LLemma 34B 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 34B:
Reasoning: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121 1. (40 * s) - (91 * q) = 37 2. (30 * 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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for b: 4 * b + 44 * m = 180 4 * b = 180 - 44 * m b = (180 - 44 * m)/4 Now we can substitute this expression for b into the second equation: 79 * m - 6 * b = 48 79 * m - 6 * ((180 - 44 * m)/4) = 48 79 * m - (180 - 44 * m)/2 = 48 79 * m - 180/2 + 44 * m/2 = 48 79 * m + 44 * m/2 = 48 + 180/2 123 * m/2 = 48 + 180/2 123 * m = 48 * 2 + 180 123 * m = 96 + 180 123 * m = 276 m = 276/123 m = <<276/123=2>>2 Now we can substitute this value for m into the first equation to solve for b: 4 * b + 44 * m = 180 4 * b + 44 * 2 = 180 4 * b + 88 = 180 4 * b = 180 - 88 4 * b = 92 b = 92/4 b = <<92/4=23>>23 ### Answer: b = 23, m = 2
Expected answer: {b: 3027/145, m: 318/145}
Correct? No
Problem: Solve the following system of equations to find the values of o and g: 42 * o - 15 * g = 28 47 * o + 29 * g = 141.
LLemma 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for o: 42 * o - 15 * g = 28 42 * o = 28 + 15 * g o = (28 + 15 * g)/42 We can substitute this expression for o into the second equation: 47 * o + 29 * g = 141 47 * ((28 + 15 * g)/42) + 29 * g = 141 (47 * (28 + 15 * g))/42 + 29 * g = 141 (47 * 28 + 47 * 15 * g)/42 + 29 * g = 141 (47 * 28)/42 + (47 * 15 * g)/42 + 29 * g = 141 (47 * 28)/42 + (47 * 15 * g + 29 * 42)/42 = 141 (47 * 28)/42 + (47 * 15 * g + 29 * 42)/42 = 141 (47 * 28 + 47 * 15 * g + 29 * 42)/42 = 141 (47 * (28 + 15 * g) + 29 * 42)/42 = 141 (47 * ((28 + 15 * g)/42) + 29 * 42)/42 = 141 (47 * o + 29 * 42)/42 = 141 47 * o + 29 * 42 = 141 * 42 47 * o + 1238 = 5882 47 * o = 5882 - 1238 47 * o = 4644 o = 4644/47 o = <<4644/47=98.85185185185185>>98.85185185185185 We can now substitute this value for o into the first equation to solve for g: 42 * o - 15 * g = 28 42 * 98.85185185185185 - 15 * g = 28 4137.23723723724 - 15 * g = 28 -15 * g = 28 - 4137.23723723724 -15 * g = -4109.23723723724 g = -4109.23723723724/-15 g = <<-4109.23723723724/-15=-273.951851851852>>-273.951851851852 ### Answer: o = 98.85185185185185, g = -273.951851851852
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for v: v = (21 + (6 * o)) / 12 We can substitute this expression for v into the second equation: ((76 * ((21 + (6 * o)) / 12)) - (45 * o)) = 50 We can solve this equation for o: o = 1 We can substitute this value for o into the first equation: v = (21 + (6 * 1)) / 12 v = 2 ### Answer: v = 2, 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 34B:
Reasoning: 75 * f + 25 * g = 138 and 2 * f + 90 * g = 143. 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75 * f + 25 * g = 138 2 * f + 90 * g = 143 75
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 34B:
Reasoning: We can solve the first equation for y: y = (194 - 14 * n)/4 = <<(194-14*n)/4=48-3.5*n>>48-3.5*n. We can substitute this expression for y into the second equation: 7 = 65 * n - 79 * (48-3.5*n) = <<65*n-79*(48-3.5*n)=-316+114*n>>-316+114*n. We can solve this equation for n: n = (7+316)/114 = <<(7+316)/114=2.7>>2.7. We can substitute this value for n into the expression for y: y = 48-3.5*2.7 = <<48-3.5*2.7=19.5>>19.5. ### Answer: y = 19.5, n = 2.7
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 34B:
Reasoning: We can solve the first equation for g: 146 = 28 * g + 26 * l 146 - 26 * l = 28 * g (146 - 26 * l) / 28 = g (146 - 26 * l) / 28 = <<(146-26*l)/28>> We can solve the second equation for l: 6 = 41 * l - 53 * g 6 + 53 * g = 41 * l (6 + 53 * g) / 41 = l (6 + 53 * g) / 41 = <<(6+53*g)/41>> ### Answer: g = <<(146-26*l)/28>>, l = <<(6+53*g)/41>>
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 34B:
Reasoning: 1) 20 * p + 56 * y = 90 2) 89 * p + 7 * y = 98 2) - 1) => 69 * p + 56 * y = 8 1) - 2) => 191 * p + 56 * y = 18 1) - 2) => 191 * p + 56 * y = 18 2) - 1) => 69 * p + 56 * y = 8 191 * p + 56 * y = 18 69 * p + 56 * y = 8 191 * p + 56 * y = 18 -69 * p - 56 * y = -8 122 * p = 10 p = 10/122 = <<10/122=0.082>>0.082 1) 20 * p + 56 * y = 90 20 * 0.082 + 56 * y = 90 4.16 + 56 * y = 90 56 * y = 85.84 y = 85.84/56 = <<85.84/56=1.53>>1.53 p = 0.082 y = 1.53 ### Answer: p = 0.082, y = 1.53
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for k: 72*k - 22*d = 17 72*k = 17 + 22*d k = <<17+22*d/72>>17/72 + 22/72*d We can substitute this expression for k into the second equation: 6*d + 69*k = 96 6*d + 69*<<17/72 + 22/72*d>> = 96 6*d + 17/8 + 22/8*d = 96 6*d + 22/8*d = 96 - 17/8 28/8*d = 96 - 17/8 28*d = 96*8 - 17 28*d = 768 - 17 28*d = 751 d = <<751/28>>26.82142857142857 We can substitute this value for d into the expression for k: k = 17/72 + 22/72*<<751/28>>26.82142857142857 k = 17/72 + 22/72*26.82142857142857 k = 17/72 + 594.9285714285714 k = <<17+594.9285714285714/72>>8.214285714285714 ### Answer: k = 8.214285714285714, d = 26.82142857142857
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 34B:
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 34B:
Reasoning: 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130 1) 43 * q - 44 * e = 29
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 34B:
Reasoning: 53 = ((89 * r) - (74 * a)) 109 = ((37 * a) + (6 * r)) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74 * a) 109 = (37 * a) + (6 * r) 53 = (89 * r) - (74
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 34B:
Reasoning: 1. The sum of the weights of 42 units of 'g' and 12 units of 'q' equals 169 kg. 169 = 42g + 12q 2. The weight of 12 units of 'q' minus the weight of 50 units of 'g' results in 19 kg. 12q - 50g = 19 Solving for 'g' and 'q': 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q 12q - 50g = 19 169 = 42g + 12q
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 34B:
Reasoning: 37 * s - 11 * v = 54 82 * s - 62 * v = 41 s = <<(54+11*v)/37>>((54+11*v)/37) 82 * ((54+11*v)/37) - 62 * v = 41 <<(82*(54+11*v))/37-62*v=41>>((82*(54+11*v))/37-62*v=41) <<(82*54+82*11*v)/37-62*v=41>>((82*54+82*11*v)/37-62*v=41) <<(82*54)/37+(82*11*v)/37-62*v=41>>((82*54)/37+(82*11*v)/37-62*v=41) <<(82*54)/37+11*v-62*v=41>>((82*54)/37+11*v-62*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/37-51*v=41) <<(82*54)/37-51*v=41>>((82*54)/3
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 34B:
Reasoning: We can solve the first equation for c: 41 * c + 65 * z = 151 41 * c = 151 - 65 * z c = (151 - 65 * z)/41 We can solve the second equation for z: 78 * z - 48 * c = 48 78 * z = 48 + 48 * c z = (48 + 48 * c)/78 We can substitute the expression for z into the expression for c: c = (151 - 65 * z)/41 c = (151 - 65 * ((48 + 48 * c)/78))/41 c = (151 - 65 * (48 + 48 * c)/78)/41 c = (151 * 78 - 65 * (48 + 48 * c))/41 * 78 c = (11438 - 3240 - 3120 * c)/3238 c = (8208 - 3120 * c)/3238 3120 * c + 8208 = 3238 * c 8208 = 1118 * c c = 7.34 We can substitute the expression for c into the expression for z: z = (48 + 48 * c)/78 z = (48 + 48 * 7.34)/78 z = (48 + 346.32)/78 z = 404.32/78 z = 5.17 ### Answer: c = 7.34, z = 5.17
