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

Gemma 2B:

 
Solving the equations, we get:

q = 11
s = 5

Therefore, the total number of clips sold in April and May is 40 * 5 = <<40 * 5 = 200>>200.

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.

Gemma 2B:

Reasoning: From the first equation, we get 4b = 180 - 44m. Substituting this expression for b into the second equation, we get 79m - 6(180 - 44m) = 48.
Simplifying the equation, we get 79m - 1080 + 192m = 48.
Combining like terms, we get 58m = 1128, so m = 19.6.
Substituting the value of m back into the equation we derived earlier, we get 4b = 180 - 44(19.6) = 180 - 836 = 544.
Therefore, b = 136.
### Answer: b = 136, m = 19.6

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.

Gemma 2B:

 
Reasoning:

From the first equation, we have 42o = 28 + 15g. 

Substituting this expression for o into the second equation, we get:

47(28) + 29g = 141

1216 + 29g = 141

29g = 141 - 1216

29g = -1075

g = -37.9

Substituting the value of g back into the first equation, we get:

42o = 28 + 15(-37.9)

42o = 28 - 555.5

42o = -527.5

o = -12.3

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.

Gemma 2B:

### Answer: v = 3, o = 5

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.

Gemma 2B:

Reasoning: From the first equation, we have 75f = 138 - 25g. Substituting this into the second equation, we get 2(138 - 25g) + 90g = 143. Simplifying this equation, we get 276 - 50g + 90g = 143. Combining like terms, we get -30g = -133, so g = <<133/30>> = 4.1.
Substituting the value of g back into the first equation, we get 75f = 138 - 25(4.1) = 138 - 100 = 38.
Therefore, f = <<38/75>> = 0.52.
### Answer: f = 0.52, g = 4.1

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.

Gemma 2B:

Reasoning: From the first equation, we get 4 * y = 194 - 14 * n. Substituting this expression for y into the second equation, we get 7 = 65 * n - 79 * (194 - 14 * n). Simplifying this equation, we get 7 = 65 * n - 1476 + 1156 * n. Combining like terms, we get 1156 * n - 65 * n = 194 + 7. This simplifies to 1091 * n = 201, so n = 2.
Substituting n = 2 back into the first equation, we get 194 = 4 * y + 14 * 2. Simplifying this equation, we get 4 * y = 194 - 28 = 166, so y = 41.
Therefore, the solution to the system of equations is n = 2 and y = 41.

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.

Gemma 2B:

Substituting the first equation into the second equation, we get:

28 * g + 26 * l = 146
41 * l - 53 * g = 6

Solving this system of linear equations, we get:

g = 4
l = 3

Therefore, the values of l and g that satisfy the given equations are l = 3 and g = 4.

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.

Gemma 2B:

Reasoning: We can use the substitution method to solve this system. First, we solve equation 1 for p:

20p = 90 - 56y
p = (90 - 56y) / 20

Substituting this expression for p into equation 2, we get:

89(90 - 56y) + 7y = 98
7260 - 4880y + 7y = 98
-4873y = -6282
y = 12.5

Substituting the value of y back into the equation we derived earlier, we get:

p = (90 - 56(12.5)) / 20 = 15

Therefore, the solution to the system of equations is p = 15 and y = 12.5.

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.

Gemma 2B:

Reasoning: From the first equation, we have 72k = 17 + 22d. Substituting this into the second equation, we get:
6(17 + 22d) + 69k = 96
102 + 132d + 69k = 96
132d - 69k = 96 - 102
d = (96 - 102)/132 = 0.75
Substituting the value of d into the first equation, we get:
72k = 17 + 22(0.75) = 17 + 15 = 32
Therefore, k = 0.4.
### Answer: k = 0.4, d = 0.75

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

Gemma 2B:

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

Gemma 2B:

Reasoning: From equation 1, we have 43q - 44e = 29. From equation 2, we have 30q + 43e = 130. Subtracting equation 1 from equation 2, we get (43q - 44e) - (30q + 43e) = 130 - 29 = 101. Solving for q, we get 43q - 30q = 101, or q = 101/13 = 7. Therefore, e = (130 - 43 * 7) / 30 = 7.
### Answer: q = 7, e = 7

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))

Gemma 2B:

Reasoning:

From the first equation, we get:

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

From the second equation, we get:

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

Substituting the first equation into the second equation, we get:

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

Solving this system of equations, we get:

r = 3
a = 4

Therefore, the solution is r = 3 and a = 4.

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.

Gemma 2B:

3. The weight of 42 units of 'g' is equal to 3 times the weight of 12 units of 'q'.

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

Gemma 2B:

Reasoning:

From the first equation, we have 37s = 11v + 54.

