Claude-3 Haiku 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
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve for q in the second equation. (30 * s) + (12 * q) = 121 12 * q = 121 - (30 * s) q = (121 - (30 * s)) / 12 Step 2: Substitute the expression for q into the first equation. (40 * s) - (91 * ((121 - (30 * s)) / 12)) = 37 (40 * s) - (91 * (121 / 12) - (91 * (30 * s) / 12)) = 37 (40 * s) - (7,533 - (227.5 * s)) = 37 (40 * s) - 7,533 + 227.5 * s = 37 267.5 * s - 7,533 = 37 267.5 * s = 7,570 s = 28.3 Step 3: Substitute the value of s into the expression for q. q = (121 - (30 * 28.3)) / 12 q = (121 - 849) / 12 q = -728 / 12 q = -60.67 Therefore, the values of the variables are: q = -60.67 s = 28.3 ### Answer: q = -60.67, s = 28.3
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Given equations: 4 * b + 44 * m = 180 79 * m - 6 * b = 48 Step 1: Solve for b in the second equation. 79 * m - 6 * b = 48 6 * b = 79 * m - 48 b = (79 * m - 48) / 6 Step 2: Substitute the expression for b in the first equation. 4 * b + 44 * m = 180 4 * ((79 * m - 48) / 6) + 44 * m = 180 (316 * m - 192) / 6 + 44 * m = 180 (316 * m - 192 + 264 * m) / 6 = 180 580 * m / 6 = 180 m = 180 * 6 / 580 m = 1.8571 Step 3: Substitute the value of m in the expression for b. b = (79 * 1.8571 - 48) / 6 b = (146.3321 - 48) / 6 b = 98.3321 / 6 b = 16.3887 Therefore, the values of b and m are: b = 16.3887 m = 1.8571 ### Answer: b = 16.3887, m = 1.8571
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Given equations: 42 * o - 15 * g = 28 47 * o + 29 * g = 141 Step 1: Solve for o in the first equation. 42 * o - 15 * g = 28 42 * o = 28 + 15 * g o = (28 + 15 * g) / 42 Step 2: Substitute the expression for o in the second equation. 47 * o + 29 * g = 141 47 * ((28 + 15 * g) / 42) + 29 * g = 141 (47 * 28 + 47 * 15 * g) / 42 + 29 * g = 141 (1316 + 705 * g) / 42 + 29 * g = 141 (1316 + 705 * g + 1218 * g) / 42 = 141 (1316 + 1923 * g) / 42 = 141 1316 + 1923 * g = 5922 1923 * g = 4606 g = 2.4 Step 3: Substitute the value of g in the expression for o. o = (28 + 15 * g) / 42 o = (28 + 15 * 2.4) / 42 o = (28 + 36) / 42 o = 64 / 42 o = 1.52 Therefore, the values of o and g are: o = 1.52 g = 2.4 ### Answer: o = 1.52, g = 2.4
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for v. (12 * v) - (6 * o) = 21 12v - 6o = 21 v = (21 + 6o) / 12 Step 2: Substitute the expression for v in the second equation. (76 * v) - (45 * o) = 50 76 * ((21 + 6o) / 12) - 45 * o = 50 (76 * (21 + 6o) / 12) - 45 * o = 50 (1596 + 456o) / 12 - 45 * o = 50 (2052 - 540o) / 12 = 50 2052 - 540o = 600 -540o = -1452 o = 2.7 Step 3: Substitute the value of o in the first equation to find the value of v. v = (21 + 6o) / 12 v = (21 + 6 * 2.7) / 12 v = (21 + 16.2) / 12 v = 37.2 / 12 v = 3.1 Therefore, the values of v and o are: v = 3.1 o = 2.7 ### Answer: v = 3.1, o = 2.7
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Let's start by solving the first equation for f: 75 * f + 25 * g = 138 75 * f = 138 - 25 * g f = (138 - 25 * g) / 75 Now, let's substitute this expression for f into the second equation: 2 * f + 90 * g = 143 2 * (138 - 25 * g) / 75 + 90 * g = 143 276 - 50 * g + 6750 * g = 10725 6700 * g = 10449 g = 10449 / 6700 g = 1.56 Now, let's substitute the value of g back into the first equation to find the value of f: f = (138 - 25 * 1.56) / 75 f = (138 - 39) / 75 f = 99 / 75 f = 1.32 Therefore, the values of f and g are: f = 1.32 g = 1.56 ### Answer: f = 1.32, g = 1.56
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. First, let's solve the second equation for y: 7 = 65 * n - 79 * y 79 * y = 65 * n - 7 y = (65 * n - 7) / 79 Now, we can substitute this expression for y into the first equation: 194 = 4 * y + 14 * n 194 = 4 * ((65 * n - 7) / 79) + 14 * n 194 = 260 * n / 79 - 28 / 79 + 14 * n 194 = (260 * n + 14 * 79 - 28) / 79 194 * 79 = 260 * n + 1106 - 28 15326 = 260 * n + 1078 14248 = 260 * n n = 54.8 Now, we can substitute the value of n into the expression for y: y = (65 * 54.8 - 7) / 79 y = 3562 / 79 y = 45 ### Answer: y = 45, n = 54.8
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.
Claude-3 Haiku:
To solve this problem, we need to find the values of l and g that satisfy the two given equations. Let's start with the first equation: 146 = 28 * g + 26 * l Rearranging the terms, we get: 26 * l = 146 - 28 * g l = (146 - 28 * g) / 26 Now, let's substitute this expression for l into the second equation: 6 = 41 * ((146 - 28 * g) / 26) - 53 * g 6 = (41 * (146 - 28 * g)) / 26 - 53 * g 156 = 41 * (146 - 28 * g) - 1378 * g 156 = 5986 - 1148 * g - 1378 * g 156 = 5986 - 2526 * g 2370 = 2526 * g g = 2370 / 2526 g = 0.9382 Now, let's substitute the value of g into the first equation to find the value of l: l = (146 - 28 * 0.9382) / 26 l = (146 - 26.2696) / 26 l = 119.7304 / 26 l = 4.6058 Therefore, the values of l and g that satisfy the two equations are: l = 4.6058 g = 0.9382 ### Answer: l = 4.6058, g = 0.9382
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.