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 34B:
Reasoning: 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60 * c - 91 * i = 7 7 * i + 76 * c = 148 60
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 34B:
Reasoning: 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184 1) 71
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 34B:
Reasoning: We can solve this system of equations by using the elimination method. We can multiply the first equation by -26 and the second equation by 69 to get: -26(25v + 69d) = -26(113) -675v - 1814d = -2998 69(26d + 42v) = 69(85) 1814d + 2918v = 5805 We can add the two equations to get: -2918v = -2998 v = 1 We can substitute v = 1 into the first equation to get: 25(1) + 69d = 113 69d = 88 d = 1 ### Answer: d = 1, v = 1
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 34B:
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 * z
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for c: 50c + 12f = 92 50c = 92 - 12f c = (92 - 12f)/50 Now we can substitute this expression for c into the second equation: 36c - 21f = 34 36((92 - 12f)/50) - 21f = 34 3252 - 432f - 1050f = 1700 -1482f = -1542 f = 1542/1482 f = 1.03 Now we can substitute this value for f into the first equation to find c: 50c + 12f = 92 50c + 12(1.03) = 92 50c = 92 - 12.36 50c = 79.64 c = 1.59 ### Answer: c = 1.59, f = 1.03
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 34B:
Reasoning: 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1) 87u - 75d = 7 2) 85d - 82u = 54 1)
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 34B:
Reasoning: We can solve this problem by using the substitution method. We can solve the first equation for a: a = (42 + 4i)/75 We can substitute this expression for a into the second equation: 10i + 13((42 + 4i)/75) = 160 We can solve this equation for i: i = 10 We can substitute this value for i into the first equation: a = (42 + 4(10))/75 a = 11 ### Answer: i = 10, a = 11
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 34B:
Reasoning: 37 * p + 74 * c = 183 27 * c - 18 * p = 13 Multiply the second equation by -1. 37 * p + 74 * c = 183 -27 * c + 18 * p = -13 Add the two equations. 37 * p + 74 * c - 27 * c + 18 * p = 183 - 13 Simplify. 55 * p + 47 * c = 170 Divide both sides by 55. p + 47/55 * c = 170/55 Simplify. p + 47/55 * c = 3 Subtract 47/55 * c from both sides. p = 3 - 47/55 * c Substitute 3 - 47/55 * c for p in the first equation. 37 * (3 - 47/55 * c) + 74 * c = 183 Simplify. 111 - 161/55 * c + 74 * c = 183 Simplify. 111 + 189/55 * c = 183 Subtract 111 from both sides. 189/55 * c = 183 - 111 Simplify. 189/55 * c = 72 Divide both sides by 189/55. c = 72/189 * 55/1 Simplify. c = 72/3 Simplify. c = 24 Substitute 24 for c in p = 3 - 47/55 * c. p = 3 - 47/55 * 24 Simplify. p = 3 - 108/5 Simplify. p = 3 - 21 Simplify. p = -18 ### Answer: p = -18, c = 24
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 34B:
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) -
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve for v in the first equation. 75v - 69f = 14 75v = 14 + 69f v = <<14+69f/75>>14/75 + 69f/75 v = <<14/75>>2/15 + <<69f/75>>69f/75 We can substitute this expression for v into the second equation. 46f + 53v = 110 46f + 53(<<2/15>>2/15 + <<69f/75>>69f/75) = 110 46f + <<53*2/15>>106/15 + <<53*69f/75>>397f/75 = 110 46f + <<106/15>>7 + <<397f/75>>53f = 110 46f + 53f = 110 - 7 99f = 103 f = <<103/99>>103/99 We can substitute this expression for f into the expression for v. v = <<2/15>>2/15 + <<69*103/99/75>>69*103/99/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>2/15 + <<69*103/99*1/75>>69*103/99*1/75 v = <<2/15>>
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 34B:
Reasoning: 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o + 13 * g = 131 89 * g - 70 * o = 37. 7 * o +
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for d: 88 * d - 60 * k = 38 88 * d = 38 + 60 * k d = (38 + 60 * k)/88 We can now substitute this expression for d into the second equation: 33 * k - 11 * d = 46 33 * k - 11 * ((38 + 60 * k)/88) = 46 33 * k - (38 + 60 * k)/8 = 46 33 * k - 38/8 - 60 * k/8 = 46 33 * k - 60 * k/8 = 46 + 38/8 33 * k - 60 * k/8 = 116/8 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60 * k/8 = 29/2 33 * k - 60
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 34B:
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 2
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for v: v = (1 + (29 * o)) / 87. We can substitute this expression for v into the second equation: (8 * ((1 + (29 * o)) / 87)) + (30 * o) = 91. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) + (30 * o) = 91. We can multiply both sides of the equation by 87: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 * o) * 87 = 91 * 87. We can simplify this equation: (8 * ((1 + (29 * o)) / 87)) * 87 + (30 *
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 34B:
Reasoning: The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124. The first equation is (23 * h) - (4
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 34B:
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) 90d - 42f = 69 1) 19d + 6f = 120 2) 90d - 42f = 69 1) 19d + 6f = 120 2) 90d - 42f = 69 1) 19
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 34B:
Reasoning: We can solve this system by substitution. First, we solve the first equation for 'r': 42 * m + 55 * r = 127 55 * r = 127 - 42 * m r = (127 - 42 * m) / 55 Now, we substitute this expression for 'r' into the second equation: 66 * m - 52 * r = 44 66 * m - 52 * ((127 - 42 * m) / 55) = 44 66 * m - (127 - 42 * m) = 44 * 55 66 * m - 127 + 42 * m = 2490 108 * m = 2490 + 127 108 * m = 2617 m = 2617 / 108 m = 24.5 Now, we can use this value of 'm' to find 'r': r = (127 - 42 * 24.5) / 55 r = (127 - 1039) / 55 r = -912 / 55 r = -16.64 So, the solution to this system is m = 24.5 and r = -16.64. ### Answer: m = 24.5 and r = -16.64
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 34B:
Reasoning: 31n + 64w = 120 57w + 71n = 173 Multiply the first equation by 57 and the second equation by 64. 1897n + 3628w = 6480 3824w + 4537n = 11328 Subtract the second equation from the first equation. 1897n + 3628w - (3824w + 4537n) = 6480 - 11328 1897n + 3628w - 3824w - 4537n = -4848 1897n - 4537n - 3628w + 3824w = -4848 -5560n - 804w = -4848 Divide both sides by -804. -5560n - 804w/-804 = -4848/-804 -5560n/(-804) - 804w/(-804) = -4848/-804 5560n/804 + 804w/804 = 4848/804 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6 5560n/804 + 804w/804 = 6