Substituting this expression for s into the second equation, we get:

82(11v + 54) - 62 * v = 41

822v + 4440 - 62v = 41

76v = -4320

v = -58

Substituting the value of v back into the first equation, we get:

37s = 11(-58) + 54

37s = -644 + 54

37s = -590

s = -16

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.

Gemma 2B:

### Answer: c = 5, z = 3

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?

Gemma 2B:

Solution:

From the first equation, we get:

c = (7/60) * 71 = 0.115

Substituting this value of c into the second equation, we get:

7 * i + 76 * 0.115 = 148

7 * i = 148 - 0.115 * 76

7 * i = 141.6

i = 20.14

Therefore, the values of c and i are:

c = 0.115

i = 20.14

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

Gemma 2B:

Reasoning:

From equation 1, we have 71v - 37e = 69.
From equation 2, we have 41e + 76v = 184.

Solving these equations simultaneously, we get:

71v - 37e = 69
41e + 76v = 184

Subtracting the first equation from the second equation, we get:

(71 - 41)v = 184 - 69

20v = 115

v = 5.75

Substituting the value of v back into either of the original equations, we get:

71v - 37e = 71(5.75) - 37e = 395.25 - 37e

Substituting the values of v and e into the first equation, we get:

71(5.75) - 37e = 395.25 - 37e

Solving for e, we get:

37e = 395.25 - 395.25

e = 0

Substituting the value of e back into either of the original equations, we get:

71v - 37e = 71(5.75) - 37(0) = 395.25 - 0

Therefore, the values of e and v that satisfy the given equations are e = 0 and v = 5.75.

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.

Gemma 2B:

### Answer: d=3, v=5

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

Gemma 2B:

Reasoning:

From the first equation, we have 43z - 91g = 33.
From the second equation, we have 72g - 3z = 68.
Subtracting the first equation from the second equation, we get:

(43z - 91g) - (72g - 3z) = 33 - 68
43z - 91g - 72g + 3z = 33 - 68
-58g = -35
g = 0.625
Substituting the value of g into either of the original equations, we get:

43z - 91(0.625) = 33
43z - 56.25 = 33
43z = 33 + 56.25
43z = 89.25
z = 2.03

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.

Gemma 2B:

Solving this system of equations, we get:

c = 12
f = 14

Therefore, the quantity of cherries is 12 and the quantity of figs is 14.

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

Gemma 2B:

 
Solving these equations simultaneously, we get:

u = 15 and d = 12

Therefore, Natalia sold 15 clips in April and 12 clips in May.

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?

Gemma 2B:

Reasoning: From the first equation, we get a = (4/75) * i + 42. Substituting this into the second equation, we get:

(10 * (4/75) * i + 13 * a) = 160

=> (40/75) * i + 13a = 160

=> 40i + 13a = 160

Solving this system of two linear equations, we get:

i = 6 and a = 4

### Answer: i = 6, a = 4

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

Gemma 2B:

Reasoning:

From the first equation, we have 37p = 183 - 74c.
Substituting this expression for p into the second equation, we get:

27c - 18(183 - 74c) = 13
27c - 324 + 132c = 13
105c = 357
c = 3.5
Substituting the value of c back into the first equation, we get:

37p = 183 - 74(3.5) = 183 - 260 = 57
Therefore, p = 1.5.
### Answer: p = 1.5, c = 3.5

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.

Gemma 2B:

Reasoning:
From the first equation, we have 48k - 30p = 37.
From the second equation, we have 16p + 33k = 105.
Substituting the first equation into the second equation, we get:
(48k - 30p) = 37
(16p + 33k) = 105
Solving for k and p, we get:
k = 4 and p = 5

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.

Gemma 2B:

Answer: v = 2 and f = 5

Reasoning: From equation 1, we have 75v = 14 + 69 = 83. Therefore, v = 1. From equation 2, we have 46(5) + 53v = 110, which simplifies to 230 + 53v = 110, or 53v = 60, or v = 1.

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.

Gemma 2B:

Reasoning: 
From the first equation, we have 7o = 131 - 13 * g. 
Substituting this expression for o into the second equation, we get 89g - 70(131 - 13g) = 37. 
Simplifying the equation, we get 89g - 9150 + 91g = 37. 
Combining like terms, we have 178g = 9167, so g = 53.

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.

Gemma 2B:

Reasoning:
From the first equation, we get:
88d = 60k + 38
From the second equation, we get:
33k = 11d - 46
Substitute the first equation into the second equation:
(88d) - (11d) = 46
7d = 46
d = 6
Substitute the value of d back into the first equation:
88 * 6 - 60 * k = 38
480 - 60k = 38
-60k = -442
k = 7.4
Therefore, the solution to the system of equations is d = 6 and k = 7.4.