Claude-3 Haiku:
To solve this system of linear equations, we can use the elimination method. Step 1: Multiply the first equation by 7 and the second equation by 56 to eliminate the variable y. 7 * (20 * p + 56 * y = 90) 140 * p + 392 * y = 630 56 * (89 * p + 7 * y = 98) 4984 * p + 392 * y = 5488 Step 2: Subtract the second equation from the first equation to eliminate the variable y. 140 * p + 392 * y = 630 -4984 * p - 392 * y = -5488 -4844 * p = -4858 Step 3: Solve for the variable p. -4844 * p = -4858 p = -4858 / -4844 p = 1 Step 4: Substitute the value of p into one of the original equations to find the value of y. Using the first equation: 20 * 1 + 56 * y = 90 20 + 56 * y = 90 56 * y = 70 y = 70 / 56 y = 5/4 = 1.25 Therefore, the values of p and y are: p = 1 y = 1.25 ### Answer: p = 1, y = 1.25
Expected answer: {p: 347/346, y: 3025/2422}
Correct? No
Problem: Given the system of equations: 72*k - 22*d = 17 and 6*d +69*k = 96, find the values of variables k and d.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve for d in the first equation. 72*k - 22*d = 17 -22*d = 17 - 72*k d = (17 - 72*k) / (-22) d = (17 - 72*k) / 22 Step 2: Substitute the expression for d into the second equation. 6*d + 69*k = 96 6*((17 - 72*k) / 22) + 69*k = 96 (102 - 432*k) / 22 + 69*k = 96 102 - 432*k + 1518*k = 2112 1518*k - 432*k = 2112 - 102 1086*k = 2010 k = 2010 / 1086 k = 1.85 Step 3: Substitute the value of k into the expression for d. d = (17 - 72*k) / 22 d = (17 - 72*1.85) / 22 d = (17 - 133.2) / 22 d = -116.2 / 22 d = -5.28 Therefore, the values of the variables are: k = 1.85 d = -5.28 ### Answer: k = 1.85, d = -5.28
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
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Step 1: Solve the first equation for d. 73 * h - 33 * d = 48 33 * d = 73 * h - 48 d = (73 * h - 48) / 33 Step 2: Substitute the expression for d in the second equation. 10 * d + 14 * h = 91 10 * ((73 * h - 48) / 33) + 14 * h = 91 (730 * h - 480) / 33 + 14 * h = 91 730 * h - 480 + 462 * h = 3003 1192 * h = 3483 h = 3483 / 1192 h = <<3483/1192=2.92>>2.92 Step 3: Substitute the value of h in the first equation to find the value of d. 73 * 2.92 - 33 * d = 48 213.16 - 33 * d = 48 -33 * d = -165.16 d = <<-165.16/-33=5>>5 Therefore, the values of d and h are: d = 5 h = 2.92 ### Answer: d = 5, h = 2.92
Expected answer: {d: 5971/1192, h: 3483/1192}
Correct? No
Problem: Solve the following system of equations for the values of q and e: 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Step 1: Rearrange the first equation to solve for q. 43 * q - 44 * e = 29 43 * q = 29 + 44 * e q = (29 + 44 * e) / 43 Step 2: Substitute the expression for q in the second equation. 30 * q + 43 * e = 130 30 * ((29 + 44 * e) / 43) + 43 * e = 130 (870 + 1320 * e) / 43 + 43 * e = 130 870 + 1320 * e + 1290 * e = 5590 2610 * e = 4720 e = 4720 / 2610 e = 1.8 Step 3: Substitute the value of e back into the first equation to find the value of q. 43 * q - 44 * 1.8 = 29 43 * q - 79.2 = 29 43 * q = 108.2 q = 108.2 / 43 q = 2.52 Therefore, the values of q and e are: q = 2.52 e = 1.8 ### Answer: q = 2.52, e = 1.8
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))
Claude-3 Haiku:
To solve this system of equations, we need to find the values of the variables r and a. Given the system of equations: 53 = ((89 * r) - (74 * a)) 109 = ((37 * a) + (6 * r)) Step 1: Solve for r in the first equation. 53 = ((89 * r) - (74 * a)) 53 + (74 * a) = 89 * r r = (53 + (74 * a)) / 89 Step 2: Substitute the expression for r into the second equation. 109 = ((37 * a) + (6 * r)) 109 = ((37 * a) + (6 * ((53 + (74 * a)) / 89))) 109 = ((37 * a) + (318 + (444 * a)) / 89) 109 * 89 = (37 * a * 89) + (318 + (444 * a)) 9,701 = (3,293 * a) + (318 + (444 * a)) 9,701 - 318 = (3,293 * a) + (444 * a) 9,383 = (3,737 * a) a = 9,383 / 3,737 a ≈ 2.51 Step 3: Substitute the value of a into the expression for r. r = (53 + (74 * a)) / 89 r = (53 + (74 * 2.51)) / 89 r = (53 + 185.74) / 89 r = 238.74 / 89 r ≈ 2.68 Therefore, the values of r and a are: r ≈ 2.68 a ≈ 2.51 ### Answer: r ≈ 2.68, a ≈ 2.51
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.
Claude-3 Haiku:
To solve for 'g' and 'q', we can use the given equations and set up a system of linear equations. Let's define the variables: - 'g' represents the quantity of one type of grain - 'q' represents the quantity of the other type of grain Given information: 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. Step 1: Set up the first equation. 42g + 12q = 169 Step 2: Set up the second equation. 12q - 50g = 19 Step 3: Solve the system of equations using substitution or elimination method. From the second equation, we can solve for 'q': 12q - 50g = 19 12q = 19 + 50g q = (19 + 50g) / 12 Substituting the value of 'q' into the first equation: 42g + 12((19 + 50g) / 12) = 169 42g + (19 + 50g) = 169 92g + 19 = 169 92g = 150 g = 150 / 92 g ≈ 1.63 units Substituting the value of 'g' into the second equation to find 'q': 12q - 50(1.63) = 19 12q - 81.5 = 19 12q = 100.5 q = 100.5 / 12 q ≈ 8.38 units ### Answer: g ≈ 1.63 units, q ≈ 8.38 units
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
Claude-3 Haiku:
To solve this system of equations, we can use the elimination method. Step 1: Multiply the first equation by 62 and the second equation by 11 to eliminate the variable v. 62 * (37 * s - 11 * v = 54) 2294 * s - 682 * v = 3348 11 * (82 * s - 62 * v = 41) 902 * s - 682 * v = 451 Step 2: Subtract the second equation from the first equation to eliminate the variable v. 2294 * s - 3348 = 902 * s - 451 1392 * s = 2897 s = 2897 / 1392 s = 2.08 Step 3: Substitute the value of s into either equation to find the value of v. Using the first equation: 37 * 2.08 - 11 * v = 54 77.96 - 11 * v = 54 -11 * v = -23.96 v = 2.18 Therefore, the values of s and v are: s = 2.08 v = 2.18 ### Answer: s = 2.08, v = 2.18
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.
Claude-3 Haiku:
To solve for the values of c and z, we can use the system of linear equations provided. The first equation is: 41 * c + 65 * z = 151 The second equation is: 78 * z - 48 * c = 48 We can solve this system of equations using the substitution method. Step 1: Solve for c in the second equation. 78 * z - 48 * c = 48 -48 * c = 78 * z - 48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) / -48 c = (78 * z - 48) /
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?
