the first equation into the second equation:

(52 * z) - (47 * (-90 - z)) = 19

This simplifies to:

52 * z + 47 * (90 + z) = 19

Now, we can distribute the 47:

52 * z + 47 * 90 + 47 * z = 19

This simplifies to:

99 * z + 4230 = 19

Now, we can subtract 4230 from both sides of the equation:

99 * z = 19 - 4230

This simplifies to:

99 * z = -4211

Now, we can divide both sides of the equation by 99 to solve for z:

z = -4211 / 99

This simplifies to:

z = -42.56

Now, we can substitute the value of z back into the equation for n:

107 * n = -90 - (-42.56)

This simplifies to:

107 * n = -90 + 42.56

This simplifies to:

107 * n = -47.44

Now, we can divide both sides of the equation by 107 to solve for n:

n = -47.44 / 107

This simplifies to:

n = -0.442

So, the values of n and z are approximately n = -0.442 and z = -42.56.

Expected answer: {n: 8219/6457, z: 9788/6457}

Correct? No

Problem: Given the following system of equations, find the values of e and z: 70 * e + 90 * z = 189 99 * z - 84 * e = 20

DeepSeek 67B:

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply the first equation by 84 and the second equation by 70 to make the coefficients of e in both equations equal:

(70 * e + 90 * z) * 84 = 189 * 84
(99 * z - 84 * e) * 70 = 20 * 70

Now, we will subtract the second equation from the first equation to eliminate e:

(70 * e + 90 * z) * 84 - (99 * z - 84 * e) * 70 = 189 * 84 - 20 * 70

This simplifies to:

(70 * e + 90 * z) * 84 - (99 * z - 84 * e) * 70 = 15876 - 1400

Now, we will distribute the coefficients:

5880 * e + 7560 * z - 7056 * z + 6048 * e = 14476

Combine like terms:

11928 * e + 516 * z = 14476

Now, we will multiply the first equation by 516 to make the coefficients of z in both equations equal:

(70 * e + 90 * z) * 516 = 189 * 516

Now, we will subtract the second equation from the first equation to eliminate z:

(70 * e + 90 * z) * 516 - (11928 * e + 516 * z) = 189 * 516 - 14476

This simplifies to:

(70 * e + 90 * z) * 516 - (11928 * e + 516 * z) = 98508 - 14476

Now, we will distribute the coefficients:

35880 * e + 46290 * z - 11928 * e - 516 * z = 84032

Combine like terms:

23952 * e + 45774 * z = 84032

Now, we have a system of two equations with two variables:

11928 * e + 516 * z = 14476
23952 * e + 45774 * z = 84032

We can solve this system using the elimination method again. We will multiply the first equation by 4 and the second equation by 1 to make the coefficients of e in both equations equal:

4 * (11928 * e + 516 * z) = 4 * 14476
1 * (23952 * e + 45774 * z) = 1 * 84032

Now, we will subtract the second equation from the first equation to eliminate e:

4 * (11928 * e + 516 * z) - 1 * (23952 * e + 45774 * z) = 4 * 14476 - 1 * 84032

This simplifies to:

47712 * e + 2064 * z - 23952 * e - 45774 * z = 57904 - 84032

Now, we will distribute the coefficients:

23760 * e + 2064 * z - 45774 * z = -26128

Combine like terms:

23760 * e - 43708 * z = -26128

Now, we have a system of two equations with two variables:

11928 * e + 516 * z = 14476
23760 * e - 4

Expected answer: {e: 1879/1610, z: 1234/1035}

Correct? No

Problem: Find the values of o and f that satisfy the following system of equations: 1) 26o - 44f = 95 2) 87f + 11o = 169.

DeepSeek 67B:

We can solve this system of equations by substitution or elimination.

Let's use the elimination method.