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.

Gemma 2B:

Solving this system of equations, we get: z = 4, y = 5.
Therefore, Natalia sold 48 + 72 = 120 clips altogether in April and May.

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.

Gemma 2B:

Reasoning: From the first equation, we get v = (1/29)(87 * o). Substituting this into the second equation, we get:

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

=> (56 * o) + (30 * o) = 91

=> 86 * o = 91

=> o = 10.2

Substituting the value of o back into the equation v = (1/29)(87 * o), we get:

v = (1/29)(87 * 10.2) = 4.2

Therefore, the solution to the system of equations is v = 4.2 and o = 10.2.

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.

Gemma 2B:

 
Solving the system by substitution, we get:

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

From the first equation, we have 23h = 58 + 46p.

Substituting this expression for h into the second equation, we get:

11p + 23(58 + 46p) = 124
11p + 1260 + 1040p = 124
11p - 980 = 124
p = 12.

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.

Gemma 2B:

Reasoning: From equation 1, we have 19d = 120 - 6f. Substituting this into equation 2, we get:
90d - 42(120 - 6f) = 69
=> 90d - 480 + 24f = 69
=> 90d + 24f = 481

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'?

Gemma 2B:

 
**Solution:**

Solving the system of linear equations, we get:

42m + 55r = 127
66m - 52r = 44

Using the substitution method, we can find the value of m in terms of r from the first equation:

42m = 127 - 55r
m = (127 - 55r) / 42

Substituting this value of m into the second equation, we get:

66(127 - 55r) - 52r = 44
774 - 335r - 52r = 44
-80r = -730
r = 9.25

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

42m + 55r = 127
42m + 55(9.25) = 127
42m + 495.25 = 127
42m = 127 - 495.25
m = 3.03

Therefore, the solution to the system of linear equations is m = 3.03 and r = 9.25.

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

Gemma 2B:

 
Answer: n = 2, w = 3

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.

Gemma 2B:

Reasoning:

From equation 1, we have:

72a + 26u = 112

From equation 2, we have:

76u - 77a = 31

Substituting the second equation into the first equation, we get:

(76u - 77a) - (72a + 26u) = 31

Simplifying the equation, we get:

4u = 31

Therefore, u = 7.75.

Substituting the value of u back into the second equation, we get:

76(7.75) - 77a = 31

Simplifying the equation, we get:

575 - 77a = 31

Adding 77a to both sides, we get:

575 = 84a

Therefore, a = 6.62.

### Answer: u = 7.75, a = 6.62

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.

Gemma 2B:

Reasoning: From the first equation, we have 54n = 143 - 18b. Substituting this expression for n into the second equation, we get 87(143 - 18b) - 93b = 48. Simplifying this equation, we get 1181 - 147b - 93b = 48. Combining like terms, we get -54b = -1133, and dividing both sides by -54, we find b = 21. Substituting this value of b back into the equation we derived from the first equation, we get 54n = 143 - 18(21) = 143 - 36 = 107. Therefore, n = 19.
### Answer: b = 21, n = 19

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)?

Gemma 2B:

Solution:

Subtracting the second equation from the first equation, we get:

(53 * e) - (69 * e) = 103 - 6
-16 * e = 37
e = -37/16

Substituting the value of e into either of the original equations, we get:

53 * (-37/16) + 7 * p = 103
-405 + 7p = 103
7p = 103 + 405
7p = 508
p = 70

Therefore, the costs of electricity (e) and petroleum (p) are $ -37/16 and $70, respectively.

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.

Gemma 2B:

Answer: p = 3 and s = 4

### Answer: p = 3, s = 4

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.

Gemma 2B:

Reasoning: From the first equation, we get 59c - 34w = 84.
From the second equation, we get 11c + 42w = 98.
Substituting the second equation into the first equation, we get 59c - (11c + 42w) = 84.
Simplifying the equation, we get 48c = 84, so c = 1.8.
Substituting the value of c back into either of the original equations, we get 59(1.8) - 34w = 84, so w = 14.
Therefore, c = 1.8 and w = 14.

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.

Gemma 2B:

Solving the system, we get:

f = 2 and z = 3.

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.

Gemma 2B:

Reasoning:
From the first equation, we have 26p = 6 + 21u.
Substituting this expression for 26p into the second equation, we get:
15(6 + 21u) + 58u = 83
90 + 315u + 58u = 83
148u = 34
u = 2.34
Substituting the value of u back into the first equation, we get:
26p = 6 + 21(2.34) = 6 + 50.08 = 56.08
Therefore, p = 2.22.
### Answer: p = 2.22, u = 2.34

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

Gemma 2B:

Reasoning:

From equation 1, we have 92n - 38s = 62.
From equation 2, we have 59n + 8s = 75.