Claude-3 Haiku:
To solve this system of linear equations, we can use the substitution method. Step 1: Solve the first equation for i. 60 * c - 91 * i = 7 91 * i = 60 * c - 7 i = (60 * c - 7) / 91 Step 2: Substitute the expression for i into the second equation. 7 * i + 76 * c = 148 7 * (60 * c - 7) / 91 + 76 * c = 148 420 * c - 49 + 6,916 * c = 13,388 7,336 * c = 13,437 c = 13,437 / 7,336 c = 1.83 Step 3: Substitute the value of c into the first equation to find the value of i. 60 * 1.83 - 91 * i = 7 109.8 - 91 * i = 7 -91 * i = -102.8 i = 1.13 Therefore, the values of c and i are: c = 1.83 i = 1.13 ### Answer: c = 1.83, i = 1.13
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
Claude-3 Haiku:
To solve this system of two simultaneous linear equations, we can use the elimination method. Step 1: Multiply the first equation by 76 and the second equation by 71 to eliminate one of the variables. 76(71*v - 37*e = 69) 5396*v - 2812*e = 5244 71(41*e + 76*v = 184) 2911*e + 5396*v = 13064 Step 2: Subtract the second equation from the first equation to eliminate the variable v. 5396*v - 2812*e = 5244 -2911*e - 5396*v = -13064 --------------------------------------------------------------------------- -8723*e = -7820 Step 3: Solve for e. -8723*e = -7820 e = -7820/-8723 e = 0.9 Step 4: Substitute the value of e into one of the original equations to solve for v. Using the first equation: 71*v - 37*e = 69 71*v - 37*0.9 = 69 71*v - 33.3 = 69 71*v = 102.3 v = 102.3/71 v = 1.44 Therefore, the values of e and v that satisfy the given system of linear equations are: e = 0.9 v = 1.44 ### Answer: e = 0.9, v = 1.44
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Let's start by solving for 'd' in the second equation: 26d + 42v = 85 26d = 85 - 42v d = (85 - 42v) / 26 Now, we can substitute this expression for 'd' into the first equation: 25v + 69((85 - 42v) / 26) = 113 25v + 2205 - 1803v / 26 = 113 25v - 1803v / 26 = -2092 -1778v / 26 = -2092 v = 26 * (-2092) / -1778 v = 31 Now that we have the value of 'v', we can substitute it back into the second equation to find the value of 'd': 26d + 42(31) = 85 26d + 1302 = 85 26d = -1217 d = -46.81 Therefore, the values of 'd' and 'v' are: ### Answer: d = -46.81, v = 31
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
Claude-3 Haiku:
To solve the given system of linear equations for g and z, we can use the substitution method. Step 1: Solve the first equation for z. 43 * z - 91 * g = 33 43 * z = 91 * g + 33 z = (91 * g + 33) / 43 Step 2: Substitute the expression for z in the second equation. 72 * g - 3 * z = 68 72 * g - 3 * (91 * g + 33) / 43 = 68 72 * g - 273 * g / 43 - 99 / 43 = 68 72 * g - 6.349 * g - 2.302 = 68 65.651 * g = 70.302 g = 1.072 Step 3: Substitute the value of g back into the first equation to find the value of z. 43 * z - 91 * 1.072 = 33 43 * z - 97.552 = 33 43 * z = 130.552 z = 3.035 Therefore, the solutions to the system of equations are: g = 1.072 z = 3.035 ### Answer: g = 1.072, z = 3.035
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.
Claude-3 Haiku:
To solve this problem, we need to use the system of equations provided: 50c + 12f = 92 36c - 21f = 34 Step 1: Multiply the second equation by 12 and the first equation by 36 to eliminate the variable c. 36(50c + 12f) = 36(92) 36c - 21f = 34 432c + 432f = 3312 36c - 21f = 34 Subtracting the second equation from the first equation: 432c + 432f - 36c + 21f = 3312 - 34 396c + 453f = 3278 Step 2: Multiply the first equation by 21 and the second equation by 50 to eliminate the variable f. 21(50c + 12f) = 21(92) 36c - 21f = 34 1050c + 252f = 1932 1800c - 1050f = 1700 Subtracting the second equation from the first equation: 1050c + 252f - 1800c + 1050f = 1932 - 1700 -750c = 232 c = -232/750 = -0.3093 Step 3: Substitute the value of c into the first equation to find the value of f. 50c + 12f = 92 50(-0.3093) + 12f = 92 -15.465 + 12f = 92 12f = 107.465 f = 107.465/12 = 8.9555 Therefore, the quantity of cherries (c) is -0.3093, and the quantity of figs (f) is 8.9555. ### Answer: c = -0.3093, f = 8.9555
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
Claude-3 Haiku:
Okay, let's solve this system of linear equations to find the values of u and d. First, we'll use the substitution method to solve for u in terms of d. From equation 1: 87u - 75d = 7 87u = 75d + 7 u = (75d + 7) / 87 Now, we'll substitute this expression for u into equation 2: 85d - 82((75d + 7) / 87) = 54 85d - (6150d + 574) / 87 = 54 (7395d - 6150d - 574) / 87 = 54 (1245d - 574) / 87 = 54 1245d - 574 = 4698 1245d = 5272 d = 4.23 Now that we have the value of d, we can plug it back into the first equation to solve for u: 87u - 75(4.23) = 7 87u - 318.25 = 7 87u = 325.25 u = 3.74 Therefore, the values of u and d are: u = 3.74 d = 4.23 ### Answer: u = 3.74, d = 4.23
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?
Claude-3 Haiku:
Okay, let's solve this step-by-step: 1) We have two equations: (75 * a) - (4 * i) = 42 (10 * i) + (13 * a) = 160 2) First, let's solve for i in the second equation: (10 * i) + (13 * a) = 160 10 * i = 160 - 13 * a i = (160 - 13 * a) / 10 3) Now, let's substitute this expression for i into the first equation: (75 * a) - (4 * (160 - 13 * a) / 10) = 42 (75 * a) - (64 - 5.2 * a) = 42 75 * a - 64 + 5.2 * a = 42 80.2 * a = 106 a = 106 / 80.2 a = 1.32 4) Now that we have the value of a, we can plug it back into the second equation to solve for i: (10 * i) + (13 * 1.32) = 160 10 * i = 160 - 17.16 i = (160 - 17.16) / 10 i = 14.28 ### Answer: i = 14.28, a = 1.32
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
Claude-3 Haiku:
To solve the given system of equations for the values of p and c, we can use the substitution method. Step 1: Solve the first equation for p. 37 * p + 74 * c = 183 37 * p = 183 - 74 * c p = (183 - 74 * c) / 37 Step 2: Substitute the expression for p in the second equation. 27 * c - 18 * p = 13 27 * c - 18 * (183 - 74 * c) / 37 = 13 27 * c - 3294 + 1332 * c / 37 = 13 27 * c - 3294 + 36 * c = 481 63 * c = 3775 c = 60 Step 3: Substitute the value of c in the first equation to find the value of p. 37 * p + 74 * 60 = 183 37 * p + 4440 = 183 37 * p = 183 - 4440 37 * p = -4257 p = -115 Therefore, the values of p and c are: p = -115 c = 60 ### Answer: p = -115, c = 60
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.
