LLemma 7B on 8.EE.C.8
Common Core Standard 8.EE.C.8
This standard covers the following:
Analyze and solve pairs of simultaneous linear equations.
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
LLemma 7B:
Reasoning: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the first equation by 12: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the second equation by 91: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the first equation by 30: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the second equation by 40: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the first equation by 12: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the second equation by 91: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the first equation by 30: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the second equation by 40: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the first equation by 12: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the second equation by 91: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the first equation by 30: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the second equation by 40: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the first equation by 12: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the second equation by 91: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the first equation by 30: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the second equation by 40: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 Multiply both sides of the first equation by 12: 1. (40 * s)
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.
LLemma 7B:
Reasoning: 4 * b + 44 * m = 180 4 * b + 44 * m = 180 4 * b = 180 - 44 * m b = (180 - 44 * m)/4 79 * m - 6 * b = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 = 48 79 * m - 6 * (180 - 44 * m)/4 =
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.
LLemma 7B:
Reasoning: 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 * o - 15 * g = 28 47 * o + 29 * g = 141 42 *
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.
LLemma 7B:
Reasoning: The first equation is: ((12 * v) - (6 * o)) = 21 The second equation is: ((76 * v) - (45 * o)) = 50. ### Answer: v = 1, o = 1
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.
LLemma 7B:
Reasoning: 75 * f + 25 * g = 138 and 2 * f + 90 * g = 143. 75 * f + 25 * g = 138 2 * f + 90 * g = 143 ### Answer: f = 1, g = 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.
LLemma 7B:
Reasoning: 194 = 4 * y + 14 * n 194 - 4 * y = 14 * n <<194-4*y=14*n>> 7 = 65 * n - 79 * y <<7=65*n-79*y>> ### Answer: y = <<y=1>>1, n = <<n=1>>1
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.
LLemma 7B:
Reasoning: The first equation is 146 = 28 * g + 26 * l. The second equation is 6 = 41 * l - 53 * g. ### Answer: l = 1, g = 1
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.
LLemma 7B:
Reasoning: 1) 20 * p + 56 * y = 90 2) 89 * p + 7 * y = 98 Multiply the first equation by 7 and the second equation by 56. 7 * (20 * p + 56 * y = 90) 56 * (89 * p + 7 * y = 98) 7 * (20 * p) + 7 * (56 * y) = 7 * (90) 56 * (89 * p) + 56 * (7 * y) = 56 * (98) 140 * p + 392 * y = 630 4752 * p + 372 * y = 5328 Subtract the second equation from the first equation. 140 * p + 392 * y = 630 - (4752 * p + 372 * y = 5328) -------------------------------- -4612 * p - 330 * y = -798 Divide both sides by -330. -4612 * p - 330 * y = -798 -330 * y = -798 ------------------ -4612 * p = 238 Divide both sides by -4612. -4612 * p = 238 -4612 * p = 238 ------------------ p = -0.052 Substitute p = -0.052 into the first equation. 1) 20 * p + 56 * y = 90 2) 89 * p + 7 * y = 98 1) 20 * (-0.052) + 56 * y = 90 2) 89 * (-0.052) + 7 * y = 98 -1.04 + 56 * y = 90 -0.478 + 7 * y = 98 56 * y = 89.96 7 * y = 98.03 Divide both sides by 56. 56 * y = 89.96 56 * y = 89.96 ------------------ y = 1.62 Substitute y = 1.62 into the first equation. 1) 20 * p + 56 * y = 90 2) 89 * p + 7 * y = 98 1) 20 * p + 56 * (1.62) = 90 2) 89 * p + 7 * (1.62) = 98 20 * p + 90.56 = 90 89 * p + 11.44 = 98 20 * p = -9.56 89 * p = 9.56 Divide both sides by 20. 20 * p = -9.56 20 * p = -9.56 ------------------ p = -0.478 Substitute p = -0.478 into the first equation. 1) 20 * p + 56 * y = 90 2) 89 * p + 7 * y = 98 1) 20 * (-0.478) + 56 * y = 90 2) 89 * (-0.478) + 7 * y = 98 -9.56 + 56 * y = 90 -42.02 + 7 * y = 98 56 * y = 99.56 7 * y = 56.02 Divide both sides by 56. 56 * y = 99.56 56 * y = 99.56 ------------------ y = 1.81
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.