Subtracting equation 1 from equation 2, we get:

(92n - 38s) - (59n + 8s) = 75 - 62
13n = 13

Therefore, n = 1.

Substituting n = 1 into either of the original equations, we get:

92 * 1 - 38 * s = 62
-38s = 62 - 92
-38s = -30
s = 30/38
s = 0.78

Therefore, the values of n and s are n = 1 and s = 0.78.

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.

Gemma 2B:

Reasoning: From the first equation, we have 35 * e = 87 * z - 63. Substituting this into the second equation, we get 81 * z + 58 * (87 - z) = 154. Simplifying this equation, we get 81 * z + 4714 - 58z = 154. Combining like terms, we get -7z = -1270. Therefore, z = <<1270/7>> = 180. Substituting z = 180 into the first equation, we get 35 * e = 87 * 180 - 63 = 1405. Therefore, e = <<1405/35>> = 40.

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.

Gemma 2B:

Reasoning:

From the first equation, we have 11t = 110 - 44h.
Substituting this expression for t into the second equation, we get:

(98 * (110 - 44h)) - (36 * h) = 18
1100 - 440h - 36h = 18
-476h = -1022
h = 2.14

Substituting the value of h back into the first equation, we get:

11t = 110 - 44(2.14) = 110 - 95.76 = 14.24

Therefore, t = 1.43 and h = 2.14.

### Answer: t = 1.43, h = 2.14

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.

Gemma 2B:

Reasoning:
From the first equation, we get 69b - 50o = 55.
From the second equation, we get 41o + 18b = 85.
Substituting the first equation into the second equation, we get:
69b - 50o = 55
41o + 18b = 85
Subtracting the first equation from the second equation, we get:
(69b - 50o) - (41o + 18b) = 85 - 55
28b = 30
b = 1.125
Substituting the value of b back into either of the original equations, we get:
69 * 1.125 - 50 * o = 55
69.25 - 50o = 55
-50o = -14.75
o = 0.295
Therefore, b = 1.125 and o = 0.295.

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.

Gemma 2B:

Reasoning: From the first equation, we get 80p = 67q - 65. Substituting this into the second equation, we get 60 + 5p = 177, or 5p = 117, or p = 23.
Substituting the value of p into the first equation, we get 80q = 67(23) - 65 = 1545, or q = 19.
Therefore, the solution to the system of equations is p = 23 and q = 19.

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.

Gemma 2B:

Reasoning: From the first equation, we get 48f = 161 - 10k. Substituting this into the second equation, we get 33k - 41f = 31.
Solving this system of equations, we get k = 3 and f = 5.

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

Gemma 2B:

Substituting the first equation into the second equation, we get:

58 * l - n = 103
n = 58 * l - 103

Substituting the second equation into the first equation, we get:

31 = 58 * l - (58 * l - 103)
31 = 58 * l - 58 * l + 103
31 = 103
l = 3

Substituting the value of l back into the second equation, we get:

n = 58 * 3 - 103 = 162

Therefore, the solution to the system of equations is l = 3 and n = 162.

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

Gemma 2B:

Reasoning:

Add the two equations together:

(29y - 28j) + (25y + 37j) = 39 + 116 = 155

Equate the coefficients of y:

29 - 25 = 4

Equate the coefficients of j:

-28 = 37

Reasoning:

y = 4
j = -7

Therefore, the solution to the system of equations is y = 4 and j = -7.

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.

Gemma 2B:

### Answer: u = 5, a = 10

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

Gemma 2B:

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.

Gemma 2B:

 
**Solution:**

Subtracting the first equation from the second equation, we get:

(85 * r) - (17 * r) = 187 - 103

58 * r = 84

r = 1.5

Substituting the value of r back into the first equation, we get:

(17 * 1.5) + (67 * b) = 103

25 + 67b = 103

67b = 78

b = 1.2

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.

Gemma 2B:

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.

Gemma 2B:

Reasoning:

1) 75u - (69q) = 5

2) 48u + 25q = 140

Solving the system of equations, we get:

u = 0.6

q = 3.2

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

Correct? No

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

Gemma 2B:

Reasoning:

From the first equation, we have 20v + 17x = 149.
From the second equation, we have 24x - 49v = 22.

Substituting the first equation into the second equation, we get:

(20v + 17x) - (24x - 49v) = 22
20v + 17x - 24x + 49v = 22
29v - 3x = 22

Solving for v and x, we get:

v = 22/29 and x = 3

### Answer: v = 22/29, x = 3

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.