Claude-3 Haiku:
To solve this system of equations, we need to use the substitution method. Let's start by solving the first equation for k: (48 * k) - (30 * p) = 37 48k - 30p = 37 k = (37 + 30p) / 48 Now, let's substitute this expression for k into the second equation: (16 * p) + (33 * k) = 105 (16 * p) + (33 * (37 + 30p) / 48) = 105 (16 * p) + (1221 + 990p) / 48 = 105 (16 * p) + 2211 / 48 = 105 (16 * p) + 46.0625 = 105 16p = 59.9375 p = 3.74609375 Now, let's substitute the value of p back into the first equation to find the value of k: (48 * k) - (30 * 3.74609375) = 37 48k - 112.3828125 = 37 48k = 149.3828125 k = 3.11630859375 Therefore, the values of k and p are: k = 3.11630859375 p = 3.74609375 ### Answer: k = 3.11630859375, p = 3.74609375
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.
Claude-3 Haiku:
To solve this system of simultaneous equations, we can use the elimination method. Step 1: Multiply the first equation by 53 and the second equation by 75 to eliminate one of the variables. Multiplying the first equation by 53: 53(75v - 69f = 14) 3,975v - 3,657f = 742 Multiplying the second equation by 75: 75(46f + 53v = 110) 3,450f + 3,975v = 8,250 Step 2: Subtract the first equation from the second equation to eliminate the variable v. 3,450f + 3,975v = 8,250 -3,975v - 3,657f = -742 ----------------- -207f = 7,508 Step 3: Solve for the variable f. -207f = 7,508 f = -7,508/207 f = -36.25 Step 4: Substitute the value of f into one of the original equations to solve for the variable v. Using the first equation: 75v - 69f = 14 75v - 69(-36.25) = 14 75v + 2,501.25 = 14 75v = -2,487.25 v = -33.165 Therefore, the values of the variables that satisfy the system of equations are: v = -33.165 f = -36.25 ### Answer: v = -33.165, f = -36.25
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Given the system of equations: 7 * o + 13 * g = 131 89 * g - 70 * o = 37 Step 1: Solve for o in the first equation. 7 * o + 13 * g = 131 7 * o = 131 - 13 * g o = (131 - 13 * g) / 7 Step 2: Substitute the expression for o in the second equation. 89 * g - 70 * o = 37 89 * g - 70 * (131 - 13 * g) / 7 = 37 89 * g - 70 * (131 / 7 - 13 * g / 7) = 37 89 * g - 70 * (18.71 - 1.86 * g) = 37 89 * g - 1309.7 + 130.2 * g = 37 219.2 * g = 1346.7 g = 6.15 Step 3: Substitute the value of g in the expression for o. o = (131 - 13 * g) / 7 o = (131 - 13 * 6.15) / 7 o = (131 - 79.95) / 7 o = 51.05 / 7 o = 7.29 Therefore, the values of o and g are: o = 7.29 g = 6.15 ### Answer: o = 7.29, g = 6.15
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Step 1: Solve the first equation for d. 88 * d - 60 * k = 38 88 * d = 38 + 60 * k d = (38 + 60 * k) / 88 Step 2: Substitute the expression for d in the second equation. 33 * k - 11 * d = 46 33 * k - 11 * (38 + 60 * k) / 88 = 46 33 * k - 418 / 88 - 660 * k / 88 = 46 33 * k - 4.75 - 7.5 * k = 46 25.5 * k = 50.75 k = 50.75 / 25.5 k = 2 Step 3: Substitute the value of k in the expression for d. d = (38 + 60 * k) / 88 d = (38 + 60 * 2) / 88 d = (38 + 120) / 88 d = 158 / 88 d = 1.8 Therefore, the values of d and k are: d = 1.8 k = 2 ### Answer: d = 1.8, k = 2
Expected answer: {d: 669/374, k: 203/102}
Correct? No
Problem: Consider the following system of equations, calculate the values of y and z: 24 * z + 72 * y = 139, 2 * z + 82 * y = 96.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for z. 24 * z + 72 * y = 139 24 * z = 139 - 72 * y z = (139 - 72 * y) / 24 Step 2: Substitute the expression for z into the second equation. 2 * z + 82 * y = 96 2 * (139 - 72 * y) / 24 + 82 * y = 96 (278 - 144 * y) / 24 + 82 * y = 96 278 - 144 * y + 1968 * y = 2304 1824 * y = 2026 y = 2026 / 1824 y = 1.11 Step 3: Substitute the value of y back into the first equation to find the value of z. z = (139 - 72 * 1.11) / 24 z = (139 - 79.92) / 24 z = 59.08 / 24 z = 2.46 Therefore, the values of y and z are: y = 1.11 z = 2.46 ### Answer: y = 1.11, z = 2.46
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Given equations: (87 * o) - (29 * v) = 1 (8 * v) + (30 * o) = 91 Step 1: Solve for v in the second equation. (8 * v) + (30 * o) = 91 8 * v = 91 - (30 * o) v = (91 - 30 * o) / 8 Step 2: Substitute the expression for v in the first equation. (87 * o) - (29 * ((91 - 30 * o) / 8)) = 1 (87 * o) - (29 * (91 - 30 * o) / 8) = 1 (87 * o) - (29 * 91 / 8) + (29 * 30 * o / 8) = 1 (87 * o) - 333.375 + 112.5 * o = 1 199.5 * o - 333.375 = 1 199.5 * o = 334.375 o = 334.375 / 199.5 o = <<o = 1.675>>1.675 Step 3: Substitute the value of o in the expression for v. v = (91 - 30 * 1.675) / 8 v = (91 - 50.25) / 8 v = 40.75 / 8 v = <<v = 5.09>>5.09 ### Answer: v = 5.09, o = 1.675
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.
