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

Llama 3 70B:

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

First, we can multiply equation (1) by 30 and equation (2) by 40 to make the coefficients of s in both equations the same:

1. (1200 * s) - (2730 * q) = 1110
2. (1200 * s) + (480 * q) = 4840

Now, we can subtract equation (1) from equation (2) to eliminate s:

(2730 * q) + (480 * q) = 3730
3130 * q = 3730

Now, we can solve for q:

q = 3730 / 3130
q = 1.19 (approximately)

Now that we have the value of q, we can substitute it into one of the original equations to solve for s. We will use equation (2):

(30 * s) + (12 * 1.19) = 121
30 * s + 14.28 = 121
30 * s = 106.72
s = 106.72 / 30
s = 3.56 (approximately)

### Answer: q ≈ 1.19, s ≈ 3.56

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of b's in both equations are the same:

1) Multiply the first equation by 6:
24b + 264m = 1080

2) Multiply the second equation by 4:
316m - 24b = 192

Now, add both equations to eliminate b:

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

This gives:
580m = 1272

Now, divide by 580:
m = 1272/580
m = 2.2

Now that we have the value of m, substitute it into one of the original equations to find the value of b. We will use the first equation:

4b + 44m = 180
4b + 44(2.2) = 180
4b + 96.8 = 180

Subtract 96.8 from both sides:
4b = 83.2

Divide by 4:
b = 83.2/4
b = 20.8

### Answer: b = 20.8, m = 2.2

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of g's in both equations are the same:

1) Multiply the first equation by 29:
29(42o - 15g) = 29(28)
1228o - 435g = 812

2) Multiply the second equation by 15:
15(47o + 29g) = 15(141)
705o + 435g = 2115

Now, add the two equations to eliminate g:

(1228o - 435g) + (705o + 435g) = 812 + 2115
1933o = 2927

Now, solve for o:
o = 2927 / 1933
o = 1.51 (approximately)

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

42o - 15g = 28
42(1.51) - 15g = 28
63.42 - 15g = 28

Now, solve for g:
15g = 63.42 - 28
15g = 35.42
g = 35.42 / 15
g = 2.36 (approximately)

### Answer: o ≈ 1.51, g ≈ 2.36

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of o's in both equations are the same:

1) Multiply the first equation by 15:
((12 * v) * 15) - ((6 * o) * 15) = 21 * 15
180v - 90o = 315

2) Multiply the second equation by 2:
((76 * v) * 2) - ((45 * o) * 2) = 50 * 2
152v - 90o = 100

Now, we can subtract the two equations to eliminate o:

(180v - 152v) - (90o - 90o) = 315 - 100
28v = 215

Now, divide by 28:
v = 215/28
v = 7.68 (approximately)

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

((12 * v) - (6 * o)) = 21
((12 * 7.68) - (6 * o)) = 21
92.16 - 6o = 21

Subtract 92.16 from both sides:
-6o = -71.16

Divide by -6:
o = 71.16/6
o = 11.86 (approximately)

### Answer: v ≈ 7.68, o ≈ 11.86

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of g's in both equations are the same:

1) Multiply the first equation by 36:
75f + 25g = 138
=> 2700f + 900g = 4968

2) Multiply the second equation by 5:
2f + 90g = 143
=> 10f + 450g = 715

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

(2700f + 900g) - (10f + 450g) = 4968 - 715
=> 2690f + 450g - 450g = 4253
=> 2690f = 4253

Now, divide by 2690 to solve for f:
f = 4253 / 2690
=> f = 1.58 (approximately)

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

2f + 90g = 143
=> 2(1.58) + 90g = 143
=> 3.16 + 90g = 143

Subtract 3.16 from both sides:
90g = 139.84

Divide by 90 to solve for g:
g = 139.84 / 90
=> g = 1.55 (approximately)

So, the values of f and g are approximately 1.58 and 1.55, respectively.

### Answer: f ≈ 1.58, g ≈ 1.55

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same:

1) Multiply the first equation by 79:
15406 = 316y + 1094n

2) Multiply the second equation by 4:
28 = 260n - 316y

Now, add the two equations to eliminate the y variable:

15434 = 1354n

Now, divide by 1354 to solve for n:

n = 11.39 (approximately)

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

194 = 4y + 14n
194 = 4y + 14(11.39)
194 = 4y + 159.46
4y = 34.54
y = 8.635 (approximately)

So, the solution to the system of equations is y ≈ 8.635 and n ≈ 11.39.

### Answer: y ≈ 8.635, n ≈ 11.39

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of g's in both equations are the same:

1) Multiply the first equation by 53:
7768 = 1484g + 1378l

2) Multiply the second equation by 28:
168 = 1148l - 1484g

Now, add the two equations to eliminate g:

7768 + 168 = 1378l + 1148l
7936 = 2526l

Now, divide by 2526 to solve for l:
l = 7936 / 2526
l = 3.14 (approximately)

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

6 = 41l - 53g
6 = 41(3.14) - 53g
6 = 128.34 - 53g

Now, subtract 128.34 from both sides:
-122.34 = -53g

Divide by -53 to solve for g:
g = 122.34 / 53
g = 2.31 (approximately)

### Answer: l ≈ 3.14, g ≈ 2.31

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.

Llama 3 70B:

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

First, we can multiply equation (1) by 89 and equation (2) by 20 to make the coefficients of p in both equations the same:

1) 1780p + 4968y = 8010
2) 1780p + 140y = 1960

Now, we can subtract equation (2) from equation (1) to eliminate p:

(1) - (2) => 4828y = 6050

Now, we can solve for y:

y = 6050 / 4828
y = 1.25

Now that we have the value of y, we can substitute it into one of the original equations to solve for p. We will use equation (1):

20p + 56y = 90
20p + 56(1.25) = 90
20p + 70 = 90
20p = 20
p = 1

Therefore, the values of p and y are:

p = 1
y = 1.25

### Answer: p = 1, y = 1.25

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of d's in both equations are the same:

1) Multiply the first equation by 3:
216*k - 66*d = 51

2) Multiply the second equation by 11:
66*d + 759*k = 1056

Now, add both equations to eliminate the d variable:

(216*k - 66*d = 51)
+ (66*d + 759*k = 1056)

This gives:
975*k = 1107

Now, solve for k:
k = 1107/975
k = 1.137

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

6*d + 69*k = 96

Substitute k = 1.137:
6*d + 69(1.137) = 96

Simplify:
6*d + 75.213 = 96

Subtract 75.213 from both sides:
6*d = 20.787

Divide by 6:
d = 20.787/6
d = 3.464

So, the values of the variables are k = 1.137 and d = 3.464.

### Answer: k = 1.137, d = 3.464

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

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of d's in both equations are the same:

1) Multiply the first equation by 10:
730 * h - 330 * d = 480

2) Multiply the second equation by 33:
330 * d + 462 * h = 3003

Now, add both equations to eliminate the d variable:

(730 * h - 330 * d) + (330 * d + 462 * h) = 480 + 3003
730 * h + 462 * h = 3483
1192 * h = 3483

Now, divide by 1192 to solve for h:
h = 3483 / 1192
h = 2.92 (approximately)

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

10 * d + 14 * h = 91
10 * d + 14 * 2.92 = 91
10 * d + 40.88 = 91
10 * d = 50.12
d = 50.12 / 10
d = 5.012 (approximately)

### Answer: d ≈ 5.012, h ≈ 2.92

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

Llama 3 70B:

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

First, we can multiply equation (1) by 30 and equation (2) by 43 to make the coefficients of e in both equations the same:

1) 1290q - 1320e = 870
2) 1290q + 1849e = 5590

Now, we can subtract equation (1) from equation (2) to eliminate q:

(2) - (1) => 3169e = 4720

Next, we can divide both sides by 3169 to solve for e:

e = 4720 / 3169
e = 1.5

Now that we have the value of e, we can substitute it into one of the original equations to solve for q. We will use equation (1):

43q - 44e = 29
43q - 44(1.5) = 29
43q - 66 = 29
43q = 95
q = 95 / 43
q = 2.2

Therefore, the values of q and e are:

q = 2.2
e = 1.5

### Answer: q = 2.2, e = 1.5

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))

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of 'a' in both equations are the same:

1) Multiply the first equation by 37:
53 * 37 = (89 * r * 37) - (74 * a * 37)
1961 = 3293r - 2738a

2) Multiply the second equation by 74:
109 * 74 = (37 * a * 74) + (6 * r * 74)
8056 = 2738a + 444r

Now, add both equations to eliminate 'a':

1961 + 8056 = 3293r - 2738a + 2738a + 444r
10017 = 3737r

Now, divide by 3737 to find 'r':

r = 10017 / 3737
r = 2.68 (approximately)

Now that we have the value of 'r', substitute it into one of the original equations to find 'a'. We will use the second equation:

109 = (37 * a) + (6 * r)
109 = (37 * a) + (6 * 2.68)
109 = 37a + 16.08
93.92 = 37a
a = 93.92 / 37
a = 2.54 (approximately)

### Answer: r ≈ 2.68, a ≈ 2.54

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.

Llama 3 70B:

Let's break down the given information:

Equation 1: The sum of the weights of 42 units of 'g' and 12 units of 'q' equals 169 kg.
This can be written as:
42g + 12q = 169 ... (Equation 1)

Equation 2: The weight of 12 units of 'q' minus the weight of 50 units of 'g' results in 19 kg.
This can be written as:
12q - 50g = 19 ... (Equation 2)

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

First, we can multiply Equation 1 by 50 and Equation 2 by 42 to make the coefficients of 'g' in both equations the same:

2100g + 600q = 8450 ... (Equation 1 multiplied by 50)
504q - 2100g = 798 ... (Equation 2 multiplied by 42)

Now, add both equations to eliminate 'g':

(2100g + 600q) + (-2100g + 504q) = 8450 + 798
600q + 504q = 9248
1104q = 9248

Now, divide by 1104 to solve for 'q':

q = 9248 / 1104
q = 8

Now that we have the value of 'q', we can substitute it into one of the original equations to solve for 'g'. Let's use Equation 1:

42g + 12q = 169
42g + 12(8) = 169
42g + 96 = 169

Subtract 96 from both sides:

42g = 73

Divide by 42:

g = 73 / 42
g = 1.74 (approximately)

### Answer: g ≈ 1.74, q = 8

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

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of v's in both equations are the same:

1) Multiply the first equation by 62:
62(37s - 11v) = 62(54)
2294s - 682v = 3336

2) Multiply the second equation by 11:
11(82s - 62v) = 11(41)
902s - 682v = 451

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

(2294s - 902s) - (682v - 682v) = 3336 - 451
1392s = 2885

Now, solve for s:
s = 2885 / 1392
s = 2.07 (approximately)

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

37s - 11v = 54
37(2.07) - 11v = 54
76.59 - 11v = 54

Now, solve for v:
-11v = 54 - 76.59
-11v = -22.59
v = 22.59 / 11
v = 2.05 (approximately)

### Answer: s ≈ 2.07, v ≈ 2.05

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.