Gemma 2B:

Reasoning:

From the first equation, we have 39t = 147 - 41g.
Substituting this expression for t into the second equation, we get:

94 - 91g = 42.
Solving for g, we get g = (42 - 94)/(-91) = -2.

Substituting the value of g back into the first equation, we get:

39t = 147 - 41(-2) = 147 + 82 = 239.
Therefore, t = 6.

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.

Gemma 2B:

Reasoning: From the first equation, we get 55o = 112 - 6c. Substituting this into the second equation, we get 55c - 48(112 - 6c) = 36.
Simplifying the equation, we get 55c - 528 + 24c = 36.
Combining like terms, we get 79c = 564.
Dividing both sides by 79, we find c = 7.
Substituting c = 7 into the first equation, we get 55o = 112 - 6(7) = 112 - 42 = 66.
Therefore, o = 6.
### Answer: c = 7, o = 6

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.

Gemma 2B:

Reasoning:
From the first equation, we have 68u = 170 - 8w.
Substituting this expression for u into the second equation, we get:
71(170 - 8w) - 94u = 8.
Simplifying the equation, we get:
1260 - 560w - 94u = 8.
Adding 94u to both sides, we get:
1260 = 554u + 8.
Subtracting 8 from both sides, we get:
1252 = 554u.
Dividing both sides by 554, we get:
u = 22.
Substituting the value of u back into the equation we derived earlier, we get:
68u = 68(22) = 1488.
Therefore, w = 22.
### Answer: u = 22, w = 22

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

Gemma 2B:

Reasoning:

From the first equation, we have 68r = 92 - 8f.
Substituting this expression for r into the second equation, we get:

97f + 38(92 - 8f) = 182
97f + 348 - 32f = 182
5f = 66
f = 13.2
Substituting the value of f back into the equation we derived earlier, we get:

r = (92 - 8f) / 68 = (92 - 8(13.2)) / 68 = 1.6
Therefore, the solution to the system of equations is r = 1.6 and f = 13.2.

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).

Gemma 2B:

### Answer: d = 5, r = 10

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.

Gemma 2B:

Solving this system, we get x = 3 and a = 12.

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.

Gemma 2B:

Reasoning:
From the first equation, we have 67p = 110 - 18b.
Substituting this expression for p into the second equation, we get:
82b - 44(110 - 18b) = 58
82b - 440 + 56b = 58
128b = 58 + 440
128b = 498
b = 4.06
Substituting the value of b back into the first equation, we get:
67p = 110 - 18(4.06) = 110 - 72 = 38
Therefore, the values of p and b are p = 3 and b = 4.06.

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.

Gemma 2B:

Reasoning:

From the first equation, we have 63c - 36e = 27.
From the second equation, we have 8c + 32e = 69.

Subtracting the first equation from the second equation, we get:

(8c + 32e) - (63c - 36e) = 69 - 27
16e = 42
e = 2.6

Substituting the value of e into the first equation, we get:

63c - 36(2.6) = 27
63c - 93.6 = 27
63c = 27 + 93.6
63c = 120.6
c = 1.9

Therefore, the values of c and e are c = 1.9 and e = 2.6, respectively.

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

Gemma 2B:

 
Answer: x = 4, p = 3

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.

Gemma 2B:

### Answer: g = 2, l = 4

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.

Gemma 2B:

Reasoning:

From the first equation, we have 33a = 54c + 20.
Substituting this expression for a into the second equation, we get 24(54c + 20) - 26*c = 29.
Simplifying the equation, we get 1260c + 480 - 26c = 29.
Solving for c, we get 1234 = 234c.
Dividing both sides by 234, we find c = 5.
Substituting the value of c back into the first equation, we get 33a = 54(5) + 20 = 260.
Solving for a, we find a = 8.
Therefore, the values for a and c are a = 8 and c = 5.

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.

Gemma 2B:

Reasoning: We can use the substitution method to solve this system. First, we solve one equation for one variable in terms of the other. Let's solve the second equation for j:

43j = 91 + 84d
j = (91 + 84d) / 43

Now we can substitute this expression for j into the first equation:

71d + 9((91 + 84d) / 43) = 132
71d + (84d + 816) / 43 = 132
(71d + 84d) / 43 = 132 - 816 = -684
d = -8.5

Finally, we can substitute the value of d back into either of the original equations to solve for j. Let's use the second equation:

43j - 84d = 91
43j - 84(-8.5) = 91
43j + 720 = 91
43j = 91 - 720 = -629
j = -15.5

Therefore, the solution to the system of equations is d = -8.5 and j = -15.5.