Claude-3 Haiku:
To solve the given system of equations for h and p, we can use the substitution method. Step 1: Rearrange the second equation to express h in terms of p. (11 * p) + (23 * h) = 124 23 * h = 124 - (11 * p) h = (124 - (11 * p)) / 23 Step 2: Substitute the expression for h in the first equation. (23 * h) - (46 * p) = 58 (23 * ((124 - (11 * p)) / 23)) - (46 * p) = 58 (124 - (11 * p)) - (46 * p) = 58 124 - 11 * p - 46 * p = 58 124 - 57 * p = 58 -57 * p = -66 p = 1.16 Step 3: Substitute the value of p in the expression for h to find the value of h. h = (124 - (11 * p)) / 23 h = (124 - (11 * 1.16)) / 23 h = (124 - 12.76) / 23 h = 111.24 / 23 h = 4.84 Therefore, the solution to the system of equations is: h = 4.84 p = 1.16 ### Answer: h = 4.84, p = 1.16
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for d. 19d + 6f = 120 19d = 120 - 6f d = (120 - 6f) / 19 Step 2: Substitute the expression for d into the second equation. 90d - 42f = 69 90 * (120 - 6f) / 19 - 42f = 69 (90 * 120 - 90 * 6f) / 19 - 42f = 69 (10800 - 540f) / 19 - 42f = 69 10800 - 540f - 798f = 1311 -1338f = -9489 f = 7 Step 3: Substitute the value of f into the first equation to find d. 19d + 6(7) = 120 19d + 42 = 120 19d = 78 d = 4 Therefore, the values of the integers d and f are: d = 4 f = 7 ### Answer: d = 4, f = 7
Expected answer: {d: 909/223, f: 3163/446}
Correct? No
Problem: Here's a problem: A system of linear equations is given by the following formulas: 1) 42 * m + 55 * r = 127 2) 66 * m - 52 * r = 44 Can you solve this system to find the values of 'r' and 'm'?
Claude-3 Haiku:
To solve this system of linear equations, we can use the substitution method or the elimination method. Using the substitution method: From the first equation, we can solve for m: 42m + 55r = 127 42m = 127 - 55r m = (127 - 55r) / 42 Substituting this expression for m into the second equation: 66((127 - 55r) / 42) - 52r = 44 (66/42)(127 - 55r) - 52r = 44 (66/42)127 - (66/42)55r - 52r = 44 (66/42)127 - 118r = 44 118r = (66/42)127 - 44 r = ((66/42)127 - 44) / 118 r = 1 Substituting r = 1 back into the first equation to find m: 42m + 55(1) = 127 42m + 55 = 127 42m = 72 m = 72/42 m = 12/7 Therefore, the solution to the system of linear equations is: r = 1 m = 12/7 ### Answer: r = 1, m = 12/7
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
Claude-3 Haiku:
To solve this system of linear equations, we can use the elimination method. Step 1: Multiply the first equation by 71 and the second equation by 31 to eliminate the variable n. 71(31n + 64w = 120) 2201n + 4544w = 8520 31(57w + 71n = 173) 1767w + 2201n = 5363 Step 2: Subtract the second equation from the first equation to eliminate the variable n. 2201n + 4544w = 8520 1767w + 2201n = 5363 -------------- 2777w = 3157 Step 3: Solve for w. w = 3157 / 2777 w = <<3157/2777=1.14>>1.14 Step 4: Substitute the value of w into the first equation to solve for n. 31n + 64(1.14) = 120 31n + 72.96 = 120 31n = 47.04 n = <<47.04/31=1.52>>1.52 Therefore, the values of n and w that satisfy the system of linear equations are: n = 1.52 w = 1.14 ### Answer: n = 1.52, w = 1.14
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.
Claude-3 Haiku:
To solve this system of linear equations, we can use the substitution method. From the first equation, we can solve for u: 72 * a + 26 * u = 112 26 * u = 112 - 72 * a u = (112 - 72 * a) / 26 Now, we can substitute this expression for u into the second equation: 76 * ((112 - 72 * a) / 26) - 77 * a = 31 (76 * (112 - 72 * a)) / 26 - 77 * a = 31 (8,512 - 5,472 * a) / 26 - 77 * a = 31 (3,040 - 5,472 * a) / 26 = 31 + 77 * a 3,040 - 5,472 * a = 806 + 2,002 * a -7,474 * a = -2,234 a = 0.2985 (rounded to 4 decimal places) Now, we can substitute the value of a back into the first equation to find the value of u: 72 * 0.2985 + 26 * u = 112 21.492 + 26 * u = 112 26 * u = 112 - 21.492 u = 3.4577 (rounded to 4 decimal places) ### Answer: u = 3.4577, a = 0.2985
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. First, let's solve for n in the first equation: 54n + 18b = 143 54n = 143 - 18b n = (143 - 18b) / 54 Now, let's substitute the expression for n into the second equation: 87n - 93b = 48 87 * (143 - 18b) / 54 - 93b = 48 (7761 - 1566b) / 54 - 93b = 48 7761 - 1566b - 5022b = 2592 -6588b = -5169 b = 0.785 Now, let's substitute the value of b back into the first equation to find the value of n: 54n + 18(0.785) = 143 54n + 14.13 = 143 54n = 128.87 n = 2.39 ### Answer: b = 0.785, n = 2.39
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)?
Claude-3 Haiku:
To solve this system of linear equations, we can use the substitution method. Step 1: Solve for e in the first equation. 53 * e + 7 * p = 103 53 * e = 103 - 7 * p e = (103 - 7 * p) / 53 Step 2: Substitute the expression for e in the second equation. 69 * e - 88 * p = 6 69 * ((103 - 7 * p) / 53) - 88 * p = 6 (69 * 103 - 69 * 7 * p) / 53 - 88 * p = 6 (7107 - 483 * p) / 53 - 88 * p = 6 7107 - 483 * p - 4664 * p = 318 -5147 * p = -6789 p = 1.32 Step 3: Substitute the value of p in the first equation to find the value of e. 53 * e + 7 * 1.32 = 103 53 * e + 9.24 = 103 53 * e = 93.76 e = 1.77 Therefore, the cost of electricity (e) is $1.77, and the cost of petroleum (p) is $1.32. ### Answer: Electricity cost (e) = $1.77, Petroleum cost (p) = $1.32
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.