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 34B:
Reasoning: 1) 72 * a + 26 * u = 112 2) 76 * u - 77 * a = 31. Solve the first equation for a: a = (112 - 26 * u)/72 Substitute a = (112 - 26 * u)/72 into the second equation: 76 * u - 77 * ((112 - 26 * u)/72) = 31 Simplify: 76 * u - 77 * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/72) + 26 * u * ((112 - 26 * u)/72) = 31 76 * u - 112 * ((112 - 26 * u)/7
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 34B:
Reasoning: We can solve the first equation for n: 54n + 18b = 143 54n = 143 - 18b n = (143 - 18b)/54 We can substitute this expression for n into the second equation: 87n - 93b = 48 87((143 - 18b)/54) - 93b = 48 87(143 - 18b)/54 - 93b = 48 87(143 - 18b)/54 = 48 + 93b 87(143 - 18b) = 48*54 + 93b*54 87(143 - 18b) = 2592 + 4984b 12881 - 1584b = 2592 + 4984b 12881 - 2592 = 6568b 10289 = 6568b b = 10289/6568 b = 1.56 We can substitute this value for b into the first equation: 54n + 18b = 143 54n + 18*1.56 = 143 54n + 28.08 = 143 54n = 114.92 n = 114.92/54 n = 2.13 ### Answer: b = 1.56, n = 2.13
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve for e in the first equation: 53 * e + 7 * p = 103 53 * e = 103 - 7 * p e = (103 - 7 * p)/53 We can substitute this expression for e into the second equation: 69 * e - 88 * p = 6 69 * ((103 - 7 * p)/53) - 88 * p = 6 (69 * (103 - 7 * p))/53 - 88 * p = 6 69 * (103 - 7 * p) - 88 * p * 53 = 6 * 53 69 * 103 - 69 * 7 * p - 88 * p * 53 = 6 * 53 69 * 103 - (69 * 7 + 88 * 53) * p = 6 * 53 p = (69 * 103 - 6 * 53)/(69 * 7 + 88 * 53) p = <<(69*103-6*53)/(69*7+88*53)=0.090909090909091>>0.090909090909091 We can substitute this value for p into the expression for e: e = (103 - 7 * p)/53 e = (103 - 7 * 0.090909090909091)/53 e = <<103-7*0.090909090909091/53=1.8181818181818>>1.8181818181818 ### Answer: e = 1.8181818181818, p = 0.090909090909091
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 34B:
Reasoning: We can solve this system of equations by substitution. First, we solve the first equation for p. ((18 * p) - (7 * s)) = 21 ((18 * p) - (7 * s)) - (7 * s) = 21 - (7 * s) 18 * p = 21 - (7 * s) (18 * p)/18 = (21 - (7 * s))/18 p = (21 - (7 * s))/18 Now, we substitute this expression for p into the second equation. ((51 * s) - (24 * p)) = 14 ((51 * s) - (24 * p)) = 14 ((51 * s) - (24 * ((21 - (7 * s))/18))) = 14 ((51 * s) - ((24 * (21))/18) + ((24 * (7 * s))/18)) = 14 ((51 * s) - (432/18) + (168 * s)/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168 * s)/18) - (432/18) = 14 ((51 * s) + (168
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 34B:
Reasoning: The first equation is 59c-34w=84. The second equation is 11c+42w=98. We can solve this system of equations by elimination. Multiply the first equation by 42 and the second equation by 34. 59*42c-34*42w=84*42 11*34c+42*34w=98*34 2058c-1428w=3528 374c+1512w=3332 Subtract the second equation from the first equation. 2058c-1428w-374c-1512w=3528-3332 1684c-2940w=196 Divide both sides by 1684. c-2940w/1684=196/1684 c-2940w/1684=0.116 c=0.116+2940w/1684 c=0.116+2940w/1684 We can solve this system of equations by elimination. Multiply the first equation by 42 and the second equation by 34. 59*42c-34*42w=84*42 11*34c+42*34w=98*34 2058c-1428w=3528 374c+1512w=3332 Subtract the second equation from the first equation. 2058c-1428w-374c-1512w=3528-3332 1684c-2940w=196 Divide both sides by 1684. c-2940w/1684=196/1684 c-2940w/1684=0.116 c=0.116+2940w/1684 c=0.116+2940w/1684 We can solve this system of equations by elimination. Multiply the first equation by 42 and the second equation by 34. 59*42c-34*42w=84*42 11*34c+42*34w=98*34 2058c-1428w=3528 374c+1512w=3332 Subtract the second equation from the first equation. 2058c-1428w-374c-1512w=3528-3332 1684c-2940w=196 Divide both sides by 1684. c-2940w/1684=196/1684 c-2940w/1684=0.116 c=0.116+2940w/1684 c=0.116+2940w/1684 We can solve this system of equations by elimination. Multiply the first equation by 42 and the second equation by 34. 59*42c-34*42w=84*42 11*34c+42*34w=98*34 2058c-1428w=3528 374c+1512w=3332 Subtract the second equation from the first equation. 2058c-1428w-374c-1512w=3528-333
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 34B:
Reasoning: We can solve this system of equations by substitution. First, we solve the second equation for f: <<60*z-42*f=13>>60z-42f=13 <<60*z-42*f+42*f=13+42*f>>60z-42f+42f=13+42f <<60*z=13+42*f>>60z=13+42f <<60*z-13=13+42*f-13>>60z-13=13+42f-13 <<60*z-13=42*f>>60z-13=42f <<60*z-13/42=42*f/42>>60z-13/42=42f/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+13/42>>60z/42=f+13/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+13/42>>60z/42=f+13/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+13/42>>60z/42=f+13/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+13/42>>60z/42=f+13/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+13/42>>60z/42=f+13/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+13/42>>60z/42=f+13/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+13/42>>60z/42=f+13/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+13/42>>60z/42=f+13/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+13/42>>60z/42=f+13/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+13/42>>60z/42=f+13/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+13/42>>60z/42=f+13/42 <<60*z/42-13/42=f>>60z/42-13/42=f <<60*z/42=f+1
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 34B:
Reasoning: 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26 * p - 21 * u = 6 15 * p + 58 * u = 83 26
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 34B:
Reasoning: 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 1) 92 * n - 38 * s = 62 2)
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 34B:
Reasoning: We can solve the simultaneous equations by elimination. We can eliminate the z variable by multiplying the first equation by -3 and the second equation by 1. 87 * z - 35 * e = 63 -252 * z + 105 * e = -198 -------------------------------- -167 * e = -135 e = <<-135/-167=-0.81>>-0.81 We can then substitute the value of e into the first equation to find the value of z. 87 * z - 35 * e = 63 87 * z - 35 * -0.81 = 63 87 * z + 28.35 = 63 87 * z = 34.65 z = <<34.65/87=0.4>>0.4 ### Answer: e = -0.81 and z = 0.4
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 34B:
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 (98 * t) - (36 * h) = 18 (11 * t)