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.

Gemma 2B:

Reasoning:

From the first equation, we get: 

26n = 163 - 81 = 82
Therefore, n = 3.2

Substituting n = 3.2 into the second equation, we get:

87(3.2) + 23(o) = 143
252 + 23o = 143
23o = 143 - 252 = -109
Therefore, o = 4.5

### Answer: o = 4.5, n = 3.2

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.

Gemma 2B:

Reasoning:

From the first equation, we have 46s = 2 + 61w.
Substituting this into the second equation, we get 61s + 67w = 178.
Solving for s in the first equation, we get s = (2 + 61w)/46.
Substituting this into the second equation, we get 61(2 + 61w)/46 + 67w = 178.
Simplifying the equation, we get 136 + 371w - 67w = 178.
Solving for w, we get w = 4.
Substituting w = 4 into the equation we derived earlier, we get s = 10.
Therefore, the solution to the system of equations is w = 4 and s = 10.

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.

Gemma 2B:

### Answer: i = 5, k = 7

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

Gemma 2B:

Reasoning:

From the first equation, we have 26m = 146 - 17l.
Substituting this expression for m into the second equation, we get:

93(146 - 17l) + l = 126
1288 - 146l + l = 126
-135l = -162
l = <<162/135>>
l = <<1.18>>

Substituting the value of l back into the equation we found earlier, we get:

26m = 146 - 17(1.18) = 146 - 19.56 = 126
m = <<126/26>>
m = <<5>>

Therefore, the solution to the system of equations is l = 1.18 and m = 5.

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

Gemma 2B:

 
**Solution:**

From equation 1:

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

Substitute this expression for (8 * f) into equation 2:

(29 * f) + (46 * h) = 119

Solving this system of equations, we get:

f = 3

h = 4

Therefore, the values of 'h' and 'f' are 4 and 3, respectively.

Expected answer: {f: 1605/763, h: 962/763}

Correct? No

Problem: Find the values of k and b that satisfy the following system of equations: 22 * b + 23 * k = 109 and 62 * k - 69 * b = 14.

Gemma 2B:

Reasoning: From the first equation, we get 22b + 23k = 109. From the second equation, we get 62k - 69b = 14. Subtracting the first equation from the second equation, we get (22b + 23k) - (62k - 69b) = 109 - 14 = 95. Simplifying the equation, we get 16b = 95, so b = 5. Substituting the value of b into the first equation, we get 22k = 109, so k = 5.
### Answer: k = 5, b = 5

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.

Gemma 2B:

Reasoning:

From equation 1, we have 84m = 55s + 1.
Substituting this expression for m in equation 2, we get:

63(55s + 1) - 2s = 98.
Simplifying the equation:

325s + 63 - 2s = 98.
Combining like terms:

323s = 80.
Dividing both sides by 323:

s = 25.
Substituting the value of s back into either equation 1 or 2, we find that m = 4.
Therefore, the solution to the system of linear equations is m = 4 and s = 25.

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.

Gemma 2B:

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.

Gemma 2B:

Reasoning:

Add the two equations together:

(33d + 62u) + (32u + 40d) = 114 + 83
65d = 197
d = 3.14

Substitute the value of d into one of the original equations to find u:

33(3.14) + 62u = 114
99.2 + 62u = 114
62u = 114 - 99.2 = 15.8
u = 0.26

Therefore, the solution to the system of equations is d = 3.14 and u = 0.26.

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

Gemma 2B:

### Answer: c = 6, p = 2

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.

Gemma 2B:

Reasoning:
From the first equation, we have 58r = 147 - 12q.
Substituting this expression for r into the second equation, we get:
97q - 76(147 - 12q) = 76.
Simplifying the equation, we get:
97q - 1008 + 92q = 76.
Combining like terms, we get:
17q = 1084.
Dividing both sides by 17, we find q = 64.
Substituting the value of q back into the equation we derived earlier, we get:
r = (147 - 12q) / 58 = (147 - 12(64)) / 58 = 10.
Therefore, r = 10 and q = 64.

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.

Gemma 2B:

Solving this system, we get:
v = 4 and k = 8.

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?

Gemma 2B:

Reasoning:
From the first equation, we get:
89p - 41q = 60.
From the second equation, we get:
27p + 91q = 124.
Subtracting the first equation from the second equation, we get:
(89p - 41q) - (27p + 91q) = 124 - 60.
Simplifying the equation, we get:
62p = 64.
Dividing both sides by 62, we get:
p = 1.06.
Substituting the value of p into either of the original equations, we get:
89(1.06) - 41q = 60.
Simplifying the equation, we get:
87.2 - 41q = 60.
Adding 41q to both sides, we get:
87.2 = 60 + 41q.
Solving for q, we get:
41q = 27.2.
Dividing both sides by 41, we get:
q = 0.66.
Therefore, the values of p and q are p = 1.06 and q = 0.66, respectively.