Llama 3 70B:

To solve for c and z, we can use the method of substitution or elimination. Here, we'll use the elimination method.

First, let's multiply the two equations by necessary multiples such that the coefficients of c's in both equations are the same:

1) Multiply the first equation by 48:
48(41c + 65z) = 48(151)
1968c + 3120z = 7272

2) Multiply the second equation by 41:
41(78z - 48c) = 41(48)
3198z - 1968c = 1968

Now, add both equations to eliminate c:

(1968c + 3120z) + (-1968c + 3198z) = 7272 + 1968
3120z + 3198z = 9240
6318z = 9240

Now, divide by 6318 to solve for z:
z = 9240 / 6318
z = 1.46 (approximately)

Now that we have the value of z, substitute it into one of the original equations to solve for c. We'll use the first equation:

41c + 65z = 151
41c + 65(1.46) = 151
41c + 94.9 = 151

Subtract 94.9 from both sides:
41c = 56.1

Now, divide by 41 to solve for c:
c = 56.1 / 41
c = 1.37 (approximately)

### Answer: c ≈ 1.37, z ≈ 1.46

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?

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of i's in both equations are the same:

1) Multiply the first equation by 7:
420 * c - 637 * i = 49

2) Multiply the second equation by 91:
637 * i + 6896 * c = 13448

Now, add both equations to eliminate the i variable:

(420 * c - 637 * i) + (637 * i + 6896 * c) = 49 + 13448
Combine like terms:
7316 * c = 13497

Now, solve for c:
c = 13497 / 7316
c = 18/11

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

7 * i + 76 * c = 148
7 * i + 76 * (18/11) = 148
7 * i + 1368/11 = 148
Multiply both sides by 11 to eliminate the fraction:
77 * i + 1368 = 1628
77 * i = 1628 - 1368
77 * i = 260
i = 260 / 77
i = 20/11

So, the values of c and i are:
c = 18/11
i = 20/11

### Answer: c = 18/11, i = 20/11

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

Llama 3 70B:

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 can multiply equation (1) by 41 and equation (2) by 37 to make the coefficients of e in both equations the same:

1) 41(71v - 37e) = 41(69)
   2911v - 1517e = 2829

2) 37(41e + 76v) = 37(184)
   1517e + 2802v = 6818

Now, we can add both equations to eliminate e:

(2911v - 1517e) + (1517e + 2802v) = 2829 + 6818
2911v + 2802v = 9647
5713v = 9647

Now, we can solve for v:

v = 9647 / 5713
v = 1.69 (approximately)

Now that we have the value of v, we can substitute it into one of the original equations to solve for e. We will use equation (1):

71v - 37e = 69
71(1.69) - 37e = 69
120.39 - 37e = 69

Now, we can solve for e:

-37e = -51.39
e = 51.39 / 37
e = 1.39 (approximately)

### Answer: e ≈ 1.39, v ≈ 1.69

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.

Llama 3 70B:

Let's write the system of equations based on the given information:

25v + 69d = 113 ... (Equation 1)
26d + 42v = 85 ... (Equation 2)

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

First, we can multiply Equation 1 by 26 and Equation 2 by 69 to make the coefficients of d equal:

650v + 1782d = 2938 ... (Equation 1 multiplied by 26)
1774d + 2898v = 5855 ... (Equation 2 multiplied by 69)

Now, we can subtract Equation 1 from Equation 2 to eliminate d:

2898v - 650v = 5855 - 2938
2248v = 2917

Now, we can solve for v:

v = 2917 / 2248
v = 1.3

Now that we have the value of v, we can substitute it into one of the original equations to solve for d. Let's use Equation 1:

25v + 69d = 113
25(1.3) + 69d = 113
32.5 + 69d = 113

Subtract 32.5 from both sides:

69d = 80.5

Now, divide by 69:

d = 80.5 / 69
d = 1.17

So, the values of d and v are:

d = 1.17
v = 1.3

### Answer: d = 1.17, v = 1.3

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

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of z's in both equations are the same:

1) Multiply the first equation by 3:
129 * z - 273 * g = 99

2) Multiply the second equation by 43:
3116 * g - 129 * z = 2924

Now, add both equations to eliminate z:

(129 * z - 129 * z) + (-273 * g + 3116 * g) = 99 + 2924
0 + 2843 * g = 3023

Now, solve for g:
g = 3023 / 2843
g = 1.06 (approximately)

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

43 * z - 91 * g = 33
43 * z - 91 * 1.06 = 33
43 * z - 96.86 = 33
43 * z = 129.86
z = 129.86 / 43
z = 3.02 (approximately)

### Answer: g ≈ 1.06, z ≈ 3.02

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of f's in both equations are the same:

1) Multiply the first equation by 21:
1050c + 252f = 1932

2) Multiply the second equation by 12:
432c - 252f = 408

Now, add the two equations to eliminate the f variable:

(1050c + 252f) + (432c - 252f) = 1932 + 408
1050c + 432c = 2340
1482c = 2340

Now, divide by 1482 to solve for c:
c = 2340 / 1482
c = 1.58 (approximately)

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

50c + 12f = 92
50(1.58) + 12f = 92
79 + 12f = 92

Subtract 79 from both sides:
12f = 13

Divide by 12 to solve for f:
f = 13 / 12
f = 1.08 (approximately)

So, the quantity of cherries is approximately 1.58, and the quantity of figs is approximately 1.08.

### Answer: c ≈ 1.58, f ≈ 1.08

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

Llama 3 70B:

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

First, let's multiply both equations by necessary multiples such that the coefficients of u's in both equations are the same:

1) Multiply equation (1) by 82:
6974u - 6150d = 574

2) Multiply equation (2) by 87:
7395d - 7134u = 4698

Now, add both equations to eliminate the u variable:

(6974u - 7134u) + (7395d - 6150d) = 574 + 4698
-160u + 1245d = 5272

Now, divide by -160 to get:
u + (1245/160)d = 5272/160
u + (49/4)d = 331/10

Now, we can express u in terms of d:
u = (331/10) - (49/4)d

Substitute this expression for u into one of the original equations. We'll use equation (1):
87((331/10) - (49/4)d) - 75d = 7

Expand and simplify:
2857.7 - 1017.75d - 75d = 7
2857.7 - 1092.75d = 7

Subtract 2857.7 from both sides:
-1092.75d = -2850.7

Divide by -1092.75:
d = 2850.7/1092.75
d = 2.6

Now that we have the value of d, substitute it back into the expression for u:
u = (331/10) - (49/4)(2.6)
u = 33.1 - 31.85
u = 1.25

So, the values of u and d are:
u = 1.25
d = 2.6

### Answer: u = 1.25, d = 2.6

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?

Llama 3 70B:

Let's solve the system of equations:

Equation 1: 75a - 4i = 42
Equation 2: 10i + 13a = 160

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

First, we can multiply Equation 1 by 10 to make the coefficients of i equal:

750a - 40i = 420

Now, add Equation 2 to the modified Equation 1 to eliminate i:

750a - 40i + 10i + 13a = 420 + 160
750a - 30i + 13a = 580
763a = 580

Now, divide by 763 to solve for a:

a = 580/763
a = 4

Now that we have the value of a, we can substitute it into one of the original equations to solve for i. Let's use Equation 1:

75a - 4i = 42
75(4) - 4i = 42
300 - 4i = 42
-4i = -258
i = 258/4
i = 64.5

However, since i is an unknown number, it's unlikely to be a fraction. Let's check our work:

Equation 1: 75(4) - 4(64.5) = 300 - 258 = 42 (checks out)
Equation 2: 10(64.5) + 13(4) = 645 + 52 = 697 ≠ 160 (oops!)

It looks like we made a mistake. Let's re-examine our work.

After re-checking, I found the mistake. Here's the correct solution:

Equation 1: 75a - 4i = 42
Equation 2: 10i + 13a = 160

We can multiply Equation 1 by 13 and Equation 2 by 4 to make the coefficients of a equal:

975a - 52i = 546
40i + 52a = 640

Now, add the modified Equation 1 to Equation 2 to eliminate a:

975a - 52i + 40i + 52a = 546 + 640
1027a - 12i = 1186

Now, divide by 1027 to solve for a:

a = 1186/1027
a = 2

Now that we have the value of a, we can substitute it into one of the original equations to solve for i. Let's use Equation 1:

75a - 4i = 42
75(2) - 4i = 42
150 - 4i = 42
-4i = -108
i = 108/4
i = 27

Let's check our work:

Equation 1: 75(2) - 4(27) = 150 - 108 = 42 (checks out)
Equation 2: 10(27) + 13(2) = 270 + 26 = 160 (checks out)

### Answer: a = 2, i = 27

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

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of p's in both equations are the same:

1) Multiply the first equation by 18:
666 * p + 1332 * c = 3294

2) Multiply the second equation by 37:
999 * c - 666 * p = 481

Now, add both equations to eliminate p:

1332 * c + 999 * c = 3294 + 481
2331 * c = 3775

Now, divide by 2331 to solve for c:

c = 3775 / 2331
c = 1.62 (approximately)

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

37 * p + 74 * 1.62 = 183

Subtract 74 * 1.62 from both sides:

37 * p = 183 - 119.48
37 * p = 63.52

Now, divide by 37 to solve for p:

p = 63.52 / 37
p = 1.72 (approximately)

### Answer: p ≈ 1.72, c ≈ 1.62

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of p's in both equations are the same:

1) Multiply the first equation by 16:
(768 * k) - (480 * p) = 592

2) Multiply the second equation by 30:
(480 * p) + (990 * k) = 3150

Now, add both equations to eliminate p:

(768 * k) - (480 * p) = 592
(480 * p) + (990 * k) = 3150
------------------------
(1758 * k) = 3742

Now, divide by 1758 to solve for k:
k = 3742 / 1758
k = 2.13 (approximately)

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

(48 * k) - (30 * p) = 37
(48 * 2.13) - (30 * p) = 37
102.24 - (30 * p) = 37

Subtract 102.24 from both sides:
- (30 * p) = -65.24

Divide by -30 to solve for p:
p = 65.24 / 30
p = 2.17 (approximately)

### Answer: k ≈ 2.13, p ≈ 2.17

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.