LLemma 7B:
Reasoning: 72*k - 22*d = 17 and 6*d +69*k = 96 72*k - 22*d = 17 6*d +69*k = 96 ### Answer: k = 1, d = 2
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
LLemma 7B:
Reasoning: 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73 * h - 33 * d = 48 10 * d + 14 * h = 91 73
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
LLemma 7B:
Reasoning: 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 Multiply equation 1 by 43 and equation 2 by 44: 43 * 43 * q - 44 * 43 * e = 43 * 29 30 * 43 * q + 43 * 44 * e = 130 * 44 Simplify: 1849 * q - 1844 * e = 1213 30 * 43 * q + 43 * 44 * e = 5520 Multiply equation 1 by 44 and equation 2 by 43: 1849 * 44 * q - 1844 * 44 * e = 1213 * 44 30 * 43 * 43 * q + 43 * 44 * 43 * e = 5520 * 43 Simplify: 81116 * q - 81152 * e = 52124 30 * 43 * 43 * q + 43 * 44 * 43 * e = 239120 Multiply equation 1 by 43 and equation 2 by 44: 81116 * 43 * q - 81152 * 43 * e = 52124 * 43 30 * 43 * 43 * 43 * q + 43 * 44 * 43 * 43 * e = 239120 * 44 Simplify: 3501152 * q - 3501986 * e = 2218452 30 * 43 * 43 * 43 * q + 43 * 44 * 43 * 43 * e = 10368800 Multiply equation 1 by 44 and equation 2 by 43: 3501152 * 44 * q - 3501986 * 44 * e = 2218452 * 44 30 * 43 * 43 * 43 * 43 * q + 43 * 44 * 43 * 43 * 43 * e = 10368800 * 43 Simplify: 15005664 * q - 15023876 * e = 9323728 30 * 43 * 43 * 43 * 43 * q + 43 * 44 * 43 * 43 * 43 * e = 43144000 Multiply equation 1 by 43 and equation 2 by 44: 15005664 * 43 * q - 15023876 * 43 * e = 9323728 * 43 30 * 43 * 43 * 43 * 43 * 43 * q + 43 * 44 * 43 * 43 * 43 * 43 * e = 43144000 * 44 Simplify: 675309952 * q - 676274888 * e = 40391244 30 * 43 * 43 * 43 * 43 * 43 * q + 43 * 44 * 43 * 43 * 43 * 43 * e = 1843200000 Multiply equation 1 by 44 and equation 2 by 43: 675309952 *
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))
LLemma 7B:
Reasoning: 53 = ((89 * r) - (74 * a)) 109 = ((37 * a) + (6 * r)) 53 = 89 * r - 74 * a 109 = 37 * a + 6 * r 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 * a 53 + 74 * a = 89 * r 109 - 6 * r = 37 *
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.
LLemma 7B:
Reasoning: 1. g + q = 169 2. q - 50g = 19 Multiply the first equation by 50 and the second equation by 1: 50g + 50q = 8450 q - 50g = 19 Subtract the second equation from the first equation: 51q = 8261 Divide both sides by 51: q = 162 Substitute q = 162 into the first equation: g + 162 = 169 Subtract 162 from both sides: g = 7 ### Answer: g = 7, q = 162
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
LLemma 7B:
Reasoning: 37 * s - 11 * v = 54 37 * s - 11 * v = 54 82 * s - 62 * v = 41 ### Answer: s = 1, v = 2
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.
LLemma 7B:
Reasoning: The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48. The first equation is
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?
LLemma 7B:
Reasoning: 60 * c - 91 * i = 7 7 * i + 76 * c = 148 7 * i + 76 * c = 148 - 60 * c - 91 * i = 7 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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
LLemma 7B:
Reasoning: 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 Multiply equation 1 by 41 and equation 2 by 37: 41*71*v - 41*37*e = 41*69 37*41*e + 37*76*v = 37*184 Subtract the two equations: (41*71*v - 41*37*e) - (37*41*e + 37*76*v) = 41*69 - 37*184 Simplify: (41*71*v - 41*37*e) - (37*41*e + 37*76*v) = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*184 41*71*v - 41*37*e - 37*41*e - 37*76*v = 41*69 - 37*1
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.
LLemma 7B:
Reasoning: 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113 26d+42v=85 25v+69d=113
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
LLemma 7B:
Reasoning: 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 * z - 91 * g = 33 72 * g - 3 * z = 68 43 *
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.
LLemma 7B:
Reasoning: 50c + 12f = 92 50c = 92 - 12f c = (92 - 12f)/50 36c - 21f = 34 36c = 34 + 21f c = (34 + 21f)/36 c = (92 - 12f)/50 = (34 + 21f)/36 36(92 - 12f) = 50(34 + 21f) 3296 - 368f = 1700 + 1050f 1700 + 1050f = 3296 - 368f 1700 + 1050f + 368f = 3296 1050f + 368f = 3296 - 1700 1418f = 1596 f = 1596/1418 f = 1.08 c = (92 - 12f)/50 c = (92 - 12(1.08))/50 c = (92 - 12.96)/50 c = 79.04/50 c = 1.58 ### Answer: 1.58, 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
LLemma 7B:
Reasoning: We can solve this system of equations using substitution. Let's start by solving the first equation for u: 87u - 75d = 7 87u = 75d + 7 u = (75d + 7)/87 Now, let's substitute (75d + 7)/87 for u in the second equation: 85d - 82u = 54 85d - 82((75d + 7)/87) = 54 85d - (82(75d + 7))/87 = 54 85d - (82(75d))/87 - (82(7))/87 = 54 85d - (6500d)/87 - (574)/87 = 54 (85(87)d - 6500d - 574)/87 = 54 (7495d - 6500d - 574)/87 = 54 (795d - 574)/87 = 54 795d = 54(87) + 574 795d = 4908 + 574 795d = 5482 d = 5482/795 d = 7 Now, let's substitute 7 for d in the first equation: 87u - 75d = 7 87u - 75(7) = 7 87u - 525 = 7 87u = 525 + 7 87u = 532 u = 532/87 u = 6 ### Answer: u = 6, d = 7
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?