Expected answer: {p: 5272/4603, q: 4708/4603}

Correct? No

Problem: Find the values of u and d that make these equations true: The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71

Gemma 2B:

 
Solving the system of equations, we get:

(88 * u) - (96 * d) = 12
(27 * u) + (36 * d) = 71

Using the substitution method, we get:

(88 * u) - (96 * d) = 12
(27 * u) + (36 * d) = 71

Subtracting the first equation from the second equation, we get:

(88 * u) - (96 * d) - (27 * u) + (36 * d) = 71 - 12
52 * u = 59

Dividing both sides by 52, we get:

u = 1.125

Substituting the value of u back into one of the original equations, we get:

(88 * 1.125) - (96 * d) = 12
92 - 96d = 12

Adding 96d to both sides, we get:

96d + 92 = 12
96d = 12 - 92
96d = -70
d = -7

Therefore, u = 1.125 and d = -7.

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.

Gemma 2B:

Reasoning: From the first equation, we get 20n = 18f + 9. Substituting this into the second equation, we get 3n - 2f = 70. Solving this system of equations, we get f = 15 and n = 3.
### Answer: f = 15, n = 3

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.

Gemma 2B:

Reasoning:
From the first equation, we have 7l = 133 - 97j.
Substituting this expression for l into the second equation, we get:
22j + 37(133 - 97j) = 76
22j + 4595 - 347j = 76
112j = -4529
j = -41.25
Substituting the value of j back into the first equation, we get:
7l = 133 - 97(-41.25) = 133 + 3975 = 4208
l = 60.8
### Answer: j = -41.25, l = 60.8

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.

Gemma 2B:

Reasoning:

Add the two equations together:

(21d - 46f) + (8d + 84f) = 53 + 133
29d = 286
d = 9.8
Substitute the value of d into one of the original equations to find f:

21(9.8) - 46f = 53
196 - 46f = 53
-46f = -143
f = 3.1
Therefore, the values of d and f are d = 9.8 and f = 3.1.

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.

Gemma 2B:

Reasoning: We can solve this system by using substitution or elimination. Here's the elimination method:

Add the two equations together:

(74u + 11w) + (66u - 68w) = 116 + 21
140u = 137

Now solve for u by dividing both sides by 140:

u = <<137/140>>

Substitute the value of u back into one of the original equations to find w:

66(137/140) - 68w = 21
851 - 88w = 21
-88w = -830
w = <<-830/-88>>

Therefore, the values of u and w that satisfy the system of equations are u = 137/140 and w = -830/-88.

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.

Gemma 2B:

Reasoning: From the first equation, we get f = (96 + 42j)/76. Substituting this value of f into the second equation, we get:
47j + 34 * (96 + 42j)/76 = 136
Simplifying the equation, we get:
(47j + 34 * 96) - (136 * 76) = 0
47j + 3168 - 968 = 0
47j - 872 = 0
j = 18.
Substituting the value of j back into the equation we found earlier, we get:
f = (96 + 42(18))/76 = 48.
Therefore, the solution to the system of equations is f = 48 and j = 18.

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.

Gemma 2B:

Solving this system of equations, we get:
u = 4 and e = 3.

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.

Gemma 2B:

Reasoning: Subtracting the first equation from the second equation, we get:

(57p + 4n) - (24p + 39n) = 129 - 154

33p = 25

p = 8

Substituting the value of p into either of the original equations, we get:

57(8) + 4n = 129

480 + 4n = 129

4n = 70

n = 17.5

Therefore, the values of p and n that satisfy the system of linear equations are p = 8 and n = 17.5.

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.

Gemma 2B:

Reasoning:

From the first equation, we have 66z = 44y + 32.
Substituting this expression for z in the second equation, we get:

60y + 14(44y + 32) = 116
60y + 596y + 48 = 116
756y = 68
y = 0.092
Substituting the value of y back into the first equation, we get:

66z = 44(0.092) + 32
66z = 3.92 + 32
66z = 35.92
z = 0.56
Therefore, the values of y and z are y = 0.092 and z = 0.56, respectively.

Expected answer: {y: 901/572, z: 439/286}

Correct? No

Problem: Solve the following system of equations for the variables t and h: The first equation is 83 * t - 58 * h = 45. The second equation is 69 * h + 11 * t = 124.