Claude-3 Haiku:
To solve this problem, we need to use the method of substitution or elimination to find the values of p and s that satisfy both linear equations. Let's start by rearranging the first equation to solve for p: (18 * p) - (7 * s) = 21 18 * p = 21 + (7 * s) p = (21 + 7 * s) / 18 Now, we can substitute this expression for p into the second equation: (51 * s) - (24 * ((21 + 7 * s) / 18)) = 14 (51 * s) - (24 * (21 + 7 * s) / 18) = 14 (51 * s) - (24 * 21 / 18) - (24 * 7 * s / 18) = 14 (51 * s) - 28 - (28 * s / 6) = 14 (51 * s) - 28 - (4.67 * s) = 14 (46.33 * s) - 28 = 14 46.33 * s = 42 s = 0.91 Now, we can substitute the value of s back into the first equation to find the value of p: p = (21 + 7 * 0.91) / 18 p = (21 + 6.37) / 18 p = 27.37 / 18 p = 1.52 Therefore, the values of p and s that satisfy both equations are: p = 1.52 s = 0.91 ### Answer: p = 1.52, s = 0.91
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Given equations: 59c - 34w = 84 11c + 42w = 98 Step 1: Solve for c in the first equation. 59c - 34w = 84 59c = 84 + 34w c = (84 + 34w) / 59 Step 2: Substitute the expression for c in the second equation. 11c + 42w = 98 11((84 + 34w) / 59) + 42w = 98 (924 + 374w) / 59 + 42w = 98 924 + 374w + 2478w = 5782 2852w = 4858 w = 4858 / 2852 w = 1.7 Step 3: Substitute the value of w back into the first equation to find c. 59c - 34(1.7) = 84 59c - 57.8 = 84 59c = 141.8 c = 141.8 / 59 c = 2.4 Therefore, the solution to the system of equations is: c = 2.4 w = 1.7 ### Answer: c = 2.4, w = 1.7
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. First equation: 45 * f + 42 * z = 101 Rearranging the first equation to solve for f: 45 * f = 101 - 42 * z f = (101 - 42 * z) / 45 Substituting the expression for f into the second equation: 60 * z - 42 * ((101 - 42 * z) / 45) = 13 60 * z - (4242 - 1764 * z) / 45 = 13 60 * z - 2478 / 45 + 39.2 * z = 13 99.2 * z - 2478 / 45 = 13 99.2 * z = 2491 / 45 z = 25 Substituting the value of z back into the first equation to find the value of f: 45 * f + 42 * 25 = 101 45 * f + 1050 = 101 45 * f = -949 f = -21.09 Rounding the values to two decimal places: f = -21.09 z = 25.00 ### Answer: f = -21.09, z = 25.00
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.
Claude-3 Haiku:
To solve for the values of p and u, we can use the system of linear equations provided. Given: 26 * p - 21 * u = 6 15 * p + 58 * u = 83 Step 1: Solve for p in the first equation. 26 * p - 21 * u = 6 26 * p = 6 + 21 * u p = (6 + 21 * u) / 26 Step 2: Substitute the expression for p in the second equation. 15 * p + 58 * u = 83 15 * ((6 + 21 * u) / 26) + 58 * u = 83 (90 + 315 * u) / 26 + 58 * u = 83 (90 + 315 * u + 1508 * u) / 26 = 83 (90 + 1823 * u) / 26 = 83 90 + 1823 * u = 2158 1823 * u = 2068 u = 2068 / 1823 u = <<u = 1.13>>1.13 Step 3: Substitute the value of u in the expression for p. p = (6 + 21 * u) / 26 p = (6 + 21 * 1.13) / 26 p = (6 + 23.73) / 26 p = 29.73 / 26 p = <<p = 1.14>>1.14 ### Answer: p = 1.14, u = 1.13
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
Claude-3 Haiku:
To solve for the values of n and s using the given simultaneous equations, we can use the substitution method. Equation 1: 92 * n - 38 * s = 62 Equation 2: 59 * n + 8 * s = 75 Step 1: Solve for s in terms of n using Equation 1. 92 * n - 38 * s = 62 38 * s = 92 * n - 62 s = (92 * n - 62) / 38 Step 2: Substitute the expression for s in Equation 2. 59 * n + 8 * ((92 * n - 62) / 38) = 75 59 * n + 8 * (92 * n - 62) / 38 = 75 59 * n + 736 * n - 496 / 38 = 75 795 * n - 496 / 38 = 75 795 * n - 13 = 75 795 * n = 88 n = 88 / 795 n = 0.1107 Step 3: Substitute the value of n in the expression for s to find the value of s. s = (92 * n - 62) / 38 s = (92 * 0.1107 - 62) / 38 s = (10.1044 - 62) / 38 s = -51.8956 / 38 s = -1.3657 Therefore, the values of n and s are: ### Answer: n = 0.1107, s = -1.3657
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.
Claude-3 Haiku:
To solve the given system of simultaneous equations, we can use the substitution method. Given equations: 87 * z - 35 * e = 63 81 * z + 58 * e = 154 Step 1: Solve for e in the first equation. 87 * z - 35 * e = 63 35 * e = 87 * z - 63 e = (87 * z - 63) / 35 Step 2: Substitute the expression for e in the second equation. 81 * z + 58 * ((87 * z - 63) / 35) = 154 81 * z + 58 * (87 * z - 63) / 35 = 154 81 * z + 58 * (87 * z - 63) / 35 = 154 81 * z + 4986 * z - 3654 / 35 = 154 5067 * z - 3654 / 35 = 154 5067 * z - 104 = 154 5067 * z = 258 z = 258 / 5067 z = 0.051 Step 3: Substitute the value of z in the expression for e. e = (87 * 0.051 - 63) / 35 e = (4.437 - 63) / 35 e = -58.563 / 35 e = -1.673 Therefore, the values for e and z are: e = -1.673 z = 0.051 ### Answer: e = -1.673, z = 0.051
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.
Claude-3 Haiku:
To solve the given system of simultaneous linear equations, we can use the substitution method or the elimination method. Using the substitution method: From the first equation, we can express t in terms of h: (11 * t) + (44 * h) = 110 11 * t = 110 - 44 * h t = (110 - 44 * h) / 11 Substituting the expression for t in the second equation: (98 * (110 - 44 * h) / 11) - (36 * h) = 18 (1078 - 4312 * h) / 11 - 36 * h = 18 1078 - 4312 * h - 396 * h = 198 * 11 -4708 * h = 2178 h = -2178 / 4708 h = -0.4625 Substituting the value of h in the expression for t: t = (110 - 44 * (-0.4625)) / 11 t = (110 + 204.5) / 11 t = 314.5 / 11 t = 28.59 Therefore, the values of h and t are: h = -0.4625 t = 28.59 ### Answer: h = -0.4625, t = 28.59
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.