LLemma 7B:
Reasoning: (75 * a) - (4 * i) = 42 (75 * a) - (4 * i) = 42 (10 * i) + (13 * a) = 160 ### Answer: a = 16, i = 10
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
LLemma 7B:
Reasoning: 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 * p + 74 * c = 183 27 * c - 18 * p = 13 37 *
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.
LLemma 7B:
Reasoning: (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105 (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) =
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.
LLemma 7B:
Reasoning: 75v - 69f = 14 75v = 14 + 69f v = (14 + 69f)/75 46f + 53v = 110 46f + 53((14 + 69f)/75) = 110 46f + 14 + 69f = 110 115f = 110 - 14 115f = 96 f = 96/115 = <<96/115=0.836>>0.836 75v - 69(0.836) = 14 75v - 57.91 = 14 75v = 14 + 57.91 75v = 71.91 v = 71.91/75 = <<71.91/75=0.955>>0.955 ### Answer: v = 0.955, f = 0.836
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.
LLemma 7B:
Reasoning: 7 * o + 13 * g = 131 89 * g - 70 * o = 37 ### Answer: o = 1, g = 2
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.
LLemma 7B:
Reasoning: 88 * d - 60 * k = 38 33 * k - 11 * d = 46 ### Answer: d = 1, k = 2
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.
LLemma 7B:
Reasoning: 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 * z + 72 * y = 139 2 * z + 82 * y = 96 24 *
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.
LLemma 7B:
Reasoning: (87 * o) - (29 * v) = 1 and (8 * v) + (30 * o) = 91. (87 * o) - (29 * v) = 1 (8 * v) + (30 * o) = 91 ### Answer: v = 1, o = 3
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.
LLemma 7B:
Reasoning: The first equation is (23 * h) - (46 * p) = 58. The second equation is (11 * p) + (23 * h) = 124. (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124
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.
LLemma 7B:
Reasoning: 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90d - 42f = 69. 1) 19d + 6f = 120, 2) 90
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'?
LLemma 7B:
Reasoning: 1) 42 * m + 55 * r = 127 2) 66 * m - 52 * r = 44 Multiply equation 1 by 52 and equation 2 by 55 to get: 52 * (42 * m + 55 * r = 127) 55 * (66 * m - 52 * r = 44) 52 * (42 * m) + 52 * (55 * r) = 52 * (127) 55 * (66 * m) - 55 * (52 * r) = 55 * (44) 2184 * m + 2840 * r = 6628 3330 * m - 2840 * r = 2220 Add equation 1 to equation 2 to get: 2184 * m + 2840 * r = 6628 3330 * m - 2840 * r = 2220 -------------------------------- 5510 * m = 8848 Divide both sides by 5510 to get: m = 1.6129032258064516 Substitute m = 1.6129032258064516 into equation 1 to get: 42 * (1.6129032258064516) + 55 * r = 127 Multiply both sides by 100 to get: 42 * (1.6129032258064516 * 100) + 55 * r = 127 * 100 42 * (161.29032258064516) + 55 * r = 12700 42 * (161.29032258064516) + 55 * r = 12700 7095.86 + 55 * r = 12700 Subtract 7095.86 from both sides to get: 7095.86 + 55 * r - 7095.86 = 12700 - 7095.86 55 * r = 5604.14 Divide both sides by 55 to get: r = 103.618243248046875 Substitute r = 103.618243248046875 into equation 1 to get: 42 * m + 55 * (103.618243248046875) = 127 Multiply both sides by 100 to get: 42 * m + 55 * (103.618243248046875 * 100) = 127 * 100 42 * m + 55 * (10361.8243248046875) = 12700 42 * m + 55 * (10361.8243248046875) = 12700 42 * m + 55670.91 = 12700 Subtract 55670.91 from both sides to get: 42 * m + 55670.91 - 55670.91 = 12700 - 55670.91 42 *
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
LLemma 7B:
Reasoning: 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w = 120 57w + 71n = 173 31n + 64w =
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.
LLemma 7B:
Reasoning: 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 * a + 26 * u = 112 76 * u - 77 * a = 31 72 *
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.
LLemma 7B:
Reasoning: 54n + 18b = 143 87n - 93b = 48 ### Answer: b = 1, n = 2
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)?
LLemma 7B:
Reasoning: 53 * e + 7 * p = 103 69 * e - 88 * p = 6 Multiply the first equation by 88 and the second equation by 7. 4824 * e + 608 * p = 8348 513 * e - 666 * p = 42 Add the two equations together. 513 * e + 4824 * e = 8348 + 42 513 * e + 4824 * e = 8390 5937 * e = 8390 e = 8390/5937 = <<8390/5937=1.41>>1.41 Substitute e = 1.41 into the first equation. 53 * 1.41 + 7 * p = 103 77.73 + 7 * p = 103 7 * p = 103 - 77.73 7 * p = 25.27 p = 25.27/7 = <<25.27/7=3.63>>3.63 ### Answer: e = 1.41, p = 3.63
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.