Gemma 2B:

Reasoning: We can use substitution to solve this system of equations. First, we solve the second equation for h:

69 * h = 124 - 11 * t
h = (124 - 11 * t) / 69.

Then, we substitute this expression for h into the first equation:

83 * t - 58 * (124 - 11 * t) = 45
83 * t - 696 + 645 * t = 45
55 * t = 645
t = 12.16.

Finally, we substitute the value of t back into the equation we derived earlier to find h:

h = (124 - 11 * 12.16) / 69 = 5.56.

Therefore, the solution to the system of equations is t = 12.16 and h = 5.56.

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?

Gemma 2B:

Reasoning: We can solve this system by using substitution. First, we solve the second equation for h:
(16 * h) - (12 * o) = 9
=> 16 * h = 12 * o + 9
=> h = (12/16) * o + 9/16
=> h = 3/4 * o + 9/16

Now we substitute this expression for h into the first equation:
(79 * h) - (61 * o) = 9
=> 79 * (3/4 * o + 9/16) - 61 * o = 9
=> 231/4 * o + 655 - 61 * o = 9
=> 20/4 * o = 8
=> o = 4

Finally, we substitute the value of o back into either of the original equations to solve for h. Using the second equation:
(16 * h) - (12 * o) = 9
=> 16 * h - (12 * 4) = 9
=> 16 * h - 48 = 9
=> 16 * h = 49
=> h = 7.625

Therefore, the values of o and h are o = 4 and h = 7.625.

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

Gemma 2B:

 
Reasoning:

From the first equation, we have 4y = 139 - 56s.

Substituting this expression for y into the second equation, we get:

65 * (139 - 56s) - 90 * s = 16

Simplifying the equation:

8065 - 340s - 90s = 16

Combining like terms:

-120s = -792

s = 6.5

Substituting the value of s back into the equation we derived earlier, we get:

4y = 139 - 56(6.5) = 139 - 320 = 107

### Answer: s = 6.5, y = 26

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.

Gemma 2B:

Solution:

From the first equation, we have 42d - 4z = 75.
From the second equation, we have 40d + 45z = 162.

Adding the two equations together, we get:

(42d - 4z) + (40d + 45z) = 75 + 162
82d = 237
d = 28.5

Substituting the value of d back into one of the original equations, we get:

42 * 28.5 - 4z = 75
1120 - 4z = 75
-4z = -1045
z = 26.375

Therefore, the solution to the system of equations is d = 28.5 and z = 26.375.

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.

Gemma 2B:

Reasoning: From the first equation, we have 42o = 142 - 24a. Substituting this into the second equation, we get 42a + 6o = 75.
Solving this system of equations, we get a = 3 and o = 5.

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.

Gemma 2B:

Reasoning: From the first equation, we have 39r = 2p + 45. Substituting this into the second equation, we get 16p - 84r = 36. Solving this system of equations, we get p = 3 and r = 4.

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

Gemma 2B:

### Answer: d = 4, w = 3

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.

Gemma 2B:

Reasoning:

From the first equation, we have 71z = 184 - 60n.
Substituting this expression for z into the second equation, we get:

(52 * (184 - 60n)) - (47 * n) = 19
968 - 320n - 47n = 19
-367n = -959
n = 2.7

Substituting the value of n back into the first equation, we get:

71z = 184 - 60(2.7) = 184 - 162 = 22
### Answer: n = 2, z = 22

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

Gemma 2B:

### Answer: e = 0.2, z = 0.3

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.

Gemma 2B:

Reasoning:

From equation 1, we have 26o = 44f - 95.
Substituting this expression for o in equation 2, we get:

87(44f - 95) + 11o = 169
352f - 815 + 11o = 169
352f - 11o = 169 + 815
352f - 11o = 1084
Therefore, the values of o and f that satisfy the system of equations are o = 20 and f = 34.

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.

Gemma 2B:

Solving this system of equations, we get z = 3 and p = 5.

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.

Gemma 2B:

Reasoning:

From the first equation, we have 71s - 89f = 5.
From the second equation, we have 3f + 10s = 93.
Substitute the first equation into the second equation:

(71s - 89f) = 5
(3f + 10s) = 93
71s - 89f = 5
3f + 10s = 93
Add the two equations together:

(71s - 89f) + (3f + 10s) = 5 + 93
74s = 108
s = 1.5
Substitute the value of s back into one of the original equations to find f:

71 * 1.5 - 89f = 5
155 - 89f = 5
-89f = -140
f = 1.56
Therefore, the values of s and f are s = 1.5 and f = 1.56, respectively.

Expected answer: {f: 6553/1103, s: 8292/1103}

Correct? No