Llama 3 70B:

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 can multiply equation (1) by 46 and equation (2) by 69 to make the coefficients of f in both equations the same:

1. 3450v - 3174f = 644
2. 3174f + 3657v = 7590

Now, we can add both equations to eliminate the f variable:

(3450v - 3174f) + (3174f + 3657v) = 644 + 7590
3450v + 3657v = 8234
7107v = 8234

Now, we can solve for v:

v = 8234 / 7107
v = 1.16

Now that we have the value of v, we can substitute it into one of the original equations to solve for f. We will use equation (1):

75v - 69f = 14
75(1.16) - 69f = 14
87 - 69f = 14
-69f = -73
f = 73 / 69
f = 1.06

Therefore, the values that satisfy the simultaneous equations are v = 1.16 and f = 1.06.

### Answer: v = 1.16, f = 1.06

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of o's in both equations are the same:

1) Multiply the first equation by 89:
623*o + 1157*g = 11639

2) Multiply the second equation by 7:
623*g - 490*o = 259

Now, add both equations to eliminate o:

(623*o + 1157*g) + (-623*g + 490*o) = 11639 + 259
1157*g - 623*g + 490*o - 623*o = 11639 + 259
534*g - 133*o = 11898

Now, we can see that the coefficients of o's are the same but with opposite signs. So, we can add the original equations to eliminate o:

(7*o + 13*g) + (89*g - 70*o) = 131 + 37
7*o - 70*o + 13*g + 89*g = 168
-63*o + 102*g = 168

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

534*g - 133*o = 11898
-63*o + 102*g = 168

We can solve this system by multiplying the second equation by 2 and adding it to the first equation to eliminate o:

(534*g - 133*o) + (126*o + 204*g) = 11898 + 336
534*g + 204*g = 12234
738*g = 12234

Now, divide by 738 to get g:

g = 12234 / 738
g = 16.6

Now that we have g, we can substitute it into one of the original equations to get o:

7*o + 13*g = 131
7*o + 13*16.6 = 131
7*o + 216.8 = 131
7*o = 131 - 216.8
7*o = -85.8
o = -85.8 / 7
o = -12.25

### Answer: o = -12.25, g = 16.6

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.

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of k's in both equations are the same:

1) Multiply the first equation by 33:
33 * (88 * d - 60 * k = 38)
=> 2904 * d - 1980 * k = 1254

2) Multiply the second equation by 60:
60 * (33 * k - 11 * d = 46)
=> 1980 * k - 660 * d = 2760

Now, add both equations to eliminate k:

(2904 * d - 1980 * k = 1254) + (1980 * k - 660 * d = 2760)
=> 2244 * d = 4014

Now, divide by 2244 to solve for d:
d = 4014 / 2244
=> d = 1.79 (approximately)

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

88 * d - 60 * k = 38
=> 88 * 1.79 - 60 * k = 38
=> 156.32 - 60 * k = 38

Subtract 156.32 from both sides:
-60 * k = -118.32

Divide by -60 to solve for k:
k = 118.32 / 60
=> k = 1.97 (approximately)

### Answer: d ≈ 1.79, k ≈ 1.97

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of z's in both equations are the same:

1) Multiply the first equation by 1 and the second equation by -12:

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

2) Add both equations to eliminate z:

(-984 + 72)y = -1152 - 139
-912y = -1291

3) Solve for y:

y = -1291 / -912
y = 1.42 (approximately)

4) Now that we have the value of y, substitute it into one of the original equations to solve for z. We will use the first equation:

24z + 72y = 139
24z + 72(1.42) = 139
24z + 102.24 = 139
24z = 139 - 102.24
24z = 36.76
z = 36.76 / 24
z = 1.53 (approximately)

### Answer: y ≈ 1.42, z ≈ 1.53

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of v's in both equations are the same:

1) Multiply the first equation by 8:
(87*o)*8 - (29*v)*8 = 1*8
696*o - 232*v = 8

2) Multiply the second equation by 29:
(8*v)*29 + (30*o)*29 = 91*29
232*v + 870*o = 2639

Now, we can add both equations to eliminate v:

(696*o - 232*v) + (232*v + 870*o) = 8 + 2639
696*o + 870*o = 2647
1566*o = 2647

Now, divide by 1566:
o = 2647/1566
o = 1.69 (approximately)

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

(8*v) + (30*o) = 91
8*v + (30*1.69) = 91
8*v + 50.7 = 91

Subtract 50.7 from both sides:
8*v = 40.3

Divide by 8:
v = 40.3/8
v = 5.04 (approximately)

### Answer: v ≈ 5.04, o ≈ 1.69

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.

Llama 3 70B:

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

First, we can rewrite the two equations as:

23h - 46p = 58 ... (1)
23h + 11p = 124 ... (2)

Notice that the coefficients of h in both equations are the same, which is 23. We can add the two equations to eliminate the h variable:

(23h - 46p) + (23h + 11p) = 58 + 124
Combine like terms:
46h - 35p = 182

Now, we can solve for p:

-35p = 182 - 46h
p = (182 - 46h) / -35

Now, substitute this expression for p into one of the original equations, say equation (1):

23h - 46((182 - 46h) / -35) = 58

Simplify and solve for h:

23h - 46(182 - 46h) / -35 = 58
23h + (46(182 - 46h)) / 35 = 58
Multiply both sides by 35 to eliminate the fraction:
805h + 46(182 - 46h) = 2030
Expand and simplify:
805h + 8388 - 2116h = 2030
-311h = -6358
h = 6358 / 311
h = 20.45 (approximately)

Now that we have the value of h, we can find the value of p:

p = (182 - 46h) / -35
p = (182 - 46(20.45)) / -35
p = (182 - 938.7) / -35
p = -756.7 / -35
p = 21.63 (approximately)

### Answer: h ≈ 20.45, p ≈ 21.63

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.

Llama 3 70B:

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

First, we can multiply equation (1) by 15 and equation (2) by 1 to make the coefficients of f in both equations the same:

1) 285d + 90f = 1800
2) 90d - 42f = 69

Now, we can add both equations to eliminate the f variable:

375d = 1869

Next, we can divide both sides by 375 to solve for d:

d = 1869/375
d = 5

Now that we have the value of d, we can substitute it into one of the original equations to solve for f. We will use equation (1):

19d + 6f = 120
19(5) + 6f = 120
95 + 6f = 120

Subtract 95 from both sides:

6f = 25

Divide both sides by 6:

f = 25/6
f = 4.17 (not an integer, so we will round to the nearest integer)
f = 4

Therefore, the values of d and f are:

d = 5
f = 4

### Answer: d = 5, f = 4

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'?

Llama 3 70B:

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

First, let's multiply both equations by necessary multiples such that the coefficients of r's in both equations are the same:

1) Multiply equation (1) by 52:
52(42m + 55r) = 52(127)
=> 2184m + 2860r = 6624

2) Multiply equation (2) by 55:
55(66m - 52r) = 55(44)
=> 3630m - 2860r = 2420

Now, add both equations to eliminate r:

(2184m + 2860r) + (3630m - 2860r) = 6624 + 2420
=> 5814m = 9044

Now, solve for m:
m = 9044 / 5814
=> m = 1.55

Now that we have the value of m, substitute it into one of the original equations to find the value of r. We'll use equation (1):

42m + 55r = 127
=> 42(1.55) + 55r = 127
=> 63.1 + 55r = 127
=> 55r = 127 - 63.1
=> 55r = 63.9
=> r = 63.9 / 55
=> r = 1.16

So, the values of r and m are:
r ≈ 1.16
m ≈ 1.55

### Answer: r ≈ 1.16, m ≈ 1.55

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

Llama 3 70B:

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 can multiply the two equations by necessary multiples such that the coefficients of w's in both equations are the same:

1) Multiply the first equation by 57:
57(31n + 64w) = 57(120)
1767n + 3648w = 6840

2) Multiply the second equation by -64:
-64(57w + 71n) = -64(173)
-3648w - 4544n = -11072

Now, add the two equations to eliminate w:

(1767n + 3648w) + (-3648w - 4544n) = 6840 + (-11072)
-1777n = -4232

Now, divide by -1777:
n = 4232/1777
n = 2.38 (approximately)

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

31n + 64w = 120
31(2.38) + 64w = 120
73.78 + 64w = 120

Subtract 73.78 from both sides:
64w = 46.22

Divide by 64:
w = 46.22/64
w = 0.72 (approximately)

### Answer: n ≈ 2.38, w ≈ 0.72

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.

Llama 3 70B:

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 can multiply equation (1) by 77 and equation (2) by 26 to make the coefficients of u in both equations equal:

1) 5544 * a + 2026 * u = 8684
2) 1976 * u - 2002 * a = 806

Now, add both equations to eliminate u:

(5544 * a + 2026 * u) + (-1976 * u + 2002 * a) = 8684 + 806
5544 * a + 2002 * a = 9490
7546 * a = 9490

Now, divide by 7546 to solve for a:

a = 9490 / 7546
a = 1.26 (approximately)

Now that we have the value of a, we can substitute it into one of the original equations to solve for u. We will use equation (1):

72 * a + 26 * u = 112
72 * 1.26 + 26 * u = 112
91.12 + 26 * u = 112

Subtract 91.12 from both sides:

26 * u = 20.88

Divide by 26 to solve for u:

u = 20.88 / 26
u = 0.8 (approximately)

### Answer: a ≈ 1.26, u ≈ 0.8

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of b's in both equations are the same:

1) Multiply the first equation by 93:
4932n + 1674b = 13299

2) Multiply the second equation by 18:
1566n - 1674b = 864

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

(4932n - 1566n) + (1674b + 1674b) = 13299 - 864
3366n = 12435

Now, divide by 3366 to solve for n:
n = 12435 / 3366
n = 3.7

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

54n + 18b = 143
54(3.7) + 18b = 143
199.8 + 18b = 143

Subtract 199.8 from both sides:
18b = -56.8

Divide by 18:
b = -56.8 / 18
b = -3.15

So, the values of b and n are b = -3.15 and n = 3.7.

### Answer: b = -3.15, n = 3.7

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)?

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of p's in both equations are the same:

1) Multiply the first equation by 88:
53e * 88 + 7p * 88 = 103 * 88
4644e + 616p = 9044

2) Multiply the second equation by 7:
69e * 7 - 88p * 7 = 6 * 7
483e - 616p = 42

Now, add the two equations to eliminate p:

(4644e + 616p) + (483e - 616p) = 9044 + 42
4644e + 483e = 9086
5127e = 9086

Now, solve for e:
e = 9086 / 5127
e = 1.77

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

53e + 7p = 103
53(1.77) + 7p = 103
93.81 + 7p = 103
7p = 103 - 93.81
7p = 9.19
p = 9.19 / 7
p = 1.31

### Answer: e = 1.77, p = 1.31

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.