Claude-3 Haiku:
To solve the given system of linear equations, we can use the substitution method. Step 1: Rearrange the first equation to solve for b. (69 * b) - (50 * o) = 55 69b - 50o = 55 b = (55 + 50o) / 69 Step 2: Substitute the expression for b in the second equation. (41 * o) + (18 * ((55 + 50o) / 69)) = 85 (41 * o) + (990 + 900o) / 69 = 85 (41 * o) + (1890 + 900o) / 69 = 85 (41 * o + 1890 + 900o) / 69 = 85 (941 * o + 1890) / 69 = 85 941 * o + 1890 = 5865 941 * o = 3975 o = 4.22 Step 3: Substitute the value of o in the expression for b. b = (55 + 50 * 4.22) / 69 b = (55 + 211) / 69 b = 266 / 69 b = 3.85 Therefore, the values of b and o are: b = 3.85 o = 4.22 ### Answer: b = 3.85, o = 4.22
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Given equations: 67 * q - 80 * p = 65 6 * q + 5 * p = 177 Step 1: Solve for p in the second equation. 6 * q + 5 * p = 177 5 * p = 177 - 6 * q p = (177 - 6 * q) / 5 Step 2: Substitute the expression for p in the first equation. 67 * q - 80 * ((177 - 6 * q) / 5) = 65 67 * q - 80 * (177 / 5 - 6 * q / 5) = 65 67 * q - 80 * (35.4 - 1.2 * q) = 65 67 * q - 2832 + 96 * q = 65 163 * q = 2897 q = 2897 / 163 q = 17.77 (rounded to two decimal places) Step 3: Substitute the value of q in the expression for p. p = (177 - 6 * 17.77) / 5 p = (177 - 106.62) / 5 p = 70.38 / 5 p = 14.08 (rounded to two decimal places) ### Answer: p = 14.08, q = 17.77
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for f. 10 * k + 48 * f = 161 48 * f = 161 - 10 * k f = (161 - 10 * k) / 48 Step 2: Substitute the expression for f into the second equation. 33 * k - 41 * ((161 - 10 * k) / 48) = 31 33 * k - (41 * 161 - 410 * k) / 48 = 31 33 * k - (6601 - 410 * k) / 48 = 31 33 * k - 137.52 + 8.54 * k = 31 41.54 * k = 168.52 k = 4 Step 3: Substitute the value of k into the first equation to find the value of f. 10 * 4 + 48 * f = 161 40 + 48 * f = 161 48 * f = 121 f = 121 / 48 f = 2.52 Therefore, the values of k and f that satisfy the given system of equations are: k = 4 f = 2.52 ### Answer: k = 4, f = 2.52
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
Claude-3 Haiku:
To solve the given system of equations, we need to find the values of l and n. Given equations: 31 = 58 * l - n 103 = n + 73 * l Step 1: Solve for n in the first equation. 31 = 58 * l - n n = 58 * l - 31 Step 2: Substitute the expression for n in the second equation. 103 = n + 73 * l 103 = (58 * l - 31) + 73 * l 103 = 131 * l - 31 Step 3: Solve for l. 103 = 131 * l - 31 134 = 131 * l l = 134 / 131 l = 1.0229 Step 4: Substitute the value of l in the first equation to find the value of n. n = 58 * l - 31 n = 58 * 1.0229 - 31 n = 59.3282 - 31 n = 28.3282 Therefore, the values of l and n are: l = 1.0229 n = 28.3282 ### Answer: l = 1.0229, n = 28.3282
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
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for j. 29y - 28j = 39 28j = 29y - 39 j = (29y - 39) / 28 Step 2: Substitute the expression for j into the second equation. 25y + 37j = 116 25y + 37((29y - 39) / 28) = 116 25y + (1033y - 1443) / 28 = 116 25y + 1033y / 28 - 1443 / 28 = 116 1058y / 28 - 1443 / 28 = 116 1058y - 1443 = 3248 1058y = 4691 y = 4.44 Step 3: Substitute the value of y into the first equation to find the value of j. 29(4.44) - 28j = 39 128.76 - 28j = 39 -28j = -89.76 j = 3.2 Therefore, the values of y and j that satisfy the system of equations are: y = 4.44 j = 3.2 ### Answer: y = 4.44, j = 3.2
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.
Claude-3 Haiku:
To solve for u and a, we can use the substitution method. Step 1: Solve the first equation for a. 93u - 85a = 74 -85a = 74 - 93u a = (74 - 93u) / -85 a = (74 - 93u) / -85 a = (93u - 74) / 85 Step 2: Substitute the expression for a into the second equation. 96u + 17a = 186 96u + 17((93u - 74) / 85) = 186 96u + (1581u - 1258) / 85 = 186 96u + 1581u / 85 - 1258 / 85 = 186 96u + 18.6u - 14.8 = 186 114.6u - 14.8 = 186 114.6u = 200.8 u = 200.8 / 114.6 u = 1.75 Step 3: Substitute the value of u into the expression for a. a = (93u - 74) / 85 a = (93(1.75) - 74) / 85 a = (162.75 - 74) / 85 a = 88.75 / 85 a = 1.04 Therefore, the solution to the system of equations is: u = 1.75 a = 1.04 ### Answer: u = 1.75, a = 1.04
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
Claude-3 Haiku:
To solve for the values of u and g, we can use the method of substitution or elimination. Let's use the method of elimination. Step 1: Multiply the first equation by 23 and the second equation by 61 to eliminate the variable g. 23 * (97 * u - 61 * g = 15) 2231 * u - 1403 * g = 345 61 * (85 * u + 23 * g = 186) 5185 * u + 1403 * g = 11346 Step 2: Subtract the first equation from the second equation to eliminate the variable g. 5185 * u + 1403 * g = 11346 -2231 * u + 1403 * g = -345 ----------------- 2954 * u = 11691 Step 3: Solve for the variable u. u = 11691 / 2954 u = 3.96 Step 4: Substitute the value of u into one of the original equations to find the value of g. Let's use the first equation: 97 * u - 61 * g = 15 97 * 3.96 - 61 * g = 15 384.12 - 61 * g = 15 -61 * g = -369.12 g = 6.05 ### Answer: u = 3.96, g = 6.05
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Let's start by solving the first equation for r: (17 * r) + (67 * b) = 103 17r + 67b = 103 r = (103 - 67b) / 17 Now, we can substitute this expression for r into the second equation: (85 * r) + (67 * b) = 187 85 * ((103 - 67b) / 17) + 67b = 187 (85 * 103 - 85 * 67b) / 17 + 67b = 187 (8755 - 5795b) / 17 + 67b = 187 (2960 - 5795b) / 17 = 120 2960 - 5795b = 2040 -5795b = -920 b = 0.159 Now, we can substitute the value of b back into the first equation to find the value of r: r = (103 - 67b) / 17 r = (103 - 67 * 0.159) / 17 r = (103 - 10.653) / 17 r = 92.347 / 17 r = 5.432 Therefore, the values of r and b that satisfy the given system of equations are: r = 5.432 b = 0.159 ### Answer: r = 5.432, b = 0.159
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for t. (85 * n) - (54 * t) = 41 54 * t = (85 * n) - 41 t = ((85 * n) - 41) / 54 Step 2: Substitute the expression for t in the second equation. (38 * ((85 * n) - 41) / 54) + (30 * n) = 126 (3230 * n - 1558) / 54 + (30 * n) = 126 (3230 * n - 1558 + 1620 * n) / 54 = 126 4850 * n / 54 = 126 n = 126 * 54 / 4850 n = 1.4 Step 3: Substitute the value of n in the first equation to find the value of t. t = ((85 * 1.4) - 41) / 54 t = (119 - 41) / 54 t = 78 / 54 t = 1.44 Therefore, the values of t and n are: t = 1.44 n = 1.4 ### Answer: t = 1.44, n = 1.4
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.