LLemma 7B:
Reasoning: First equation: ((18 * p) - (7 * s)) = 21 Second equation: ((51 * s) - (24 * p)) = 14 First equation: ((18 * p) - (7 * s)) = 21 Second equation: ((51 * s) - (24 * p)) = 14 ### Answer: p = <<p=3>>3, s = <<s=2>>2
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.
LLemma 7B:
Reasoning: 59 times c minus 34 times w is equal to 84. 11 times c plus 42 times w is equal to 98. ### Answer: c = 11, w = 2
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.
LLemma 7B:
Reasoning: First equation is 45 * f + 42 * z = 101, Second equation is 60 * z - 42 * f = 13. Multiply the first equation by 42 and the second equation by 45. 45 * 42 * f + 42 * 42 * z = 45 * 101, 60 * 42 * z - 42 * 45 * f = 60 * 13. Simplify the equations. 1890 * f + 1764 * z = 4545, 2520 * z - 1890 * f = 780. Add the first equation to the second equation. 1890 * f + 1764 * z = 4545, 2520 * z - 1890 * f = 780. 1890 * f + 1764 * z + 2520 * z - 1890 * f = 4545 + 780. Simplify the equation. 1764 * z + 2520 * z = 5325. Simplify the equation. 4284 * z = 5325. Divide both sides by 4284. z = 5325 / 4284 = <<5325/4284=1.231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231231
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.
LLemma 7B:
Reasoning: 26 * p - 21 * u = 6 26 * p = 6 + 21 * u p = (6 + 21 * u)/26 15 * p + 58 * u = 83 15 * p = 83 - 58 * u p = (83 - 58 * u)/15 ### Answer: p = (6 + 21 * u)/26, u = (83 - 58 * p)/15
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
LLemma 7B:
Reasoning: 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 Multiply equation 1 by 38 and equation 2 by 92: 38 * (92 * n - 38 * s = 62) 92 * (59 * n + 8 * s = 75) 38 * (92 * n - 38 * s = 62) = 38 * 62 = <<38*62=2376>>2376 92 * (59 * n + 8 * s = 75) = 92 * 75 = <<92*75=6900>>6900 38 * (92 * n - 38 * s = 62) = 38 * 62 = 2376 92 * (59 * n + 8 * s = 75) = 92 * 75 = 6900 Subtract equation 1 from equation 2: 92 * (59 * n + 8 * s = 75) - 38 * (92 * n - 38 * s = 62) = 6900 - 2376 = <<6900-2376=4524>>4524 92 * (59 * n + 8 * s = 75) - 38 * (92 * n - 38 * s = 62) = 4524 Divide both sides by 110: (92 * (59 * n + 8 * s = 75) - 38 * (92 * n - 38 * s = 62)) / 110 = 4524 / 110 = <<4524/110=40.23>>40.23 (92 * (59 * n + 8 * s = 75) - 38 * (92 * n - 38 * s = 62)) / 110 = 40.23 92 * (59 * n + 8 * s = 75) - 38 * (92 * n - 38 * s = 62) = 40.23 * 110 = <<40.23*110=4430.6>>4430.6 92 * (59 * n + 8 * s = 75) - 38 * (92 * n - 38 * s = 62) = 4430.6 92 * (59 * n + 8 * s = 75) - 4430.6 = 38 * (92 * n - 38 * s = 62) 92 * (59 * n + 8 * s = 75) - 4430.6 = 38 * 62 = 2376 92 * (59 * n + 8 * s = 75) - 4430.6 = 2376 Add 4430.6 to both sides: 92 * (59 * n + 8 * s = 75) - 4430.6 + 4430.6 = 2376 + 4430.6 = <<2376+4430.6=6806.6>>6806.6 92 * (59 * n + 8 * s = 75) = 6806.6 Divide both sides by 92: (92 * (59 * n + 8 * s = 75)) / 92 = 6806.6 / 92 = <<6806.6/92=74.25>>74.25 (92 * (59 * n + 8 * s =
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.
LLemma 7B:
Reasoning: 87 * z - 35 * e = 63 and 81 * z + 58 * e = 154. 87 * z - 35 * e = 63 81 * z + 58 * e = 154 ### Answer: e = 1, z = 2
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.
LLemma 7B:
Reasoning: (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18. (11 * t) + (44 * h) = 110
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.
LLemma 7B:
Reasoning: 69*b-50*o=55 41*o+18*b=85 ### Answer: b=<<b=1.5>>1.5, o=<<o=1.5>>1.5
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.
LLemma 7B:
Reasoning: 67 * q - 80 * p = 65, and 6 * q + 5 * p = 177. ### Answer: p = <<p=10>>10, q = <<q=11>>11
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.