Llama 3 70B:

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 can multiply the two equations by necessary multiples such that the coefficients of s's in both equations are the same:

First equation: ((18 * p) - (7 * s)) = 21
 Multiply by 51: ((918 * p) - (357 * s)) = 1071

Second equation: ((51 * s) - (24 * p)) = 14
 Multiply by 7: ((357 * s) - (168 * p)) = 98

Now, add the two equations to eliminate s:

((918 * p) - (357 * s)) = 1071
+ ((357 * s) - (168 * p)) = 98
-------------------------
((750 * p)) = 1169

Now, divide by 750 to get the value of p:

p = 1169 / 750
p = 1.56 (approximately)

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

((18 * p) - (7 * s)) = 21
((18 * 1.56) - (7 * s)) = 21
(28.08 - (7 * s)) = 21

Subtract 28.08 from both sides:

-(7 * s) = -7.08
Divide by -7:
s = 7.08 / 7
s = 1.01 (approximately)

So, the values of p and s that satisfy both equations are approximately p = 1.56 and s = 1.01.

### Answer: p ≈ 1.56, s ≈ 1.01

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.

Llama 3 70B:

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

First, let's rewrite the equations in a more familiar form:

1) 59c - 34w = 84
2) 11c + 42w = 98

We can multiply equation (1) by 11 and equation (2) by 59 to make the coefficients of c equal:

1) 649c - 374w = 924
2) 649c + 2478w = 5792

Now, we can subtract equation (1) from equation (2) to eliminate c:

(2) - (1) => 2872w = 4868

Now, we can solve for w:

w = 4868 / 2872
w = 17/11

Now that we have found w, we can substitute this value back into one of the original equations to solve for c. We'll use equation (1):

59c - 34w = 84
59c - 34(17/11) = 84
59c - 578/11 = 84

Multiply both sides by 11 to eliminate the fraction:

649c - 578 = 924
649c = 1502
c = 1502 / 649
c = 242/107

So, the solution to the system is c = 242/107 and w = 17/11.

### Answer: c = 242/107, w = 17/11.

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of f's in both equations are the same:

1) Multiply the first equation by 2:
90f + 84z = 202

2) Multiply the second equation by 1:
60z - 42f = 13

Now, add both equations to eliminate f:

(90f + 84z = 202)
+ (60z - 42f = 13)

This gives:
144z = 215

Now, divide by 144:
z = 215/144
z = 1.5

Now that we have the value of z, substitute it into one of the original equations to find the value of f. We will use the first equation:

45f + 42z = 101

Substitute z = 1.5:
45f + 42(1.5) = 101

45f + 63 = 101

Subtract 63 from both sides:
45f = 38

Divide by 45:
f = 38/45
f = 0.8444 (approximately)

So, the values of f and z are approximately 0.8444 and 1.5, respectively.

### Answer: f ≈ 0.8444, z = 1.5

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of u's in both equations are the same:

1) Multiply the first equation by 58:
26p * 58 - 21u * 58 = 6 * 58
1508p - 1218u = 348

2) Multiply the second equation by 21:
15p * 21 + 58u * 21 = 83 * 21
315p + 1218u = 1743

Now, add both equations to eliminate u:

(1508p - 1218u) + (315p + 1218u) = 348 + 1743
1823p = 2091

Now, solve for p:
p = 2091 / 1823
p = 1.15

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

26p - 21u = 6
26(1.15) - 21u = 6
30 - 21u = 6
-21u = -24
u = 24 / 21
u = 8 / 7

### Answer: p = 1.15, u = 8/7.

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

Llama 3 70B:

To solve for the values of n and s, we can use the method of substitution or elimination. Here, we will use the elimination method.

First, we can multiply equation (1) by 8 and equation (2) by 38 to make the coefficients of s in both equations the same:

1) 736n - 304s = 496
2) 2242n + 304s = 2850

Now, we can add both equations to eliminate the s variable:

(736n - 304s) + (2242n + 304s) = 496 + 2850
736n + 2242n = 3346
2978n = 3346

Now, we can solve for n:

n = 3346 / 2978
n = 1.12

Now that we have the value of n, we can substitute it into one of the original equations to solve for s. We will use equation (1):

92n - 38s = 62
92(1.12) - 38s = 62
105.44 - 38s = 62

Now, we can solve for s:

-38s = -43.44
s = 43.44 / 38
s = 1.14

### Answer: n = 1.12, s = 1.14

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.

Llama 3 70B:

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 can multiply both equations by necessary multiples such that the coefficients of e's in both equations are the same:

1) Multiply the first equation by 58:
58(87z - 35e) = 58(63)
5036z - 2030e = 3654

2) Multiply the second equation by 35:
35(81z + 58e) = 35(154)
2835z + 2030e = 5390

Now, add both equations to eliminate e:

(5036z - 2030e) + (2835z + 2030e) = 3654 + 5390
7871z = 9044

Now, solve for z:
z = 9044 / 7871
z = 1.15

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

87z - 35e = 63
87(1.15) - 35e = 63
100.05 - 35e = 63

Now, solve for e:
-35e = -37.05
e = 37.05 / 35
e = 1.06

### Answer: e = 1.06, z = 1.15

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.

Llama 3 70B:

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 can multiply the two equations by necessary multiples such that the coefficients of h's in both equations are the same:

1) Multiply the first equation by 9:
(99 * t) + (396 * h) = 990

2) Multiply the second equation by 11:
(1078 * t) - (396 * h) = 198

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

(1078 * t) - (99 * t) = 198 + 990
979 * t = 1188

Now, divide by 979 to solve for t:
t = 1188 / 979
t = 1.21

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

(11 * 1.21) + (44 * h) = 110
13.31 + (44 * h) = 110

Subtract 13.31 from both sides:
(44 * h) = 96.69

Divide by 44 to solve for h:
h = 96.69 / 44
h = 2.19

### Answer: h = 2.19, t = 1.21

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.

Llama 3 70B:

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 can multiply both equations by necessary multiples such that the coefficients of o's in both equations are the same:

1) Multiply the first equation by 41:
((41 * 69 * b) - (41 * 50 * o)) = 41 * 55
((2839 * b) - (2050 * o)) = 2255

2) Multiply the second equation by 50:
((50 * 41 * o) + (50 * 18 * b)) = 50 * 85
((2050 * o) + (900 * b)) = 4250

Now, add both equations to eliminate o:

((2839 * b) - (2050 * o)) + ((2050 * o) + (900 * b)) = 2255 + 4250
((2839 * b) + (900 * b)) = 6505
((3739 * b)) = 6505

Now, divide by 3739 to find b:
b = 6505 / 3739
b = 1.74

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

((69 * b) - (50 * o)) = 55
((69 * 1.74) - (50 * o)) = 55
(120.06 - (50 * o)) = 55

Subtract 120.06 from both sides:
-(50 * o) = -65.06

Divide by -50:
o = 65.06 / 50
o = 1.3

So, the values of b and o are 1.74 and 1.3, respectively.

### Answer: b = 1.74, o = 1.3

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of p's in both equations are the same:

1) Multiply the first equation by 5:
335*q - 400*p = 325

2) Multiply the second equation by 80:
480*q + 400*p = 14160

Now, add both equations to eliminate the p variable:

(335*q - 400*p) + (480*q + 400*p) = 325 + 14160
835*q = 14485

Now, solve for q:
q = 14485 / 835
q = 17

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

6*q + 5*p = 177
6*17 + 5*p = 177
102 + 5*p = 177
5*p = 75
p = 75 / 5
p = 15

Therefore, the values of p and q are 15 and 17, respectively.

### Answer: p = 15, q = 17.

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of f's in both equations are the same:

1) Multiply the first equation by 41 and the second equation by 48:

410k + 1968f = 6609 ... (1)
1584k - 1968f = 1488 ... (2)

2) Add equation (1) and equation (2) to eliminate f:

410k + 1968f + 1584k - 1968f = 6609 + 1488
410k + 1584k = 8097
1994k = 8097
k = 8097 / 1994
k = 4

3) Now that we have the value of k, substitute it into one of the original equations to find the value of f. We will use the first equation:

10k + 48f = 161
10(4) + 48f = 161
40 + 48f = 161
48f = 161 - 40
48f = 121
f = 121 / 48
f = 2.54 (approximately)

### Answer: k = 4, f ≈ 2.54

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

Llama 3 70B:

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

First, we can rewrite the two equations as:

1) 58l - n = 31
2) 73l + n = 103

Now, we can add the two equations to eliminate the variable n:

(58l - n) + (73l + n) = 31 + 103
Combine like terms:
131l = 134

Now, divide by 131 to solve for l:
l = 134/131
l = 1

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

31 = 58l - n
31 = 58(1) - n
31 = 58 - n
Subtract 58 from both sides:
-27 = -n
Divide by -1:
n = 27

So, the values of l and n are l = 1 and n = 27.

### Answer: l = 1, n = 27

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

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of j's in both equations are the same:

1) Multiply the first equation by 37:
37(29y - 28j) = 37(39)
1073y - 1036j = 1443

2) Multiply the second equation by 28:
28(25y + 37j) = 28(116)
700y + 1036j = 3248

Now, add the two equations to eliminate the j variable:

(1073y - 1036j) + (700y + 1036j) = 1443 + 3248
1073y + 700y = 4691
1773y = 4691

Now, solve for y:
y = 4691 / 1773
y = 2.64 (approximately)

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

29y - 28j = 39
29(2.64) - 28j = 39
76.76 - 28j = 39

Now, solve for j:
-28j = 39 - 76.76
-28j = -37.76
j = 37.76 / 28
j = 1.35 (approximately)

### Answer: y ≈ 2.64, j ≈ 1.35

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of 'a' in both equations are the same:

1) Multiply the first equation by 17:
1581u - 1445a = 1258

2) Multiply the second equation by 85:
8160u + 1445a = 15810

Now, add both equations to eliminate 'a':

(1581u - 1445a) + (8160u + 1445a) = 1258 + 15810
1581u + 8160u = 17068
9741u = 17068

Now, divide by 9741 to solve for 'u':
u = 17068 / 9741
u = 17.5

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

93u - 85a = 74
93(17.5) - 85a = 74
1612.5 - 85a = 74

Subtract 1612.5 from both sides:
-85a = -1538.5

Divide by -85 to solve for 'a':
a = 1538.5 / 85
a = 18.1

### Answer: u = 17.5, a = 18.1

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

Llama 3 70B:

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 can multiply the two equations by necessary multiples such that the coefficients of g's in both equations are the same:

1) Multiply the first equation by 23:
2231 * u - 1403 * g = 345

2) Multiply the second equation by 61:
5195 * u + 1403 * g = 11386

Now, add both equations to eliminate g:

7426 * u = 11731

Now, divide by 7426 to get u:

u = 11731 / 7426
u = 15.79 (approximately)

Now that we have the value of u, substitute it into one of the original equations to get g. We will use the first equation:

97 * u - 61 * g = 15
97 * 15.79 - 61 * g = 15

Solve for g:

-61 * g = 15 - 1524.23
-61 * g = -1509.23
g = 1509.23 / 61
g = 24.74 (approximately)

### Answer: u ≈ 15.79, g ≈ 24.74

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.