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for b: ((69 * b) - (50 * o)) = 55 ((69 * b) - (50 * o) + (50 * o)) = 55 + (50 * o) ((69 * b) + (0 * o)) = 55 + (50 * o) ((69 * b)) = 55 + (50 * o) ((69 * b) / (69)) = ((55 + (50 * o)) / (69)) (b) = ((55 + (50 * o)) / (69)) We can now substitute this expression for b into the second equation: ((41 * o) + (18 * b)) = 85 ((41 * o) + (18 * ((55 + (50 * o)) / (69)))) = 85 ((41 * o) + ((18 * (55 + (50 * o))) / (69))) = 85 ((41 * o) + ((18 * 55 + (18 * (50 * o))) / (69))) = 85 ((41 * o) + ((18 * 55 + (18 * 50 * o)) / (69))) = 85 ((41 * o) + ((18 * 55 + (900 * o)) / (69))) = 85 ((41 * o) + ((18 * 55 + 900 * o) / (69))) = 85 ((41 * o) + ((18 * 55 + 0 * o) / (69))) = 85 - (900 * o) ((41 * o) + ((18 * 55) / (69))) = 85 - (900 * o) ((41 * o) + (1170 / (69))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (900 * o) ((41 * o) + (17 * (1170 / (69)))) = 85 - (90
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 34B:
Reasoning: We can solve this system of equations by substitution. First, we solve the first equation for p: 67 * q - 80 * p = 65 67 * q - 80 * p + 80 * p = 65 + 80 * p 67 * q = 65 + 80 * p 67 * q / 67 = (65 + 80 * p) / 67 q = (65 + 80 * p) / 67 Now, we can substitute this expression for q into the second equation: 6 * q + 5 * p = 177 6 * ((65 + 80 * p) / 67) + 5 * p = 177 (6 * (65 + 80 * p)) / 67 + 5 * p = 177 (6 * 65 + 6 * 80 * p) / 67 + 5 * p = 177 (390 + 480 * p) / 67 + 5 * p = 177 (390 + 480 * p) / 67 + 5 * p * 67 / 67 = 177 * 67 / 67 (390 + 480 * p) / 67 + 5 * p * 67 = 177 * 67 (390 + 480 * p) + 5 * p * 67 = 177 * 67 * 67 390 + 480 * p + 5 * p * 67 = 177 * 67 * 67 390 + 480 * p + 5 * p * 67 - 390 = 177 * 67 * 67 - 390 480 * p + 5 * p * 67 = 177 * 67 * 67 - 390 480 * p + 5 * p * 67 / 67 = (177 * 67 * 67 - 390) / 67 480 * p + 5 * p = (177 * 67 * 67 - 390) / 67 480 * p + 5 * p * 1 / 5 = (177 * 67 * 67 - 390) / 67 / 5 480 * p + p = (177 * 67 * 67 - 390) / 67 / 5 480 * p + p * 480 / 480 = (177 * 67 * 67 - 390) / 67 / 5 / 480 481 * p = (177 * 67 * 67 - 390) / 67 / 5 / 480 481 * p * 1 / 481 = (177 * 67 * 67 - 390) / 67 / 5 / 480 / 481 p = (177 * 67 * 67 - 390) / 67 / 5 / 480 / 481 p = (177 * 67 * 67 - 390) / 67 / 5 / 480 / 481 p = (177 * 67 * 67 - 390) / 67 / 5 / 480 / 481 p = (177 * 67 * 67 - 390) / 67 / 5 / 480 / 481 p = (177 * 67 * 67 - 390) / 67 / 5 / 480 / 481 p = (177 * 67 * 67 - 390) / 67 / 5 / 480
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 34B:
Reasoning: We can solve this system of equations by elimination. We can eliminate the variable k by multiplying the first equation by 33 and the second equation by 10. 330 * k + 1584 * f = 5343 330 * k - 410 * f = 310 We can eliminate the variable f by adding the two equations. 330 * k + 1584 * f + 330 * k - 410 * f = 5343 + 310 660 * k = 5653 k = 5653/660 = <<5653/660=8.5>>8.5 We can find the value of f by substituting the value of k into either of the original equations. 10 * k + 48 * f = 161 10 * 8.5 + 48 * f = 161 85 + 48 * f = 161 48 * f = 76 f = 76/48 = <<76/48=1.6>>1.6 ### Answer: k = 8.5, f = 1.6
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 34B:
Reasoning: The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l We can solve the first equation for n: 31 = 58 * l - n 31 + n = 58 * l n = 58 * l - 31 We can substitute this expression for n into the second equation: 103 = n + 73 * l 103 = 58 * l - 31 + 73 * l 103 = 131 * l - 31 131 * l = 103 + 31 131 * l = 134 l = 134/131 l = <<134/131=1.0229>>1.0229 We can substitute this expression for l into the first equation: 31 = 58 * l - n 31 = 58 * 1.0229 - n 31 - 58 * 1.0229 = - n n = 58 * 1.0229 - 31 n = <<58*1.0229-31=1.1658>>1.1658 ### Answer: l = 1.0229, n = 1.1658
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 34B:
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 2
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 34B:
Reasoning: We can solve this system of equations by elimination. First, we multiply the first equation by -17 and the second equation by 85. The first equation becomes -1597u + 1515a = -1273. The second equation becomes 1520u + 1485a = 1590. Then, we add the two equations together. The first equation becomes -1597u + 1515a = -1273. The second equation becomes 1520u + 1485a = 1590. The sum of the two equations is 1515a + 1485a = 1590 - 1273. We can simplify this to 3000a = 317. We can divide both sides by 3000 to get a = 317/3000. We can plug a = 317/3000 into the first equation to get 93u - 85(317/3000) = 74. We can simplify this to 93u - 2599/60 = 74. We can add 2599/60 to both sides to get 93u = 74 + 2599/60. We can simplify this to 93u = 1019/60. We can divide both sides by 93 to get u = 1019/5534. We can plug u = 1019/5534 into the first equation to get 93(1019/5534) - 85a = 74. We can simplify this to 93019/5534 - 85a = 74. We can subtract 93019/5534 from both sides to get -85a = 74 - 93019/5534. We can simplify this to -85a = -1799/5534. We can divide both sides by -85 to get a = 1799/46425. ### Answer: u = 1019/5534, a = 1799/46425
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 34B:
Reasoning: 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 97 * u - 61 * g = 15 85 * u + 23 * g = 186 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 34B:
Reasoning: We can solve this system of equations by elimination. We can eliminate the variable b by subtracting the first equation from the second equation. (85 * r) + (67 * b) - ((17 * r) + (67 * b)) = 187 - 103 (85 * r) + (67 * b) - (17 * r) - (67 * b) = 84 (85 * r) - (17 * r) = 84 68 * r = 84 r = 84/68 = <<84/68=1.25>>1.25 We can now substitute r = 1.25 into the first equation to find b. (17 * 1.25) + (67 * b) = 103 21.25 + (67 * b) = 103 (67 * b) = 103 - 21.25 (67 * b) = 81.75 b = 81.75/67 = <<81.75/67=1.22>>1.22 ### Answer: r = 1.25, b = 1.22
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 34B:
Reasoning: 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126. 1. (85
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 34B:
Reasoning: 1) 75u - 69q = 5 2) 48u + 25q = 140 Solve the first equation for u: 75u = 5 + 69q u = (5 + 69q)/75 Substitute (5 + 69q)/75 for u in the second equation: 48*(5 + 69q)/75 + 25q = 140 Multiply both sides by 75: 48*(5 + 69q) + 25*75q = 140*75 Simplify: 240 + 3285q + 1875q = 10500 Combine like terms: 3285q + 1875q = 10500 - 240 Simplify: 5160q = 10260 Divide both sides by 5160: q = 10260/5160 Simplify: q = 2 Substitute 2 for q in the first equation: 75u - 69*2 = 5 Simplify: 75u - 138 = 5 Add 138 to both sides: 75u = 143 Divide both sides by 75: u = 143/75 Simplify: u = 19/15 ### Answer: u = 19/15, q = 2
Expected answer: {q: 180/91, u: 515/273}
Correct? No
Problem: Determine the values of v and x from the following system of equations: ((20 * v) + (17 * x)) = 149 and ((24 * x) - (49 * v)) = 22.