LLemma 7B:
Reasoning: 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k + 48 * f = 161 33 * k - 41 * f = 31. 10 * k +
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
LLemma 7B:
Reasoning: The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l The first equation is 31
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
LLemma 7B:
Reasoning: 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25y + 37j = 116 29y - 28j = 39 25
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.
LLemma 7B:
Reasoning: The first equation is 93u - 85a = 74. The second equation is 96u + 17a = 186. Subtracting the second equation from the first equation gives us 93u - 96u - 85a - 17a = 74 - 186. Simplifying gives us -3u - 102a = -112. Dividing both sides by -3 gives us u + 34a = 37. Subtracting 34a from both sides gives us u = 37 - 34a. Substituting 37 - 34a for u in the first equation gives us 93(37 - 34a) - 85a = 74. Simplifying gives us 3351 - 3102a - 85a = 74. Simplifying further gives us 3351 - 3952a = 74. Subtracting 3351 from both sides gives us -3952a = 74 - 3351. Simplifying gives us -3952a = -2677. Dividing both sides by -3952 gives us a = -2677/-3952. Simplifying gives us a = 7/12. Substituting 7/12 for a in u = 37 - 34a gives us u = 37 - 34(7/12). Simplifying gives us u = 37 - 24. Simplifying further gives us u = 13. ### Answer: u = 13, a = 7/12
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
LLemma 7B:
Reasoning: 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u - 61 * g = 15 97 * u -
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.
LLemma 7B:
Reasoning: The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187. The first equation is (17 * r) + (67 * b) = 103.
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.
LLemma 7B:
Reasoning: 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. Multiply the first equation by 30 and the second equation by 54 to get: 1. (2550 * n) - (1620 * t) = 1230 2. (1908 * t) + (1560 * n) = 6808. Add the two equations to get: (2550 * n) - (1620 * t) + (1908 * t) + (1560 * n) = 1230 + 6808 (2550 * n) + (3388 * t) = 8038 Divide both sides by 3388 to get: (2550 * n)/3388 + (3388 * t)/3388 = 8038/3388 (2550 * n)/3388 + (t) = 2.45 Subtract (t) from both sides to get: (2550 * n)/3388 = 2.45 - (t) Multiply both sides by 3388 to get: (2550 * n) = (2.45 - (t)) * 3388 Divide both sides by 2550 to get: n = (2.45 - (t)) * 3388/2550 Multiply both sides by 2550 to get: 2550 * n = (2.45 - (t)) * 3388 Divide both sides by 3388 to get: (2550 * n)/3388 = (2.45 - (t)) Subtract (2.45 - (t)) from both sides to get: (2550 * n)/3388 - (2.45 - (t)) = 0 Multiply both sides by 3388 to get: (2550 * n) - (3388 * (2.45 - (t))) = 0 * 3388 Subtract 2550 * n from both sides to get: - (3388 * (2.45 - (t))) = - (2550 * n) Divide both sides by -3388 to get: (2.45 - (t)) = - (2550 * n)/3388 Subtract 2.45 from both sides to get: -(t) = - (2550 * n)/3388 - 2.45 Multiply both sides by -1 to get: (t) = (2550 * n)/3388 + 2.45 Subtract 2.45 from both sides to get: (t) - 2.45 = (2550 * n)/3388 Multiply both sides by 3388 to get: (3388 * (t) - 2.45) = (2550 * n) Divide both sides by 2550 to get: (3388 * (t) - 2.45)/2550 = n Multiply both sides by 2550 to get: (3388 * (t) - 2.45) = 2550 * n Divide both sides by 3388 to get: ((3388 * (t) - 2.45)/3388) = (2550 * n)/3388 Subtract 2550 * n/3388 from both sides to get: (3388 * (t) - 2.45)/
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.
LLemma 7B:
Reasoning: 1) 75u - 69q = 5 2) 48u + 25q = 140 1) 75u - 69q = 5 75u = 5 + 69q u = <<5+69q/75>> 2) 48u + 25q = 140 25q = 140 - 48u q = <<140-48u/25>> ### Answer: u = <<5+69q/75>>, q = <<140-48u/25>>
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.
LLemma 7B:
Reasoning: 20 * v + 17 * x = 149 and 24 * x - 49 * v = 22 20 * v + 17 * x = 149 24 * x - 49 * v = 22 ### Answer: v = 1, x = 2
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.
LLemma 7B:
Reasoning: 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t + 41 * g = 147 94 * t - 91 * g = 42. 39 * t +
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.
LLemma 7B:
Reasoning: 6c + 55o = 112, and 55c - 48o = 36. 6c + 55o = 112 55c - 48o = 36 ### Answer: c = 2, o = 5
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.
LLemma 7B:
Reasoning: 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u = 8. 8w + 68u = 170, 71w - 94u =
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
LLemma 7B:
Reasoning: 68*r + 8*f = 92 97*f + 38*r = 182 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 97*f + 38*r = 182 68*r + 8*f = 92 9
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).
LLemma 7B:
Reasoning: 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). ### Answer: d = $0.50, r =$1.50
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.