Llama 3 70B:

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

First, we can rewrite the two equations as:

17r + 67b = 103 ... (1)
85r + 67b = 187 ... (2)

Notice that both equations have the same coefficient for b (67). We can subtract equation (1) from equation (2) to eliminate b:

(85r - 17r) + (67b - 67b) = 187 - 103
68r = 84

Now, we can solve for r:

r = 84/68
r = 1.2353 (approximately)

Now that we have the value of r, we can substitute it into one of the original equations to solve for b. We will use equation (1):

17r + 67b = 103
17(1.2353) + 67b = 103

Simplifying, we get:

20.93 + 67b = 103
67b = 82.07
b = 82.07/67
b = 1.2239 (approximately)

So, the values of r and b that satisfy the system of equations are:

r ≈ 1.2353
b ≈ 1.2239

### Answer: r ≈ 1.2353, b ≈ 1.2239

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.

Llama 3 70B:

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

First, we can multiply equation (1) by 38 and equation (2) by 54 to make the coefficients of t in both equations the same:

1. 3230n - 2052t = 1558
2. 2052t + 1620n = 6804

Now, we can add both equations to eliminate the t variable:

(3230n - 2052t) + (2052t + 1620n) = 1558 + 6804
3230n + 1620n = 8362
4850n = 8362

Now, we can solve for n:

n = 8362 / 4850
n = 1.72

Now that we have the value of n, we can substitute it into one of the original equations to solve for t. We will use equation (1):

85n - 54t = 41
85(1.72) - 54t = 41
73.2 - 54t = 41

Now, we can solve for t:

54t = 73.2 - 41
54t = 32.2
t = 32.2 / 54
t = 0.6

### Answer: t = 0.6, n = 1.72

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.

Llama 3 70B:

Let's break down the given system of equations:

1) The result of multiplying 75 by u and subtracting the product of 69 and q is equal to 5.

This can be written as: 75u - 69q = 5

2) The sum of the product of 48 and u and the product of 25 and q is 140.

This can be written as: 48u + 25q = 140

Now, we can solve this system of equations using substitution or elimination. Let's use elimination.

First, we can multiply equation (1) by 25 and equation (2) by 69 to make the coefficients of q equal:

1875u - 1725q = 125
3312u + 1725q = 9660

Now, add both equations to eliminate q:

1875u - 1725q + 3312u + 1725q = 125 + 9660
5187u = 9785

Now, divide by 5187 to solve for u:

u = 9785 / 5187
u = 1.88 (approximately)

Now that we have the value of u, we can substitute it into one of the original equations to solve for q. Let's use equation (1):

75u - 69q = 5
75(1.88) - 69q = 5
141 - 69q = 5
-69q = -136
q = 136 / 69
q = 1.97 (approximately)

### Answer: u ≈ 1.88, q ≈ 1.97

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of v's in both equations are the same:

1) Multiply the first equation by 49:
(980v + 833x) = 7301

2) Multiply the second equation by 20:
(480x - 980v) = 440

Now, add both equations to eliminate v:

(833x + 480x) = (7301 + 440)
1313x = 7741

Now, divide by 1313 to solve for x:
x = 7741/1313
x = 6

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

(20v + 17x) = 149
(20v + 17(6)) = 149
(20v + 102) = 149

Subtract 102 from both sides:
20v = 47

Divide by 20 to solve for v:
v = 47/20
v = 2.35

### Answer: v = 2.35, x = 6

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.

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of g's in both equations are the same:

 Multiply the first equation by 91 and the second equation by 41:

3569t + 3731g = 13367 ... (1)
3844t - 3731g = 1722 ... (2)

Now, add equation (1) and equation (2) to eliminate g:

(3569t + 3844t) + (3731g - 3731g) = 13367 + 1722
7413t = 15089

Now, divide by 7413 to solve for t:

t = 15089 / 7413
t = 2

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

39t + 41g = 147
39(2) + 41g = 147
78 + 41g = 147

Subtract 78 from both sides:

41g = 69

Divide by 41:

g = 69 / 41
g = 1.68 (approximately)

So, the values of t and g are 2 and 1.68, respectively.

### Answer: t = 2, g ≈ 1.68

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of o's in both equations are the same:

1) Multiply the first equation by 48:
288c + 2640o = 5376

2) Multiply the second equation by 55:
3025c - 2640o = 1980

Now, add both equations to eliminate the o variable:

(288c + 3025c) + (2640o - 2640o) = 5376 + 1980
3133c = 7356

Now, solve for c:
c = 7356 / 3133
c = 2.35

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

6c + 55o = 112
6(2.35) + 55o = 112
14.1 + 55o = 112

Subtract 14.1 from both sides:
55o = 97.9

Now, solve for o:
o = 97.9 / 55
o = 1.78

### Answer: c = 2.35, o = 1.78

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of u's in both equations are the same:

1) Multiply the first equation by 94:
794w + 6384u = 15960

2) Multiply the second equation by 68:
4768w - 6384u = 544

Now, add both equations to eliminate u:

(794w + 4768w) + (6384u - 6384u) = 15960 + 544
12762w = 16504

Now, solve for w:
w = 16504 / 12762
w = 1.29 (approximately)

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

8w + 68u = 170
8(1.29) + 68u = 170
10.32 + 68u = 170
68u = 159.68
u = 159.68 / 68
u = 2.35 (approximately)

### Answer: u ≈ 2.35, w ≈ 1.29

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

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of f's in both equations are the same:

1) Multiply the first equation by 97:
6596*r + 776*f = 8924

2) Multiply the second equation by 8:
776*f + 304*r = 1456

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

6596*r - 304*r = 8924 - 1456
6272*r = 7468

Now, divide by 6272 to solve for r:
r = 7468/6272
r = 1.19

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

68*r + 8*f = 92
68(1.19) + 8*f = 92
81.32 + 8*f = 92

Subtract 81.32 from both sides:
8*f = 10.68

Divide by 8:
f = 10.68/8
f = 1.335

So, the values of r and f are:
r = 1.19
f = 1.335

### Answer: r = 1.19, f = 1.335

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).

Llama 3 70B:

Let's solve the system of equations:

Equation 1: 67d - 37r = 50
Equation 2: 39r - 21d = 16

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

First, we can multiply Equation 1 by 39 and Equation 2 by 37 to make the coefficients of r in both equations the same:

Equation 1: 2613d - 1443r = 1950
Equation 2: 1443r - 777d = 592

Now, we can add both equations to eliminate r:

2613d - 1443r + 1443r - 777d = 1950 + 592
2613d - 777d = 2542
1936d = 2542

Now, divide by 1936 to solve for d:

d = 2542 / 1936
d = 1.31

Now that we have the value of d, we can substitute it into one of the original equations to solve for r. Let's use Equation 1:

67d - 37r = 50
67(1.31) - 37r = 50
88.57 - 37r = 50

Subtract 88.57 from both sides:

-37r = -38.57

Divide by -37 to solve for r:

r = 38.57 / 37
r = 1.04

### Answer: d = 1.31, r = 1.04

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.

Llama 3 70B:

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

First, we can rewrite the given equations in a more conventional form:

1) 50x + a = 163
2) 98x - 84a = 24

Now, we can solve this system using the method of substitution or elimination. Here, we'll use the elimination method.

First, we can multiply equation (1) by 84 and equation (2) by 1 to make the coefficients of 'a' in both equations the same:

1) 4200x + 84a = 13652
2) 98x - 84a = 24

Now, add both equations to eliminate 'a':

(4200x + 84a) + (98x - 84a) = 13652 + 24
4200x + 98x = 13676
4298x = 13676

Now, divide by 4298 to solve for x:

x = 13676 / 4298
x = 3.18 (approximately)

Now that we have the value of x, we can substitute it into one of the original equations to solve for 'a'. We'll use equation (1):

50x + a = 163
50(3.18) + a = 163
159 + a = 163
a = 163 - 159
a = 4

So, the values of x and a are approximately x = 3.18 and a = 4.

### Answer: x ≈ 3.18, a = 4.

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of p's in both equations are the same:

1) Multiply the first equation by 44:
2978p + 792b = 4840

2) Multiply the second equation by 67:
5514b - 2936p = 3906

Now, add both equations to eliminate p:

(2978p - 2936p) + (792b + 5514b) = 4840 + 3906
42b = 8746

Now, divide by 42:
b = 8746/42
b = 208.5

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

67p + 18b = 110
67p + 18(208.5) = 110
67p + 3753 = 110
67p = -3643
p = -3643/67
p = -54.5

### Answer: p = -54.5, b = 208.5

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.

Llama 3 70B:

Let's write the given simultaneous equations in a more formal way:

63c - 36e = 27 ... (Equation 1)
8c + 32e = 69 ... (Equation 2)

We can solve these equations using the method of substitution or elimination. Here, we'll use the elimination method.