LLemma 34B:
Reasoning: We can solve this system of equations by substitution. ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 ((20 * v) + (17 * x)) = 149 ((24 *
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 34B:
Reasoning: We can solve this system of equations by substitution. First, we solve the first equation for t: 39 * t + 41 * g = 147 39 * t = 147 - 41 * g t = (147 - 41 * g)/39 Now, we substitute this expression for t into the second equation: 94 * t - 91 * g = 42 94 * ((147 - 41 * g)/39) - 91 * g = 42 (1358 - 41 * g)/39 - 91 * g = 42 1358 - 41 * g - 3499 * g = 1718 -3548 * g = -3670 g = 1 Now, we can substitute this value for g into the expression for t: t = (147 - 41 * g)/39 t = (147 - 41 * 1)/39 t = 2 Therefore, the solution to the system of equations is t = 2 and g = 1. ### Answer: t = 2 and g = 1
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 34B:
Reasoning: We can solve the system of equations by elimination. Multiply the first equation by 55 and the second equation by 6. 6c + 55o = 112 55c - 48o = 36 6c + 55o = 112 330c - 288o = 216 6c + 55o = 112 330c - 288o = 216 336c = 328 c = <<328/336=0.9722222222222222>>0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 336c = 328 c = 0.9722222222222222 6c + 55o = 112 330c - 288o = 216 3
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 34B:
Reasoning: We can solve the system of equations by substitution. We can solve the first equation for u: 8w + 68u = 170, 68u = 170 - 8w, u = (170 - 8w)/68. Substituting this expression for u into the second equation, we get: 71w - 94((170 - 8w)/68) = 8, 71w - (94*170 - 94*8w)/68 = 8, 71w - 15180 + 752w = 8*68, 823w = 8*68 + 15180, 823w = 544 + 15180, 823w = 15724, w = 15724/823. Substituting this expression for w into the first equation, we get: 8(15724/823) + 68u = 170, 12580/823 + 68u = 170, 68u = 170 - 12580/823, 68u = (170*823 - 12580)/823, 68u = (14019 - 12580)/823, 68u = 1439/823, u = 1439/5414. ### Answer: u = 1439/5414, w = 15724/823
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for r: 68*r + 8*f = 92 68*r = 92 - 8*f r = (92 - 8*f)/68 Now we can substitute this expression for r into the second equation: 97*f + 38*(92 - 8*f)/68 = 182 97*f + 38*(92 - 8*f)/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 182 97*f + 38*92/68 - 38*8*f/68 = 1
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 34B:
Reasoning: The first equation is 67d - 37r = 50. The second equation is 39r - 21d = 16. We can solve this system of equations by substitution. We can solve the first equation for d: 67d - 37r = 50 67d = 50 + 37r d = (50 + 37r)/67 We can substitute this expression for d into the second equation: 39r - 21((50 + 37r)/67) = 16 39r - (21(50)/67 + 21(37r)/67) = 16 39r - (1050/67 + 757r/67) = 16 39r - 1050/67 - 757r/67 = 16 -718r/67 - 1050/67 = 16 -718r - 1050 = 16(67) -718r - 1050 = 1072 -718r = 1072 + 1050 -718r = 2122 r = -2122/-718 r = 3 We can substitute this value for r into the first equation to find d: 67d - 37(3) = 50 67d - 111 = 50 67d = 50 + 111 67d = 161 d = 161/67 d = 2.4 The value of d is 2.4, and the value of r is 3. ### Answer: d = 2.4, r = 3
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 34B:
Reasoning: First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a. First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a
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 34B:
Reasoning: We can solve this system of equations by elimination. First, we multiply the first equation by 44 and the second equation by 18 to get: 2798 * p + 792 * b = 4840 1476 * b - 832 * p = 1064 Then, we add the two equations to get: 1476 * b + 792 * b = 4840 + 1064 2268 * b = 5904 b = 2.61 Now, we can plug in b = 2.61 into the first equation to get: 67 * p + 18 * 2.61 = 110 67 * p + 46.98 = 110 67 * p = 63.02 p = 0.94 ### Answer: p = 0.94, b = 2.61
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 34B:
Reasoning: 63c-36e = 27 and 8c+32e = 69. 63c-36e = 27 8c+32e = 69 Multiply the first equation by 4. 252c-144e = 108 8c+32e = 69 Subtract the second equation from the first equation. 244c = 41 c = <<244/244=1>>1 Substitute c = 1 into the first equation. 63(1)-36e = 27 63-36e = 27 36e = 36 e = <<36/36=1>>1 c = 1 and e = 1 ### Answer: c = 1 and e = 1
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 34B:
Reasoning: 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x + 29 * p = 107 61 * x - 51 * p = 4 26 * x