LLemma 7B:
Reasoning: 163 = 50x + a 24 = 98x - 84a ### Answer: x = 1.5, a = 1.5
Expected answer: {a: 7387/2149, x: 6858/2149}
Correct? No
Problem: Solve the following system of equations for the values of p and b: 67 * p + 18 * b = 110 82 * b - 44 * p = 58.
LLemma 7B:
Reasoning: 67 * p + 18 * b = 110 82 * b - 44 * p = 58 ### Answer: p = 1, b = 2
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.
LLemma 7B:
Reasoning: 63c-36e=27 and 8c+32e=69 63c-36e=27 8c+32e=69 63c-36e=27 -32e=-36e+27 -32e=-11e+27 -11e=-11e-8e+27 -8e=27 e=27/-8 e=-3.375 8c+32e=69 8c+32(-3.375)=69 8c-111.25=69 8c=180.25 c=180.25/8 c=22.5 c=22.5 e=-3.375
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
LLemma 7B:
Reasoning: 26 * x + 29 * p = 107 26 * x + 29 * p = 107 61 * x - 51 * p = 4 ### Answer: x = 1, p = 1
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.
LLemma 7B:
Reasoning: The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17. The first equation is: ((29 * l) + (1
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.
LLemma 7B:
Reasoning: 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*a - 54*c = 20 24*a - 26*c = 29 33*
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.
LLemma 7B:
Reasoning: 71d + 9j = 132 71d + 9j - 9j = 132 - 9j 71d = 132 - 9j 71d/71 = (132 - 9j)/71 d = (132 - 9j)/71 43j - 84d = 91 43j - 84d - 43j = 91 - 43j -84d = 91 - 43j -84d/(-84) = (91 - 43j)/(-84) d = (91 - 43j)/(-84) d = (132 - 9j)/71 and d = (91 - 43j)/(-84) (132 - 9j)/71 = (91 - 43j)/(-84) -9j/71 + 43j/(-84) = 91/71 - 132/71 (387 + 321j)/(-604) = 24/71 -604*24/71 = 387 + 321j -15584/71 = 387 + 321j -22355/71 = 321j -321j = -22355/71 j = (-22355/71)/(-321) j = 22355/22673 d = (132 - 9j)/71 d = (132 - 9*22355/22673)/71 d = (132 - 20055/22673)/71 d = (26464/22673 - 20055/22673)/71 d = 6319/158198 ### Answer: d = 6319/158198 and j = 22355/22673
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.
LLemma 7B:
Reasoning: (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143. (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) =
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.
LLemma 7B:
Reasoning: 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 46 * s - 61 * w = 2 61 * s + 67 * w = 178 4
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.
LLemma 7B:
Reasoning: 51*k - 8*i = 86 51*k = 86 + 8*i k = (86 + 8*i)/51 39*i + 8*k = 193 39*i + 8*((86 + 8*i)/51) = 193 39*i + 68 + 64*i = 193 103*i = 125 i = 125/103 i = <<125/103=1.2115>>1.2115 39*i + 8*k = 193 39*1.2115 + 8*k = 193 45.7935 + 8*k = 193 8*k = 193 - 45.7935 8*k = 147.2065 k = 147.2065/8 k = <<147.2065/8=18.03>>18.03 ### Answer: i = 1.2115, k = 18.03
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
LLemma 7B:
Reasoning: 26m + 17l = 146 93m + l = 126 26m + 17l = 146 l = 126 - 93m 26m + 17(126 - 93m) = 146 26m + 2078 - 1511m = 146 -1511m + 26m = 146 - 2078 -1455m = -1932 m = -1932/-1455 m = 1.33 93m + l = 126 93(1.33) + l = 126 126.99 + l = 126 l = 126 - 126.99 l = -0.99 ### Answer: l = -0.99, m = 1.33
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
LLemma 7B:
Reasoning: 1) (8 * f) + (39 * h) = 66 2) (29 * f) + (46 * h) = 119 Solving the first equation for f: f = (66 - 39 * h) / 8 Substituting this into the second equation: (29 * (66 - 39 * h) / 8) + (46 * h) = 119 Simplifying: (29 * 66 - 29 * 39 * h) / 8 + (46 * h) = 119 Multiplying both sides by 8: 29 * 66 - 29 * 39 * h + (46 * h) * 8 = 119 * 8 Simplifying: 29 * 66 - 29 * 39 * h + 372 * h = 952 Subtracting 29 * 66 from both sides: - 29 * 39 * h + 372 * h = 952 - 29 * 66 Simplifying: - 1041 * h = 952 - 29 * 66 Dividing both sides by -1041: h = (-952 + 29 * 66) / (-1041) Simplifying: h = 29 / 1041 Substituting this into the first equation: (8 * (29 / 1041)) + (39 * h) = 66 Simplifying: (8 * (29 / 1041)) + (39 * (29 / 1041)) = 66 Multiplying both sides by 1041: (8 * (29 / 1041)) * 1041 + (39 * (29 / 1041)) * 1041 = 66 * 1041 Simplifying: 29 + 1112 = 66 * 1041 Subtracting 29 from both sides: 1112 = 66 * 1041 - 29 Simplifying: 1112 = 66 * 1041 - 29 Dividing both sides by 66: f = (1112 - 29) / 66 Simplifying: f = 1083 / 66 f = 16.5 ### Answer: h = 29/1041, f = 16.5
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.