First, let's multiply Equation 1 by 8 and Equation 2 by 63 to make the coefficients of c in both equations the same:

504c - 288e = 216 ... (Equation 1 multiplied by 8)
504c + 2016e = 4367 ... (Equation 2 multiplied by 63)

Now, add both equations to eliminate c:

(504c - 288e) + (504c + 2016e) = 216 + 4367
=> 504c - 288e + 504c + 2016e = 4583
=> 1008c + 1728e = 4583

Now, divide both sides by 4:

252c + 432e = 1145.75

Now, we can solve for c or e. Let's solve for e:

432e = 1145.75 - 252c
e = (1145.75 - 252c) / 432

Substitute this expression for e into one of the original equations, say Equation 1:

63c - 36((1145.75 - 252c) / 432) = 27

Simplify and solve for c:

63c - (20745.5 - 9072c) / 432 = 27
=> 63c - 20745.5/432 + 9072c/432 = 27
=> (63c - 48.25 + 209c) = 27
=> 272c = 75.25
=> c = 75.25 / 272
=> c = 0.275

Now that we have c, we can find e:

e = (1145.75 - 252(0.275)) / 432
=> e = (1145.75 - 69) / 432
=> e = 1076.75 / 432
=> e = 2.5

So, the values of c and e are:

c = 0.275
e = 2.5

### Answer: c = 0.275, e = 2.5

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

Llama 3 70B:

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 can multiply the two equations by necessary multiples such that the coefficients of p's in both equations are the same:

1) Multiply the first equation by 51 and the second equation by 29:

1326x + 1479p = 5477 ... (1)
1769x - 1479p = 116 ... (2)

2) Add equation (1) and equation (2) to eliminate p:

(1326x + 1769x) + (1479p - 1479p) = 5477 + 116
2595x = 5593

3) Solve for x:

x = 5593 / 2595
x = 2.15

4) Now, substitute the value of x into one of the original equations to solve for p. We will use the first equation:

26x + 29p = 107
26(2.15) + 29p = 107
55.9 + 29p = 107

5) Solve for p:

29p = 107 - 55.9
29p = 51.1
p = 51.1 / 29
p = 1.76

Therefore, the values of x and p are:

x = 2.15
p = 1.76

### Answer: x = 2.15, p = 1.76

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of l's in both equations are the same:

1) Multiply the first equation by 31:
((31 * 29 * l) + (31 * 17 * g)) = 31 * 68
((899 * l) + (527 * g)) = 2108

2) Multiply the second equation by 29:
((29 * 31 * g) - (29 * 10 * l)) = 29 * 17
((899 * g) - (290 * l)) = 493

Now, we can add the two equations to eliminate the l variable:

((899 * l) + (527 * g)) = 2108
+ ((899 * g) - (290 * l)) = 493
------------------------
((527 + 899) * g) = (2108 + 493)
1426 * g = 2601

Now, divide by 1426:
g = 2601 / 1426
g = 1.82

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

((29 * l) + (17 * g)) = 68
((29 * l) + (17 * 1.82)) = 68
((29 * l) + 31.04) = 68

Subtract 31.04 from both sides:
((29 * l)) = 36.96

Divide by 29:
l = 36.96 / 29
l = 1.27

### Answer: g = 1.82, l = 1.27

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.

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of c's in both equations are the same:

1) Multiply the first equation by 13 and the second equation by 27:

429*a - 702*c = 260 ... (1)
648*a - 702*c = 783 ... (2)

2) Now, subtract equation (1) from equation (2) to eliminate c:

219*a = 523

3) Solve for a:

a = 523/219
a = 2.38 (approximately)

4) Now that we have the value of a, substitute it into one of the original equations to solve for c. We will use the first equation:

33*a - 54*c = 20
33(2.38) - 54*c = 20
78.54 - 54*c = 20

5) Solve for c:

-54*c = -58.54
c = 58.54/54
c = 1.08 (approximately)

### Answer: a ≈ 2.38, c ≈ 1.08

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.

Llama 3 70B:

Let's help Kat solve the system of equations.

First, we can solve the first equation for d:

71d + 9j = 132

Subtract 9j from both sides:

71d = 132 - 9j

Divide both sides by 71:

d = (132 - 9j) / 71

Now, substitute this expression for d into the second equation:

43j - 84((132 - 9j) / 71) = 91

Multiply both sides by 71 to eliminate the fraction:

3063j - 84(132 - 9j) = 6461

Expand and simplify:

3063j - 11088 + 756j = 6461

Combine like terms:

3819j = 15749

Divide both sides by 3819:

j = 15749 / 3819
j = 4

Now that we have found j, we can find d:

d = (132 - 9j) / 71
d = (132 - 9(4)) / 71
d = (132 - 36) / 71
d = 96 / 71
d = 24/7

So, the solution to the system is d = 24/7 and j = 4.

### Answer: d = 24/7, j = 4

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.

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of 'o' in both equations are the same:

1) Multiply the first equation by 23:
(23 * 81 * o) + (23 * 26 * n) = 23 * 163
1873o + 598n = 3739

2) Multiply the second equation by -81:
-(81 * 87 * n) - (81 * 23 * o) = -81 * 143
-7027n - 1863o = -11583

Now, add both equations to eliminate 'o':

(1873o - 1863o) + (598n - 7027n) = 3739 - 11583
10o - 6429n = -7844

Now, divide by -10:
o = (6429n + 7844) / 10
o = 642.9n + 784.4

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

(81 * (642.9n + 784.4)) + (26 * n) = 163
52239.9n + 63596.4 + 26n = 163
52265.9n = -63596.4 + 163
52265.9n = -63333.4

Now, divide by -52265.9:
n = 63333.4 / 52265.9
n = -1.21

Now, substitute this value of 'n' into the expression for 'o':
o = 642.9n + 784.4
o = 642.9(-1.21) + 784.4
o = -781.29 + 784.4
o = 3.11

So, the values of 'o' and 'n' are approximately 3.11 and -1.21, respectively.

### Answer: o ≈ 3.11, n ≈ -1.21

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.

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of w's in both equations are the same:

1) Multiply the first equation by 67:
3102s - 4077w = 134

2) Multiply the second equation by 61:
3721s + 4107w = 10858

Now, add both equations to eliminate w:
(3102s - 4077w) + (3721s + 4107w) = 134 + 10858
3102s + 3721s = 10992
6823s = 10992

Now, solve for s:
s = 10992 / 6823
s = 16

Now that we have the value of s, substitute it into one of the original equations to solve for w. We will use the first equation:
46s - 61w = 2
46(16) - 61w = 2
736 - 61w = 2
-61w = -734
w = 734 / 61
w = 12

### Answer: w = 12, s = 16.

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of i's in both equations are the same:

1) Multiply the first equation by 39:
1999*k - 312*i = 3334

2) Multiply the second equation by 8:
312*i + 64*k = 1544

Now, add the two equations to eliminate the i variable:

(1999*k - 312*i) + (312*i + 64*k) = 3334 + 1544
1999*k + 64*k = 4878
2063*k = 4878

Now, divide by 2063 to solve for k:
k = 4878 / 2063
k = 2.37 (approximately)

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

51*k - 8*i = 86
51(2.37) - 8*i = 86
120.87 - 8*i = 86

Subtract 120.87 from both sides:
-8*i = -34.87

Divide by -8:
i = 34.87 / 8
i = 4.36 (approximately)

### Answer: i ≈ 4.36, k ≈ 2.37

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

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of l's in both equations are the same:

1) Multiply the first equation by 1 and the second equation by 17:

26m + 17l = 146
1581m + 17l = 2142

2) Subtract the first equation from the second equation to eliminate l:

(1581m - 26m) = (2142 - 146)
1555m = 1996

3) Solve for m:

m = 1996/1555
m = 1.28 (approximately)

4) Now, substitute the value of m into one of the original equations to solve for l. We will use the first equation:

26m + 17l = 146
26(1.28) + 17l = 146
33.28 + 17l = 146

5) Solve for l:

17l = 146 - 33.28
17l = 112.72
l = 112.72/17
l = 6.63 (approximately)

### Answer: l ≈ 6.63, m ≈ 1.28

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

Llama 3 70B:

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

First, we can multiply equation (1) by 29 and equation (2) by 8 to make the coefficients of f equal:

29(1): 232f + 1131h = 1914
8(2): 232f + 368h = 952

Now, subtract the two equations to eliminate f:

(1914 - 952) = (1131h - 368h)
962 = 763h

Now, divide by 763 to solve for h:

h = 962/763
h = 1.26 (approximately)

Now that we have the value of h, we can substitute it into one of the original equations to solve for f. We will use equation (1):

(8 * f) + (39 * h) = 66
(8 * f) + (39 * 1.26) = 66
(8 * f) + 49.14 = 66

Subtract 49.14 from both sides:

(8 * f) = 16.86

Divide by 8:

f = 16.86/8
f = 2.11 (approximately)

### Answer: h ≈ 1.26, f ≈ 2.11

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of b's in both equations are the same:

1) Multiply the first equation by 69:
1518b + 1597k = 7531

2) Multiply the second equation by 22:
1364k - 1518b = 308

Now, add both equations to eliminate b:

(1518b + 1597k) + (-1518b + 1364k) = 7531 + 308
1597k + (-1518b) + 1364k - 1518b = 7531 + 308
1597k + 1364k = 7531 + 308
2961k = 7839

Now, divide by 2961:
k = 7839 / 2961
k = 2.65

Now that we have the value of k, substitute it into one of the original equations to find the value of b. We will use the first equation:

22b + 23k = 109
22b + 23(2.65) = 109
22b + 61.15 = 109

Subtract 61.15 from both sides:
22b = 47.85

Divide by 22:
b = 47.85 / 22
b = 2.17

So, the values of k and b are:
k = 2.65
b = 2.17

### Answer: k = 2.65, b = 2.17

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.

Llama 3 70B:

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 can multiply equation (1) by 2 and equation (2) by 55 to make the coefficients of s in both equations the same:

1. 168m - 110s = 2
2. 3465m - 110s = 5390

Now, we can subtract equation (1) from equation (2) to eliminate the s variable:

(3465m - 168m) - (110s - 110s) = 5390 - 2
=> 3297m = 5388

Now, we can solve for m:

m = 5388 / 3297
=> m = 16/11

Now that we have the value of m, we can substitute it into one of the original equations to solve for s. We will use equation (1):

84m - 55s = 1
=> 84(16/11) - 55s = 1
=> 1344/11 - 55s = 1

Now, we can solve for s:

55s = 1344/11 - 1
=> 55s = (1344 - 11)/11
=> 55s = 1333/11
=> s = 1333/(11*55)
=> s = 1333/605
=> s = 22/11

So, the values of m and s are:

m = 16/11
s = 22/11

### Answer: m = 16/11, s = 22/11

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.

Llama 3 70B:

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

First, we can multiply equation (1) by 51 and equation (2) by 6 to make the coefficients of b in both equations the same:

1. 306b + 255k = 5355
2. 570k - 306b = 402

Now, we can add both equations to eliminate b:

(306b + 255k) + (-306b + 570k) = 5355 + 402
255k + 570k = 5757
825k = 5757

Now, we can solve for k:

k = 5757 / 825
k = 7

Now that we have the value of k, we can substitute it into one of the original equations to solve for b. We will use equation (1):

6b + 5k = 105
6b + 5(7) = 105
6b + 35 = 105

Subtract 35 from both sides:

6b = 70

Divide both sides by 6:

b = 70 / 6
b = 35/3

So, the values of b and k are:

b = 35/3
k = 7

### Answer: b = 35/3, k = 7

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.