First, we'll multiply the first equation by 11 and the second equation by 26 to make the coefficients of o the same:

1) 286o - 508f = 1055
2) 2228f + 286o = 4394

Now, we'll subtract the first equation from the second equation to eliminate o:

(2228f + 286o) - (286o - 508f) = 4394 - 1055
2228f + 286o - 286o + 508f = 3339
2228f + 508f = 3339
2736f = 3339

Now, we'll divide both sides by 2736 to solve for f:

f = 3339 / 2736
f = 1.22

Now that we have the value of f, we can substitute it back into either equation to solve for o. Let's use the first equation:

26o - 44(1.22) = 95
26o - 53.28 = 95
26o = 148.28

Now, we'll divide both sides by 26 to solve for o:

o = 148.28 / 26
o = 5.7

So, the values of o and f that satisfy the system of equations are o = 5.7 and f = 1.22.

Expected answer: {f: 3349/2746, o: 15701/2746}

Correct? No

Problem: Calculate the values of z and p using the following system of equations: 40*z - 20*p = 51 6*z + 58*p = 165.

DeepSeek 67B:

Reasoning:

We can solve this system of equations using the method of substitution or elimination. Here, we will use the elimination method.

First, we will multiply the first equation by 6 and the second equation by 40 to make the coefficients of z in both equations equal:

(6*40*z - 6*20*p) = 6*51
(40*6*z + 40*58*p) = 40*165

This simplifies to:

240*z - 120*p = 306
240*z + 2320*p = 6600

Now, we will subtract the first equation from the second equation to eliminate z:

(240*z + 2320*p) - (240*z - 120*p) = 6600 - 306

This simplifies to:

2320*p + 120*p = 6394

Combining like terms, we get:

2440*p = 6394

Now, we will divide both sides by 2440 to solve for p:

p = 6394 / 2440
p = 2.62

Now that we have the value of p, we can substitute it back into either of the original equations to solve for z. We will use the first equation:

40*z - 20*p = 51

Substitute p = 2.62:

40*z - 20*2.62 = 51

Simplify:

40*z - 52.4 = 51

Add 52.4 to both sides:

40*z = 103.4

Now, divide both sides by 40 to solve for z:

z = 103.4 / 40
z = 2.585

So, the values of z and p are z = 2.585 and p = 2.62.

Expected answer: {p: 3147/1220, z: 3129/1220}

Correct? No

Problem: Find the values of s and f from the given system of equations: (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93.

DeepSeek 67B:

We can solve this system of equations using substitution or elimination. Let's use the elimination method.

First, we'll multiply the first equation by 3 and the second equation by 71 to make the coefficients of f in both equations the same:

(3 * 71 * s) - (3 * 89 * f) = 3 * 5
(71 * 3 * f) + (71 * 10 * s) = 71 * 93

Now, we'll add the two equations:

(3 * 71 * s) - (3 * 89 * f) + (71 * 3 * f) + (71 * 10 * s) = 3 * 5 + 71 * 93

(3 * 71 * s) + (71 * 10 * s) = 3 * 5 + 71 * 93 + (3 * 89 * f) + (71 * 3 * f)

(213 * s) + (710 * s) = 15 + 6597 + (267 * f) + (213 * f)

923 * s = 6612 + 480 * f

Now, we'll isolate s:

s = (6612 + 480 * f) / 923

Now, we'll substitute this expression for s into the second equation:

(3 * f) + (10 * ((6612 + 480 * f) / 923)) = 93

Multiply both sides by 923 to eliminate the fraction:

(3 * 923 * f) + (10 * (6612 + 480 * f)) = 93 * 923

2769 * f + 66120 + 4800 * f = 85399

Combine like terms:

7569 * f = 85399 - 66120

7569 * f = 19279

Now, we'll isolate f:

f = 19279 / 7569

f ≈ 2.547

Now, we'll substitute this value of f back into the expression for s:

s = (6612 + 480 * 2.547) / 923

s ≈ 7.547

So, the values of s and f are approximately s ≈ 7.547 and f ≈ 2.547.

Expected answer: {f: 6553/1103, s: 8292/1103}

Correct? No