LLemma 7B:
Reasoning: 22 * b + 23 * k = 109 and 62 * k - 69 * b = 14. 22 * b + 23 * k = 109 62 * k - 69 * b = 14 ### Answer: k = 1, b = 1
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.
LLemma 7B:
Reasoning: 1. 84m - 55s = 1 2. 63m - 2s = 98 Multiply the first equation by 2 and the second equation by 55. 1. 168m - 110s = 2 2. 315m - 110s = 490 Subtract the second equation from the first equation. 1. 168m - 315m = 2 - 490 2. -147s = -488 Divide the second equation by -147. 1. 168m - 315m = 2 - 490 2. s = 3.3 Substitute 3.3 for s in the first equation. 1. 168m - 315m = 2 - 490 2. s = 3.3 1. 168m - 315m = 2 - 490 2. s = 3.3 315m = 488 m = 1.5 Substitute 1.5 for m in the second equation. 1. 168m - 315m = 2 - 490 2. s = 3.3 1. 168m - 315m = 2 - 490 2. s = 3.3 s = 3.3 ### Answer: m = 1.5, s = 3.3
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.
LLemma 7B:
Reasoning: 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. Multiply the first equation by 51 and the second equation by 6. 51 * (6 * b) + 51 * (5 * k) = 51 * 105 6 * (95 * k) - 6 * (51 * b) = 6 * 67. Simplify. 306 * b + 255 * k = 51 * 105 570 * k - 306 * b = 6 * 67. Multiply the first equation by 306 and the second equation by 255. 306 * (306 * b) + 306 * (255 * k) = 306 * 51 * 105 255 * (570 * k) - 255 * (306 * b) = 255 * 6 * 67. Simplify. 306^2 * b + 306 * 255 * k = 306 * 51 * 105 255 * 570 * k - 255 * 306 * b = 255 * 6 * 67. Subtract the second equation from the first equation. 306^2 * b + 306 * 255 * k - (255 * 570 * k - 255 * 306 * b) = 306 * 51 * 105 - (255 * 570 * k - 255 * 306 * b) Simplify. 306^2 * b + 306 * 255 * k - 255 * 570 * k + 255 * 306 * b = 306 * 51 * 105 - 255 * 570 * k + 255 * 306 * b Simplify. 306^2 * b + 255 * 306 * b - 255 * 570 * k + 306 * 255 * k = 306 * 51 * 105 - 255 * 570 * k + 255 * 306 * b Simplify. 306^2 * b + 255 * 306 * b - 255 * 570 * k + 306 * 255 * k = 306 * 51 * 105 - 255 * 570 * k + 255 * 306 * b Simplify. 306^2 * b + 255 * 306 * b - 255 * 570 * k + 306 * 255 * k = 306 * 51 * 105 - 255 * 570 * k + 255 * 306 * b Simplify. 306^2 * b + 255 * 306 * b - 255 * 570 * k + 306 * 255 * k = 306 * 51 * 105 - 255 * 570 * k + 255 * 306 * b Simplify. 306^2 * b + 255 * 306 * b - 255 * 570 * k + 306 * 255 * k = 306 * 51 * 105 - 255 * 570 * k + 255 * 306 * b Simplify. 306^2 * b + 25
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.
LLemma 7B:
Reasoning: 33d + 62u = 114 and 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114 32u + 40d = 83 33d + 62u = 114
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
LLemma 7B:
Reasoning: 5 * c + 28 * p = 102 20 * c - 67 * p = 13 ### Answer: c = 1, p = 1
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.
LLemma 7B:
Reasoning: 58 * r + 12 * q = 147 97 * q - 76 * r = 76 ### Answer: r = 1, q = 2
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.
LLemma 7B:
Reasoning: 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k = 10 73 * v + 2 * k = 128 47 * v - 3 * k
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?
LLemma 7B:
Reasoning: The result of 89 multiplied by p minus 41 multiplied by q equals 60. 89p-41q=60 The result of 27 multiplied by p plus 91 multiplied by q equals 124. 27p+91q=124 ### Answer: p=2, q=1
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
LLemma 7B:
Reasoning: The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71 The first equation is: (88 * u) - (
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.
LLemma 7B:
Reasoning: 20 * n - 18 * f = 9 20 * n - 18 * f = 9 3 * n - 2 * f = 70 ### Answer: f = 11, n = 14
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.
LLemma 7B:
Reasoning: 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l + 97 * j = 133 22 * j + 37 * l = 76 7 * l
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.
LLemma 7B:
Reasoning: 21d - 46f = 53 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f = 53 8d + 84f = 133 21d - 46f =
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.
LLemma 7B:
Reasoning: 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w = 116 66u - 68w = 21. 74u + 11w =
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.
LLemma 7B:
Reasoning: 76 * f - 42 * j = 96 47 * j + 34 * f = 136. 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76 * f - 42 * j = 96 47 * j + 34 * f = 136 76
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.