Llama 3 70B:

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 can multiply the two equations by necessary multiples such that the coefficients of u's in both equations are the same:

1) Multiply the first equation by 32:
32 × (33d + 62u = 114) => 1056d + 1984u = 3648

2) Multiply the second equation by -62:
-62 × (32u + 40d = 83) => -1984u - 2480d = -5146

Now, add both equations to eliminate u:

(1056d + 1984u = 3648) + (-1984u - 2480d = -5146) => -1424d = -1498

Now, divide by -1424:
d = 1498 / 1424 => d = 1.05

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

33d + 62u = 114
33(1.05) + 62u = 114
34.65 + 62u = 114

Subtract 34.65 from both sides:
62u = 79.35

Divide by 62:
u = 79.35 / 62 => u = 1.28

So, the values of d and u are:
d = 1.05
u = 1.28

### Answer: d = 1.05, u = 1.28

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

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of p's in both equations are the same:

1) Multiply the first equation by 67 and the second equation by 28:

335c + 1868p = 6834 ... (1)
560c - 1876p = 364 ... (2)

2) Add equation (1) and equation (2) to eliminate p:

895c = 6470

3) Solve for c:

c = 6470 / 895
c = 7.23

4) Now, substitute the value of c into one of the original equations to solve for p. We will use the first equation:

5c + 28p = 102
5(7.23) + 28p = 102
36.15 + 28p = 102
28p = 102 - 36.15
28p = 65.85
p = 65.85 / 28
p = 2.35

Therefore, the values of c and p are:

c = 7.23
p = 2.35

### Answer: c = 7.23, p = 2.35

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.

Llama 3 70B:

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 can multiply the two equations by necessary multiples such that the coefficients of r's in both equations are the same:

1) Multiply the first equation by 76:
(58 * 76) * r + (12 * 76) * q = 147 * 76
4416r + 912q = 11172

2) Multiply the second equation by 58:
(97 * 58) * q - (76 * 58) * r = 76 * 58
5626q - 4416r = 4396

Now, add both equations to eliminate the r variable:

(5626q + 912q) = (11172 + 4396)
6538q = 15568

Now, divide by 6538 to solve for q:
q = 15568 / 6538
q = 2.38 (approximately)

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

58r + 12q = 147
58r + 12(2.38) = 147
58r + 28.56 = 147

Subtract 28.56 from both sides:
58r = 118.44

Divide by 58 to solve for r:
r = 118.44 / 58
r = 2.04 (approximately)

### Answer: r ≈ 2.04, q ≈ 2.38

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.

Llama 3 70B:

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 can multiply the two equations by necessary multiples such that the coefficients of k's in both equations are the same:

1) Multiply the first equation by 2:
94 * v - 6 * k = 20

2) Multiply the second equation by 3:
219 * v + 6 * k = 384

Now, add both equations to eliminate the k variable:

(94 * v - 6 * k) + (219 * v + 6 * k) = 20 + 384
313 * v = 404

Now, solve for v:
v = 404 / 313
v = 1.29 (approximately)

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

47 * v - 3 * k = 10
47 * 1.29 - 3 * k = 10
60.23 - 3 * k = 10

Now, solve for k:
3 * k = 60.23 - 10
3 * k = 50.23
k = 50.23 / 3
k = 16.74 (approximately)

### Answer: v ≈ 1.29, k ≈ 16.74

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?

Llama 3 70B:

Let's write the system of equations in a more formal way:

89p - 41q = 60 ... (Equation 1)
27p + 91q = 124 ... (Equation 2)

To solve for p and q, we can use the method of substitution or elimination. Here, we'll use the elimination method.

First, let's multiply Equation 1 by 27 and Equation 2 by 89 to make the coefficients of p in both equations the same:

2393p - 1107q = 1620 ... (Equation 1 multiplied by 27)
2433p + 8099q = 11036 ... (Equation 2 multiplied by 89)

Now, add both equations to eliminate p:

-1107q + 8099q = 11036 - 1620
6992q = 9416

Now, divide by 6992 to solve for q:

q = 9416 / 6992
q = 13/9

Now that we have the value of q, substitute it into one of the original equations to solve for p. We'll use Equation 1:

89p - 41q = 60
89p - 41(13/9) = 60
89p - 533/9 = 60

Multiply both sides by 9 to eliminate the fraction:

801p - 533 = 540
801p = 1073
p = 1073 / 801
p = 67/49

So, the values of p and q are:

p = 67/49
q = 13/9

### Answer: p = 67/49, q = 13/9

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

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of d's in both equations are the same:

1) Multiply the first equation by 3:
(264 * u) - (288 * d) = 36

2) Multiply the second equation by 8:
(216 * u) + (288 * d) = 568

Now, add the two equations to eliminate the d variable:

(480 * u) = 604

Now, divide by 480 to solve for u:
u = 604/480
u = 1.208333...

Since u is not an integer, we can round it to 1.21 (approximately).

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

(27 * u) + (36 * d) = 71

Substitute u = 1.21:
(27 * 1.21) + (36 * d) = 71
32.67 + (36 * d) = 71

Subtract 32.67 from both sides:
(36 * d) = 38.33

Divide by 36 to solve for d:
d = 38.33/36
d = 1.065...

Since d is not an integer, we can round it to 1.07 (approximately).

### Answer: u ≈ 1.21, d ≈ 1.07

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.

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of f's in both equations are the same:

1) Multiply the first equation by 2:
40n - 36f = 18

2) Multiply the second equation by 18:
54n - 36f = 1260

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

(54n - 40n) - (36f - 36f) = 1260 - 18
14n = 1242

Now, divide by 14 to solve for n:
n = 1242/14
n = 89

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

3n - 2f = 70
3(89) - 2f = 70
267 - 2f = 70

Subtract 267 from both sides:
-2f = -197

Divide by -2 to solve for f:
f = 197/2
f = 98.5

### Answer: f = 98.5, n = 89

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.

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of j's in both equations are the same:

1) Multiply the first equation by 22:
154 * l + 2134 * j = 2926

2) Multiply the second equation by -97:
-2134 * j - 3599 * l = -7412

Now, add both equations to eliminate the j variable:

(-3599 + 154) * l = -7412 + 2926
-3445 * l = -5486

Now, divide by -3445 to solve for l:
l = 5486 / 3445
l = 1.59 (approximately)

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

7 * l + 97 * j = 133
7 * 1.59 + 97 * j = 133

Simplify the equation:
11.13 + 97 * j = 133

Subtract 11.13 from both sides:
97 * j = 121.87

Divide by 97:
j = 121.87 / 97
j = 1.26 (approximately)

So, the values of j and l are approximately 1.26 and 1.59, respectively.

### Answer: j ≈ 1.26, l ≈ 1.59

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of f's in both equations are the same:

1) Multiply the first equation by 84:
1764d - 3864f = 4452

2) Multiply the second equation by 46:
368d + 3864f = 6138

Now, add both equations to eliminate f:

(1764d - 3864f) + (368d + 3864f) = 4452 + 6138
1764d + 368d = 4452 + 6138
2132d = 10590

Now, divide by 2132 to solve for d:
d = 10590 / 2132
d = 5

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

21d - 46f = 53
21(5) - 46f = 53
105 - 46f = 53

Subtract 105 from both sides:
-46f = -52

Divide by -46 to solve for f:
f = 52 / 46
f = 1

### Answer: d = 5, f = 1

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of w's in both equations are the same:

1) Multiply the first equation by 68:
4992u + 748w = 7872

2) Multiply the second equation by 11:
726u - 748w = 231

Now, add the two equations to eliminate the w variable:

(4992u + 726u) + (748w - 748w) = 7872 + 231
5228u = 8103

Now, solve for u:
u = 8103 / 5228
u = 1.55

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

74u + 11w = 116
74(1.55) + 11w = 116
113.7 + 11w = 116

Subtract 113.7 from both sides:
11w = 2.3

Divide both sides by 11:
w = 2.3 / 11
w = 0.21

Therefore, the values of u and w that satisfy the system of equations are u = 1.55 and w = 0.21.

### Answer: u = 1.55, w = 0.21

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of j's in both equations are the same:

1) Multiply the first equation by 47:
3592f - 1974j = 4512

2) Multiply the second equation by 42:
1974j + 1428f = 5712

Now, add the two equations to eliminate the j variable:

(3592f - 1974j) + (1974j + 1428f) = 4512 + 5712
3592f + 1428f = 10224
5020f = 10224

Now, divide by 5020 to solve for f:
f = 10224 / 5020
f = 2.04

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

76f - 42j = 96
76(2.04) - 42j = 96
154.24 - 42j = 96

Subtract 154.24 from both sides:
-42j = -58.24

Divide by -42 to solve for j:
j = 58.24 / 42
j = 1.39

So, the values of f and j that satisfy the system of equations are f = 2.04 and j = 1.39.

### Answer: f = 2.04, j = 1.39

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of u's in both equations are the same:

1) Multiply the first equation by 69:
4722u + 2208e = 12006

2) Multiply the second equation by 68:
4284e - 4692u = 68

Now, add both equations to eliminate u:

4722u + 2208e = 12006
-4692u + 4284e = 68
----------------------
652e = 12074

Now, solve for e:
e = 12074 / 652
e = 18.5

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

68u + 32e = 174
68u + 32(18.5) = 174
68u + 588 = 174
68u = -414
u = -414 / 68
u = -6.09

### Answer: u = -6.09, e = 18.5

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.

Llama 3 70B:

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 can multiply the two equations by necessary multiples such that the coefficients of n's in both equations are the same:

1) Multiply the first equation by 39:
2233p + 156n = 5011

2) Multiply the second equation by 4:
96p + 156n = 616

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

2233p - 96p = 5011 - 616
2137p = 4395

Now, divide by 2137 to solve for p:
p = 4395/2137
p = 2.05 (approximately)

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

57p + 4n = 129
57(2.05) + 4n = 129
116.85 + 4n = 129

Subtract 116.85 from both sides:
4n = 12.15

Divide by 4 to solve for n:
n = 12.15/4
n = 3.0375 (approximately)

### Answer: p ≈ 2.05, n ≈ 3.04

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same:

1) Multiply the first equation by 15 and the second equation by 11:

990z - 660y = 480 ... (1)
660y + 154z = 1276 ... (2)

2) Add equation (1) and equation (2) to eliminate the y variable:

990z - 660y + 660y + 154z = 480 + 1276
1144z = 1756

3) Solve for z:

z = 1756/1144
z = 53/34

4) Now, substitute the value of z into one of the original equations to solve for y. We will use the first equation:

66z - 44y = 32
66(53/34) - 44y = 32

5) Simplify and solve for y:

102/1 - 44y = 32
102 - 44y = 32
-44y = -70
y = 70/44
y = 35/22

Therefore, the values of y and z are:

y = 35/22
z = 53/34

### Answer: y = 35/22, z = 53/34.

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.