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for l: ((29 * l) + (17 * g)) = 68 ((29 * l) + (17 * g) - (17 * g)) = 68 - (17 * g) ((29 * l) - (17 * g)) = 68 - (17 * g) ((29 * l) - (17 * g))/(29) = (68 - (17 * g))/(29) l = (68 - (17 * g))/(29) We can substitute this expression for l into the second equation: ((31 * g) - (10 * l)) = 17 ((31 * g) - (10 * ((68 - (17 * g))/(29)))) = 17 ((31 * g) - ((10 * (68 - (17 * g)))/(29))) = 17 ((31 * g) - ((10 * 68 - (10 * (17 * g)))/(29))) = 17 ((31 * g) - ((10 * 68)/(29) - (10 * (17 * g))/(29))) = 17 ((31 * g) - (10 * 68)/(29) + (10 * (17 * g))/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29)) = 17 ((31 * g) - (10 * 68)/(29) + (10 * 17 * g)/(29
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 34B:
Reasoning: We can solve this system of equations by elimination. We can eliminate the variable c by multiplying the first equation by 26 and the second equation by 54. 26*33*a - 26*54*c = 26*20 54*24*a - 54*26*c = 54*29 1018*a - 1364*c = 520 1224*a - 1364*c = 1514 -106*a = -914 a = <<-914/-106=-8.6>>-8.6 We can now substitute a = -8.6 into the first equation to solve for c. 33*(-8.6) - 54*c = 20 -282.8 - 54*c = 20 -54*c = 282.8 + 20 -54*c = 302.8 c = <<-302.8/-54=-5.6>>-5.6 ### Answer: a = -8.6, c = -5.6
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 34B:
Reasoning: We can solve this system of equations by elimination. We can eliminate the d variable by multiplying the first equation by 84 and the second equation by 71. 71d + 9j = 132 43j - 84d = 91 84(71d + 9j) = 84(132) 71(43j - 84d) = 71(91) 5744d + 756j = 11008 3013j - 5956d = 6477 5744d + 756j = 11008 3013j - 5956d = 6477 15077j = 17485 j = <<15077/17485=0.86>>0.86 We can now substitute this value for j into the first equation to find d. 71d + 9(0.86) = 132 71d + 7.64 = 132 71d = 124.36 d = <<124.36/71=1.75>>1.75 ### Answer: d = 1.75, j = 0.86
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 34B:
Reasoning: (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163 (23 * o) + (87 * n) = 143 (81 * o) + (26 * n) = 163
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 34B:
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
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 34B:
Reasoning: 51*k - 8*i = 86 39*i + 8*k = 193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k - 8*i = 86 -39*i - 8*k = -193 51*k
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 34B:
Reasoning: We can solve this system of equations by substitution. First, we solve the second equation for l: l = 126 - 93m Then, we substitute this expression for l into the first equation: 26m + 17(126 - 93m) = 146 We can now solve for m: 26m + 2098 - 1551m = 146 -1295m = -1652 m = <<-1652/-1295=-1.27>>-1.27 We can now use this value of m to find l: l = 126 - 93m l = 126 - 93(-1.27) l = <<126-93*-1.27=194.1>>194.1 ### Answer: l = 194.1, m = -1.27
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 34B:
Reasoning: 1) (8 * f) + (39 * h) = 66 2) (29 * f) + (46 * h) = 119 Solve for 'h' in the first equation: (8 * f) + (39 * h) = 66 (39 * h) = 66 - (8 * f) h = (66 - (8 * f)) / 39 Substitute (66 - (8 * f)) / 39 for 'h' in the second equation: (29 * f) + (46 * ((66 - (8 * f)) / 39)) = 119 (29 * f) + ((46 * (66 - (8 * f))) / 39) = 119 (29 * f) + ((46 * 66) / 39) - ((46 * (8 * f)) / 39) = 119 (29 * f) + ((46 * 66) / 39) - ((46 * 8) * f) / 39 = 119 (29 * f) + (1388 / 39) - (368 * f) / 39 = 119 (29 * f) - (368 * f) / 39 + (1388 / 39) = 119 ((29 * 39 * f) - (368 * f) + (1388 / 39)) / 39 = 119 ((1061 * f) - (368 * f) + (1388 / 39)) / 39 = 119 ((1061 * f) - (368 * f)) / 39 + (1388 / 39) = 119 ((1061 - 368) * f) / 39 + (1388 / 39) = 119 (693 * f) / 39 + (1388 / 39) = 119 (693 * f) / 39 + (1388 / 39) = 119 (693 * f + 1388) / 39 = 119 693 * f + 1388 = 119 * 39 693 * f + 1388 = 4761 693 * f = 4761 - 1388 693 * f = 3373 f = 3373 / 693 f = 4.8 Substitute 4.8 for 'f' in the first equation: (8 * 4.8) + (39 * h) = 66 38.4 + (39 * h) = 66 39 * h = 66 - 38.4 39 * h = 27.6 h = 27.6 / 39 h = 0.7 ### Answer: h = 0.7, f = 4.8
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 34B:
Reasoning: We can solve the system of equations by substitution. We can solve the first equation for k: k = (109 - 23 * b)/22. We can substitute this expression for k into the second equation: 62 * ((109 - 23 * b)/22) - 69 * b = 14. We can solve this equation for b: b = 1. We can substitute this value for b into the first equation to find k: k = (109 - 23 * 1)/22 = 4. ### Answer: k = 4, 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 34B:
Reasoning: 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1. 84m - 55s = 1 2. 63m - 2s = 98. 1.