LLemma 7B:
Reasoning: 68u + 32e = 174 63e - 69u = 1 ### Answer: u = 1, e = 1
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.
LLemma 7B:
Reasoning: 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4n = 129 and 24p + 39n = 154. 57p + 4
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.
LLemma 7B:
Reasoning: 66 * z - 44 * y = 32 60 * y + 14 * z = 116. 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66 * z - 44 * y = 32 60 * y + 14 * z = 116 66
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.
LLemma 7B:
Reasoning: The first equation is 83 * t - 58 * h = 45. The second equation is 69 * h + 11 * t = 124. The first equation is 83 * t - 58 * h = 45. The second equation is 69 * h + 11 * t = 124. ### Answer: t = 11, h = 12
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?
LLemma 7B:
Reasoning: We can solve the system of equations by substituting one equation into the other. Let's solve the first equation for h: h = (61 * o) / 79. Now we can substitute this expression for h into the second equation: (16 * (61 * o) / 79) - (12 * o) = 9. Simplifying, we get: (972 * o) / 79 - (12 * o) = 9. Factoring out o, we get: o * (972 / 79 - 12) = 9. Dividing both sides by (972 / 79 - 12), we get: o = 9 / (972 / 79 - 12) = <<9 / (972 / 79 - 12) = 1.25>>1.25. Now we can substitute this value of o into the first equation to solve for h: h = (61 * 1.25) / 79 = <<(61 * 1.25) / 79 = 0.8>>0.8. ### Answer: o = 1.25, h = 0.8
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
LLemma 7B:
Reasoning: 4 * y + 56 * s = 139 56 * s = 139 - 4 * y s = (139 - 4 * y)/56 65 * y - 90 * s = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 16 65 * y - 90 * (139 - 4 * y)/56 = 1
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.
LLemma 7B:
Reasoning: 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 * d - 4 * z = 75 40 * d + 45 * z = 162 42 *
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.
LLemma 7B:
Reasoning: 42 * a + 6 * o = 75 42 * a + 6 * o = 142 ### Answer: a = 2, o = 3
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.
LLemma 7B:
Reasoning: 39*r - 2*p = 45 and 16*p - 84*r = 36 39*r - 2*p = 45 16*p - 84*r = 36 ### Answer: p = 10, r = 1/3
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
LLemma 7B:
Reasoning: 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 Multiply the first equation by 8 and the second equation by 52. 8 * (23 * d) - 8 * (52 * w) = 8 * 7 52 * (64 * w) - 52 * (8 * d) = 52 * 58 Simplify. 184d - 416w = 56 3328w - 416d = 2904 Add the two equations together. 184d - 416w = 56 3328w - 416d = 2904 ---------------------------------- 3512w = 3460 Divide both sides by 3512. w = 3460/3512 w = <<3460/3512=0.9816046511627906976744894092484>> Substitute w = 3460/3512 into the first equation. (23 * d) - (52 * w) = 7 (23 * d) - (52 * (3460/3512)) = 7 (23 * d) - (180720/3512) = 7 (23 * d) - 51.075 = 7 Multiply both sides by 10000. (23 * d) - 51075 = 70000 Add 51075 to both sides. (23 * d) = 12175 Divide both sides by 23. d = 12175/23 d = <<12175/23=525.21739130434782608695652173913>> ### Answer: d = 525.217, w = 0.981605
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.
LLemma 7B:
Reasoning: (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19. (71 * z) + (60 * n) = 184 (52 * z) - (47 * n)
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
LLemma 7B:
Reasoning: 99 * z - 84 * e = 20 99 * z = 20 + 84 * e z = (20 + 84 * e)/99 70 * e + 90 * z = 189 70 * e + 90 * (20 + 84 * e)/99 = 189 70 * e + 2000/3 + 840 * e/11 = 189 1540 * e + 2000/3 = 189 * 11 1540 * e = 189 * 11 - 2000/3 1540 * e = 189 * 11 - 666 1540 * e = 1123 e = 1123/1540 e = 1123/1540 * 1000/1000 e = 1123000/1540000 e = 1123/154 e = 7/11 z = (20 + 84 * e)/99 z = (20 + 84 * 7/11)/99 z = (20 + 588/11)/99 z = (20 * 11 + 588)/(11 * 99) z = 220/1089 z = 220/1089 * 10/10 z = 22/109 z = 2/109 ### Answer: e = 7/11, z = 2/109
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.
LLemma 7B:
Reasoning: 26o - 44f = 95 26o - 44f = 95 26o = 95 + 44f o = <<95+44f/26>> 87f + 11o = 169 87f + 11o = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 87f + 11(95+44f/26) = 169 8
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.
LLemma 7B:
Reasoning: 40*z - 20*p = 51 6*z + 58*p = 165. ### Answer: <<z=1.5>>1.5, p=1.5
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.
LLemma 7B:
Reasoning: (71 * s) - (89 * f) = 5 (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 * s) = 93 (3 * f) + (10 *
Expected answer: {f: 6553/1103, s: 8292/1103}
Correct? No