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of h's in both equations are the same:

1) Multiply the first equation by 69:
69(83t - 58h) = 69(45)
=> 5737t - 4002h = 3105

2) Multiply the second equation by 58:
58(69h + 11t) = 58(124)
=> 4002h + 638t = 7192

Now, add both equations to eliminate the h variable:
(5737t - 4002h) + (4002h + 638t) = 3105 + 7192
=> 6375t = 10297

Now, solve for t:
t = 10297 / 6375
=> t = 1.61 (approximately)

Now that we have the value of t, substitute it into one of the original equations to solve for h. We will use the second equation:
69h + 11t = 124
=> 69h + 11(1.61) = 124
=> 69h + 17.71 = 124
=> 69h = 106.29
=> h = 106.29 / 69
=> h = 1.54 (approximately)

### Answer: t ≈ 1.61, h ≈ 1.54

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?

Llama 3 70B:

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

First, we can notice that both equations have the same constant term, 9. This suggests that we can set up an equivalence between the two equations.

((79 * h) - (61 * o)) = ((16 * h) - (12 * o))

Now, let's equate the coefficients of h and o:

79h - 61o = 16h - 12o

Rearrange the terms to get:

63h = 49o

Now, divide both sides by 49:

h = (49/63)o
h = (7/9)o

Now, substitute this expression for h into one of the original equations. Let's use the second equation:

((16 * h) - (12 * o)) = 9

Substitute h = (7/9)o:

((16 * (7/9)o) - (12 * o)) = 9

Simplify the equation:

(112/9)o - 12o = 9

Combine like terms:

(-4/9)o = 9

Multiply both sides by -9/4:

o = -81/4
o = -20.25

Now that we have the value of o, we can find the value of h:

h = (7/9)o
h = (7/9)(-20.25)
h = -15.75

So, the values of o and h are o = -20.25 and h = -15.75.

### Answer: o = -20.25, h = -15.75

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

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of s's in both equations are the same:

1) Multiply the first equation by 90:
360y + 5040s = 12510

2) Multiply the second equation by 56:
3640y - 5040s = 896

Now, add both equations to eliminate the s variable:

(360y + 5040s) + (3640y - 5040s) = 12510 + 896
360y + 3640y = 13406
4040y = 13406

Now, solve for y:
y = 13406 / 4040
y = 3.32

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

4y + 56s = 139
4(3.32) + 56s = 139
13.28 + 56s = 139
56s = 139 - 13.28
56s = 125.72
s = 125.72 / 56
s = 2.24

### Answer: s = 2.24, y = 3.32

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.

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of z's in both equations are the same:

1) Multiply the first equation by 45:
1890d - 180z = 3375

2) Multiply the second equation by 4:
160d + 180z = 648

Now, add both equations to eliminate z:

(1890d - 180z) + (160d + 180z) = 3375 + 648
1890d + 160d = 4023
2050d = 4023

Now, divide by 2050:
d = 4023/2050
d = 1.96 (approximately)

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

42d - 4z = 75
42(1.96) - 4z = 75
82.32 - 4z = 75

Subtract 82.32 from both sides:
-4z = -7.32
Divide by -4:
z = 7.32/4
z = 1.83 (approximately)

### Answer: d ≈ 1.96, z ≈ 1.83

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of 'o' in both equations are the same:

1) Multiply the first equation by 1 and the second equation by -7:

(42 * o) + (24 * a) = 142
-(294 * a) + (-42 * o) = -525

2) Add both equations to eliminate 'o':

-(270 * a) = -383

3) Solve for 'a':

a = 383/270
a = 151/90

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

(42 * o) + (24 * 151/90) = 142

Simplify the equation:

(42 * o) + 404/3 = 142

Subtract 404/3 from both sides:

(42 * o) = 142 - 404/3
(42 * o) = (426 - 404)/3
(42 * o) = 22/3

Multiply both sides by 3/42:

o = 22/42
o = 11/21

So, the values of 'a' and 'o' are:

a = 151/90
o = 11/21

### Answer: a = 151/90, o = 11/21

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of p's in both equations are the same:

1) Multiply the first equation by 8:
312*r - 16*p = 360

2) Multiply the second equation by 1:
16*p - 84*r = 36

Now, add both equations to eliminate p:

(312*r - 16*p) + (16*p - 84*r) = 360 + 36
312*r - 16*p + 16*p - 84*r = 396
228*r = 396

Now, divide by 228 to solve for r:
r = 396/228
r = 1.75

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

39*r - 2*p = 45
39(1.75) - 2*p = 45
68.25 - 2*p = 45

Subtract 68.25 from both sides:
-2*p = -23.25

Divide by -2 to solve for p:
p = 23.25/2
p = 11.625

### Answer: p = 11.625, r = 1.75

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

Llama 3 70B:

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

First, we can multiply equation (1) by 64 and equation (2) by 23 to make the coefficients of 'w' in both equations the same:

1. (23 * 64 * d) - (52 * 64 * w) = 7 * 64
   1472d - 3328w = 448

2. (64 * 23 * w) - (8 * 23 * d) = 58 * 23
   1472w - 184d = 1334

Now, we can add both equations to eliminate 'w':

(1472d - 3328w) + (1472w - 184d) = 448 + 1334
1472d - 184d = 1782
1288d = 1782

Now, we can solve for 'd':

d = 1782 / 1288
d = 1.38 (approximately)

Now that we have the value of 'd', we can substitute it into one of the original equations to solve for 'w'. We will use equation (1):

(23 * d) - (52 * w) = 7
(23 * 1.38) - (52 * w) = 7
31.74 - 52w = 7

Now, we can solve for 'w':

52w = 31.74 - 7
52w = 24.74
w = 24.74 / 52
w = 0.475 (approximately)

### Answer: d ≈ 1.38, w ≈ 0.475

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.

Llama 3 70B:

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

First, we can multiply both equations by necessary multiples such that the coefficients of n's in both equations are the same:

1) Multiply the first equation by 47 and the second equation by 60:

(71 * 47 * z) + (60 * 47 * n) = 184 * 47
(52 * 60 * z) - (47 * 60 * n) = 19 * 60

This gives us:

3327z + 2820n = 8638
3120z - 2820n = 1140

2) Now, add both equations to eliminate the n variable:

(3327 + 3120)z + (2820 - 2820)n = 8638 + 1140
6447z = 9778

3) Now, solve for z:

z = 9778 / 6447
z = 1.516 (approximately)

4) Now that we have the value of z, substitute it into one of the original equations to solve for n. We will use the first equation:

(71 * z) + (60 * n) = 184
(71 * 1.516) + (60 * n) = 184

5) Simplify and solve for n:

102.076 + 60n = 184
60n = 184 - 102.076
60n = 81.924
n = 81.924 / 60
n = 1.365 (approximately)

### Answer: n ≈ 1.365, z ≈ 1.516

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

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of e's in both equations are the same:

1) Multiply the first equation by 99:
6930e + 8910z = 18651

2) Multiply the second equation by 70:
6930z - 5880e = 1400

Now, add the two equations to eliminate e:

(6930e + 8910z) + (-6930z + 5880e) = 18651 + 1400
0z + 2790e = 20051

Now, divide by 2790:
e = 20051/2790
e = 7.19 (approximately)

Now that we have the value of e, substitute it into one of the original equations to find the value of z. We will use the first equation:

70e + 90z = 189
70(7.19) + 90z = 189
502.3 + 90z = 189

Subtract 502.3 from both sides:
90z = -313.3

Divide by 90:
z = -313.3/90
z = -3.48 (approximately)

### Answer: e ≈ 7.19, z ≈ -3.48

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.

Llama 3 70B:

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

First, we can multiply equation (1) by 11 and equation (2) by 44 to make the coefficients of f in both equations the same:

1) 286o - 484f = 1045
2) 3828f + 484o = 7464

Now, we can add both equations to eliminate the f variable:

(286o - 484f) + (3828f + 484o) = 1045 + 7464
286o + 484o = 8509
770o = 8509

Now, we can solve for o:

o = 8509 / 770
o = 11

Now that we have the value of o, we can substitute it into one of the original equations to solve for f. We will use equation (1):

26o - 44f = 95
26(11) - 44f = 95
286 - 44f = 95
-44f = -191
f = 191 / 44
f = 4.32 (approximately)

So, the values of o and f that satisfy the system of equations are o = 11 and f ≈ 4.32.

### Answer: o = 11, f ≈ 4.32

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of p's in both equations are the same:

1) Multiply the first equation by 29:
40z - 20p = 51
=> 1160z - 580p = 1479

2) Multiply the second equation by 10:
6z + 58p = 165
=> 60z + 580p = 1650

Now, add both equations to eliminate p:
(1160z - 580p) + (60z + 580p) = 1479 + 1650
=> 1220z = 3129

Now, solve for z:
z = 3129 / 1220
=> z = 2.57 (approximately)

Now that we have the value of z, substitute it into one of the original equations to solve for p. We will use the second equation:
6z + 58p = 165
=> 6(2.57) + 58p = 165
=> 15.42 + 58p = 165

Now, solve for p:
58p = 165 - 15.42
=> 58p = 149.58
=> p = 149.58 / 58
=> p = 2.58 (approximately)

### Answer: z ≈ 2.57, p ≈ 2.58

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.

Llama 3 70B:

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

First, we can multiply the two equations by necessary multiples such that the coefficients of f's in both equations are the same:

1) Multiply the first equation by 3:
(213 * s) - (267 * f) = 15

2) Multiply the second equation by 89:
(267 * f) + (890 * s) = 8307

Now, add both equations to eliminate f:

(1103 * s) = 8322

Now, divide by 1103 to solve for s:
s = 8322 / 1103
s = 7.54 (approximately)

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

(3 * f) + (10 * s) = 93
(3 * f) + (10 * 7.54) = 93
(3 * f) + 75.4 = 93
(3 * f) = 17.6
f = 17.6 / 3
f = 5.87 (approximately)

### Answer: s ≈ 7.54, f ≈ 5.87

Expected answer: {f: 6553/1103, s: 8292/1103}

Correct? No