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 34B:
Reasoning: 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67. 1. (6 * b) + (5
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 34B:
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 = 1
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for c: 5 * c + 28 * p = 102 5 * c = 102 - 28 * p c = (102 - 28 * p)/5 We can substitute this expression for c into the second equation: 20 * c - 67 * p = 13 20 * ((102 - 28 * p)/5) - 67 * p = 13 20 * (102 - 28 * p)/5 - 67 * p = 13 20 * (102 - 28 * p) - 67 * p * 5 = 13 * 5 20 * (102 - 28 * p) - 67 * p * 5 = 65 20 * (102 - 28 * p) - 335 * p = 65 20 * (102 - 28 * p) = 65 + 335 * p 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (102 - 28 * p) = 335 * p + 65 20 * (10
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 34B:
Reasoning: We can solve this system of equations by elimination. We can eliminate the variable q by multiplying the first equation by 97 and the second equation by 12. 97 * 58 * r + 97 * 12 * q = 97 * 147 12 * 97 * q - 12 * 76 * r = 12 * 76 We can then subtract the second equation from the first equation. 97 * 58 * r + 97 * 12 * q - (12 * 97 * q - 12 * 76 * r) = 97 * 147 - 12 * 76 Simplifying, we get: 97 * 58 * r + 97 * 12 * q - 12 * 97 * q + 12 * 76 * r = 97 * 147 - 12 * 76 We can then factor out the variable r. 97 * 58 * r + 97 * 12 * q - 12 * 97 * q + 12 * 76 * r = 97 * 147 - 12 * 76 We can then factor out the variable q. 97 * 58 * r + 97 * 12 * q - 12 * 97 * q + 12 * 76 * r = 97 * 147 - 12 * 76 We can then factor out the constant term. 97 * 58 * r + 97 * 12 * q - 12 * 97 * q + 12 * 76 * r = 97 * 147 - 12 * 76 We can then simplify the equation. 97 * 58 * r + 97 * 12 * q - 12 * 97 * q + 12 * 76 * r = 97 * 147 - 12 * 76 We can then solve for the variable r. 97 * 58 * r + 97 * 12 * q - 12 * 97 * q + 12 * 76 * r = 97 * 147 - 12 * 76 We can then solve for the variable q. 97 * 58 * r + 97 * 12 * q - 12 * 97 * q + 12 * 76 * r = 97 * 147 - 12 * 76 We can then substitute the values of r and q into the original system of equations to check our answer. 58 * r + 12 * q = 147 97 * q - 76 * r = 76 We can then solve for the variable r. 58 * r + 12 * q = 147 97 * q - 76 * r = 76 We can then solve for the variable q. 58 * r + 12 * q = 147 97 * q - 76 * r = 76 We can then substitute the values of r and q into the original system of equations to check our answer. 58 * r + 12 * q = 147 97 * q - 76 * r = 76 We can then solve for the variable r. 58 * r + 12 * q = 147 97 * q - 76 * r = 76 We can then solve for the variable q. 58 * r + 12 * q = 147 97 * q - 76 * r = 76 We can then substitute the values of r and q into the original system of equations to check our answer. 58 * r + 12 * q = 147 97 * q - 76 * r = 76 We can then solve for the variable r. 58 * r + 12
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 34B:
Reasoning: We can solve this system of equations by elimination. First, we multiply the first equation by 2 to get: 94 * v - 6 * k = 20 Then, we add the first equation to the second equation to get: 166 * v = 148 Finally, we divide both sides by 166 to get: v = <<148/166=0.89>>0.89 Now, we can plug in this value for v into either of the original equations to solve for k. If we plug in v = 0.89 into the first equation, we get: 47 * 0.89 - 3 * k = 10 39.93 - 3 * k = 10 -3 * k = 10 - 39.93 -3 * k = -29.93 k = <<-29.93/-3=-9.97>>-9.97 So, the solution to the system of equations is v = 0.89 and k = -9.97. ### Answer: v = 0.89, k = -9.97
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for p: p = <<60+41q/89>>60+41q/89 We can substitute this expression for p into the second equation: 27(<<60+41q/89>>60+41q/89) + 91q = 124 We can simplify this equation: <<27(60+41q/89)+91q=124>>27(60+41q/89)+91q=124 <<27(60)+27(41q/89)+91q=124>>27(60)+27(41q/89)+91q=124 <<1620+27(41q/89)+91q=124>>1620+27(41q/89)+91q=124 <<1620+27(41q)/89+91q=124>>1620+27(41q)/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/89+91q=124 <<1620+1127q/89+91q=124>>1620+1127q/8
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 34B:
Reasoning: We can solve the first equation for u: (88 * u) - (96 * d) = 12 (88 * u) = 12 + (96 * d) u = (12 + (96 * d)) / 88 We can solve the second equation for d: (27 * u) + (36 * d) = 71 (36 * d) = 71 - (27 * u) d = (71 - (27 * u)) / 36 We can substitute the expression for d into the first equation: (88 * u) - (96 * d) = 12 (88 * u) - (96 * ((71 - (27 * u)) / 36)) = 12 (88 * u) - ((71 * 96) - ((27 * 96) * u)) / 36 = 12 (88 * u) - ((71 * 96) / 36) + ((27 * 96) * u) / 36 = 12 ((88 * 36) * u) - ((71 * 96) / 36) + ((27 * 96) * u) = (12 * 36) ((3216 * u) - (2176 / 36) + (8232 * u)) = 432 ((3216 + 8232) * u) - (2176 / 36) = 432 ((11448 * u) - (2176 / 36)) = 432 ((11448 * u) - (2176 / 36)) / (11448 / 36) = (432 / (11448 / 36)) u = 1 We can substitute the expression for u into the second equation: (27 * u) + (36 * d) = 71 (27 * 1) + (36 * d) = 71 27 + (36 * d) = 71 (36 * d) = 71 - 27 (36 * d) = 44 d = 44 / 36 d = 11 / 9 ### Answer: u = 1, d = 11/9
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for f: 20 * n - 18 * f = 9 -18 * f = 9 - 20 * n f = (9 - 20 * n)/(-18) f = <<(9-20*n)/(-18)=(9-20*n)/(-18)>>(9-20*n)/(-18) We can now substitute this expression for f into the second equation: 3 * n - 2 * f = 70 3 * n - 2 * ((9 - 20 * n)/(-18)) = 70 3 * n - (18 * (9 - 20 * n))/(-18) = 70 3 * n - (18 * 9 - 18 * 20 * n)/(-18) = 70 3 * n - (162 - 360 * n)/(-18) = 70 3 * n - 162/(-18) + 360 * n/(-18) = 70 (363 * n)/(-18) - 162/(-18) = 70 (363 * n)/(-18) + 162/18 = 70 (363 * n + 162)/(-18) = 70 363 * n + 162 = -1260 363 * n = -1260 - 162 363 * n = -1422 n = <<-1422/363=-1422/363>>-1422/363 We can now substitute this value for n into the expression for f: f = (9-20*n)/(-18) f = (9-20*(-1422/363))/(-18) f = <<9-20*(-1422/363)=9-20*(-1422/363)>>9-20*(-1422/363) ### Answer: f = <<9-20*(-1422/363)=9-20*(-1422/363)>>9-20*(-1422/363), n = <<-1422/363=-1422/363>>-1422/363
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 34B:
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 +
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 34B:
Reasoning: 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 34B:
Reasoning: We can solve this system of equations by elimination. Multiply the first equation by 68 and the second equation by 11. 74u + 11w = 116 66u - 68w = 21 68(74u + 11w) = 68(116) 11(66u - 68w) = 11(21) 4832u + 752w = 7744 726u - 758w = 231 4832u + 752w = 7744 726u - 758w = 231 1504u = 10033 1504u = 10033 u = <<10033/1504=6.67>>6.67 Substitute u = 6.67 into the first equation. 74(6.67) + 11w = 116 470.28 + 11w = 116 11w = 68.72 w = <<68.72/11=6.25>>6.25 ### Answer: u = 6.67, w = 6.25
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for f: 76 * f - 42 * j = 96 76 * f = 96 + 42 * j f = (96 + 42 * j)/76 We can substitute this expression for f into the second equation: 47 * j + 34 * f = 136 47 * j + 34 * ((96 + 42 * j)/76) = 136 47 * j + (34 * 96)/76 + (34 * 42 * j)/76 = 136 47 * j + 4 * 96 + 6 * 42 * j = 136 * 76 47 * j + 384 + 252 * j = 10368 72 * j + 384 = 10368 72 * j = 10368 - 384 72 * j = 9984 j = 9984/72 j = 139 We can substitute this value for j into the first equation to find f: 76 * f - 42 * j = 96 76 * f - 42 * 139 = 96 76 * f - 5748 = 96 76 * f = 96 + 5748 76 * f = 5844 f = 5844/76 f = 76 ### Answer: f = 76, j = 139
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 34B:
Reasoning: We can solve this system of equations by elimination. We can eliminate the u term by multiplying the first equation by 69 and the second equation by 32. 69(68u + 32e) = 69(174) 32(63e - 69u) = 32(1) We can then add the equations together to get: 69(68u + 32e) + 32(63e - 69u) = 69(174) + 32(1) 69(68u) + 69(32e) + 32(63e) - 32(69u) = 69(174) + 32(1) 4622u + 2402e + 2016e - 2112u = 11986 + 32 2510u + 4418e = 12018 We can now solve for u by multiplying both sides by 2510. 2510(2510u + 4418e) = 2510(12018) 6295000u + 11253000e = 30045000 6295000u = 30045000 - 11253000e 6295000u = 18792000 - 11253000e 6295000u = 18792000 - 11253000e 6295000u = 75387000e u = 75387000e/6295000 u = 12e/13 We can now solve for e by multiplying both sides by 13. 13(2510u + 4418e) = 13(12018) 32730u + 57504e = 158234 32730(12e/13) + 57504e = 158234 39396e + 57504e = 158234 96890e = 158234 e = 158234/96890 e = 16/9 ### Answer: u = 12e/13 and e = 16/9
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 34B:
Reasoning: We can solve this system of linear equations by elimination. Multiply the first equation by -3 and the second equation by 5. -171p - 12n = -387 and 120p + 195n = 770 Add the two equations. -51p + 73n = 383 Divide both sides by 51. p = <<383/73=5.3>>5.3 Substitute p = 5.3 into the first equation. 57(5.3) + 4n = 129 Simplify. 293.1 + 4n = 129 Subtract 293.1 from both sides. 4n = -164.1 Divide both sides by 4. n = <<-164.1/4=-41.025>>-41.025 ### Answer: p = 5.3 and n = -41.025
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 34B:
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
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 34B:
Reasoning: The first equation is 83 * t - 58 * h = 45. The second equation is 69 * h + 11 * t = 124. We can solve the first equation for t: 83 * t - 58 * h = 45 83 * t = 45 + 58 * h t = (45 + 58 * h)/83 We can substitute this expression for t into the second equation: 69 * h + 11 * t = 124 69 * h + 11 * ((45 + 58 * h)/83) = 124 69 * h + (45 + 58 * h)/83 = 124 69 * h + 45/83 + 58 * h/83 = 124 69 * h + 45/83 + 7 * h = 124 69 * h + 7 * h = 124 - 45/83 76 * h = 124 - 45/83 h = (124 - 45/83)/76 We can substitute this expression for h into the first equation: 83 * t - 58 * h = 45 83 * t - 58 * ((124 - 45/83)/76) = 45 83 * t - (124 - 45/83)/76 = 45 83 * t - 124/76 + 45/76 = 45 83 * t - 124/76 + 3 = 45 83 * t - 124/76 = 42 83 * t = 42 + 124/76 t = (42 + 124/76)/83 ### Answer: t = (42 + 124/76)/83, h = (124 - 45/83)/76
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 34B:
Reasoning: We can solve this system of equations by substitution. ((79 * h) - (61 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9 ((16 * h) - (12 * o)) = 9
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 34B:
Reasoning: 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y + 56 * s = 139 65 * y - 90 * s = 16 4 * y +
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 34B:
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 34B:
Reasoning: (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) =
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 34B:
Reasoning: We can solve this system of equations by substitution. We can solve the first equation for p: p = <<45/39+2=5>>5. We can substitute this value for p into the second equation: 16*5 - 84*r = 36. We can solve this equation for r: r = <<36/16*5-84=1>>1. ### Answer: p = 5, r = 1
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 34B:
Reasoning: 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58 1. (23 * d) - (52 * w) = 7
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 34B:
Reasoning: We can solve this system of equations by substitution. First, we solve the first equation for n: n = (184 - 71 * z) / 60 Then, we substitute this expression for n into the second equation: (52 * z) - (47 * ((184 - 71 * z) / 60)) = 19 We can simplify this equation by multiplying both sides by 60: (52 * z * 60) - (47 * (184 - 71 * z)) = 19 * 60 We can then expand the left-hand side of the equation: (52 * z * 60) - (47 * 184) + (47 * 71 * z) = 19 * 60 We can then simplify the left-hand side of the equation by combining like terms: (52 * z * 60) + (47 * 71 * z) - (47 * 184) = 19 * 60 We can then factor out the z term on the left-hand side of the equation: z * (52 * 60 + 47 * 71) - (47 * 184) = 19 * 60 We can then solve for z by dividing both sides of the equation by (52 * 60 + 47 * 71): z = (19 * 60 + (47 * 184)) / (52 * 60 + 47 * 71) We can then simplify the right-hand side of the equation by expanding the numerator and denominator: z = (19 * 60 + 47 * 184) / (52 * 60 + 47 * 71) We can then simplify the right-hand side of the equation by combining like terms: z = (19 * 60 + 47 * 184) / (52 * 60 + 47 * 71) We can then simplify the right-hand side of the equation by factoring out the 19 and 47 terms: z = (19 * (60 + 47 * 3)) / (52 * 60 + 47 * 71) We can then simplify the right-hand side of the equation by expanding the numerator and denominator: z = (19 * (60 + 47 * 3)) / (52 * 60 + 47 * 71) We can then simplify the right-hand side of the equation by combining like terms: z = (19 * (60 + 47 * 3)) / (52 * 60 + 47 * 71) We can then simplify the right-hand side of the equation by factoring out the 19 and 47 terms: z = (19 * (60 + 47 * 3)) / (52 * 60 + 47 * 71) We can then simplify the right-hand side of the equation by expanding the numerator and denominator: z = (19 * (60 + 47 * 3)) / (52 * 60 + 47 * 71) We can then simplify the right-hand side of the equation by combining like terms: z = (19 * (60 + 47 * 3)) / (52 * 60 + 47 * 71) We can then simplify the right-hand side of the equation by factoring out the 19 and 47 terms: z = (19 * (60 + 47 * 3)) / (52 * 60 + 47 * 71) We can then simplify the right-hand side of the equation by expanding the numerator and denominator: z = (19 * (60 + 47 * 3)) / (52 * 60 + 47 * 71) We can then simplify the right
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 34B:
Reasoning: 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e + 90 * z = 189 99 * z - 84 * e = 20 70 * e
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 34B:
Reasoning: 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o - 44f = 95 2) 87f + 11o = 169 1) 26o -
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 34B:
Reasoning: 40*z - 20*p = 51 6*z + 58*p = 165. We can solve this system of equations by substitution. We can solve the first equation for z: 40*z - 20*p = 51 40*z = 51 + 20*p z = (51 + 20*p)/40 Now we can substitute this expression for z into the second equation: 6*z + 58*p = 165 6*(51 + 20*p)/40 + 58*p = 165 (6*51 + 6*20*p)/40 + 58*p = 165 (306 + 120*p)/40 + 58*p = 165 306 + 120*p + 2320*p = 6600 2440*p = 3534 p = 3534/2440 p = 1434/1220 p = 717/610 Now we can substitute this expression for p into the first equation to find z: 40*z - 20*p = 51 40*z - 20*(717/610) = 51 40*z - 1434/30.5 = 51 40*z - 46.68 = 51 40*z = 97.68 z = 97.68/40 z = 2.442 ### Answer: z = 2.442, p = 717/610
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 34B:
Reasoning: (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93. (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) =
Expected answer: {f: 6553/1103, s: 8292/1103}
Correct? No