Claude-3 Haiku:
To solve this system of equations, we need to use the substitution method. Given equations: 1) 75u - 69q = 5 2) 48u + 25q = 140 Step 1: Solve for u in the second equation. 48u + 25q = 140 48u = 140 - 25q u = (140 - 25q) / 48 Step 2: Substitute the expression for u in the first equation. 75u - 69q = 5 75 * (140 - 25q) / 48 - 69q = 5 (75 * 140 - 75 * 25q) / 48 - 69q = 5 (10500 - 1875q) / 48 - 69q = 5 10500 - 1875q - 3312q = 240 -5187q = -10260 q = 2 Step 3: Substitute the value of q in the expression for u to find its value. u = (140 - 25q) / 48 u = (140 - 25 * 2) / 48 u = (140 - 50) / 48 u = 90 / 48 u = 1.875 ### Answer: u = 1.875, 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.
Claude-3 Haiku:
To solve this system of equations, we need to find the values of v and x. Given the system of equations: ((20 * v) + (17 * x)) = 149 ((24 * x) - (49 * v)) = 22 Step 1: Solve for v in the first equation. ((20 * v) + (17 * x)) = 149 20v + 17x = 149 20v = 149 - 17x v = (149 - 17x) / 20 Step 2: Substitute the expression for v into the second equation. ((24 * x) - (49 * v)) = 22 ((24 * x) - (49 * (149 - 17x) / 20)) = 22 24x - (7351 - 833x) / 20 = 22 24x - 7351 / 20 + 833x / 20 = 22 857x / 20 - 7351 / 20 = 22 857x - 7351 = 440 857x = 7791 x = 9 Step 3: Substitute the value of x into the expression for v. v = (149 - 17x) / 20 v = (149 - 17(9)) / 20 v = (149 - 153) / 20 v = -4 / 20 v = -0.2 Therefore, the values of v and x are: v = -0.2 x = 9 ### Answer: v = -0.2, x = 9
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Given equations: 39 * t + 41 * g = 147 94 * t - 91 * g = 42 Step 1: Solve for t in the first equation. 39 * t + 41 * g = 147 39 * t = 147 - 41 * g t = (147 - 41 * g) / 39 Step 2: Substitute the expression for t in the second equation. 94 * t - 91 * g = 42 94 * ((147 - 41 * g) / 39) - 91 * g = 42 (94 * 147 - 94 * 41 * g) / 39 - 91 * g = 42 (13818 - 3854 * g) / 39 - 91 * g = 42 13818 - 3854 * g - 3549 * g = 1638 -7403 * g = -12180 g = 1.65 Step 3: Substitute the value of g in the expression for t to find the value of t. t = (147 - 41 * g) / 39 t = (147 - 41 * 1.65) / 39 t = (147 - 67.65) / 39 t = 79.35 / 39 t = 2.03 Therefore, the values of t and g are: t = 2.03 g = 1.65 ### Answer: t = 2.03, g = 1.65
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. First, let's solve for c in the first equation: 6c + 55o = 112 6c = 112 - 55o c = (112 - 55o) / 6 Now, let's substitute this expression for c into the second equation: 55c - 48o = 36 55 * ((112 - 55o) / 6) - 48o = 36 (55 * 112 - 55 * 55o) / 6 - 48o = 36 (6160 - 3025o) / 6 - 48o = 36 (1027 - 504o) - 288o = 216 1027 - 792o = 216 -792o = -811 o = 1.025 Now, let's substitute the value of o back into the first equation to find the value of c: 6c + 55(1.025) = 112 6c + 56.375 = 112 6c = 55.625 c = 9.271 Therefore, the values of c and o that satisfy the system of equations are: c = 9.271 o = 1.025 ### Answer: c = 9.271, o = 1.025
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.
Claude-3 Haiku:
To solve this system of linear equations, we can use the substitution method. Step 1: Solve for u in the first equation. 8w + 68u = 170 68u = 170 - 8w u = (170 - 8w) / 68 Step 2: Substitute the expression for u into the second equation. 71w - 94u = 8 71w - 94((170 - 8w) / 68) = 8 71w - (94 * 170 - 8 * 94w) / 68 = 8 71w - (16,180 - 752w) / 68 = 8 71w - 238 + 11w = 8 82w = 246 w = 3 Step 3: Substitute the value of w into the expression for u to find the value of u. u = (170 - 8w) / 68 u = (170 - 8 * 3) / 68 u = (170 - 24) / 68 u = 146 / 68 u = 2.15 Therefore, the values of u and w that satisfy the given system of equations are: u = 2.15 w = 3 ### Answer: u = 2.15, w = 3
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
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Step 1: Solve for one variable in terms of the other. From the first equation: 68*r + 8*f = 92 8*f = 92 - 68*r f = (92 - 68*r)/8 Step 2: Substitute the expression for f into the second equation. 97*f + 38*r = 182 97*((92 - 68*r)/8) + 38*r = 182 (97*92 - 97*68*r)/8 + 38*r = 182 (8924 - 6596*r)/8 + 38*r = 182 1128 - 824.5*r + 304*r = 1456 -520.5*r = 328 r = -0.63 Step 3: Substitute the value of r back into the first equation to find the value of f. 68*r + 8*f = 92 68*(-0.63) + 8*f = 92 -42.84 + 8*f = 92 8*f = 134.84 f = 16.85 Therefore, the values of r and f are: r = -0.63 f = 16.85 ### Answer: r = -0.63, f = 16.85
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).
Claude-3 Haiku:
To solve this problem, we need to use the two given equations to find the values of r and d. Given equations: 1. 67d - 37r = 50 2. 39r - 21d = 16 Step 1: Solve for r in the second equation. 39r - 21d = 16 39r = 16 + 21d r = (16 + 21d) / 39 Step 2: Substitute the expression for r into the first equation. 67d - 37r = 50 67d - 37((16 + 21d) / 39) = 50 67d - (592 + 777d) / 39 = 50 2639d - 592 - 777d = 1950 1862d = 2542 d = 1.37 Step 3: Substitute the value of d into the expression for r to find the value of r. r = (16 + 21d) / 39 r = (16 + 21(1.37)) / 39 r = (16 + 28.77) / 39 r = 44.77 / 39 r = 1.15 Therefore, the values of d (the cost of each apple) and r (the cost of each ragdoll) are: d = $1.37 r = $1.15 ### Answer: d = $1.37, r = $1.15
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.
Claude-3 Haiku:
To solve this system of simultaneous linear equations, we need to find the values of x and a. Given: First equation: 163 = 50x + a Second equation: 24 = 98x - 84a Step 1: Solve for a in the second equation. 24 = 98x - 84a 84a = 98x - 24 a = (98x - 24) / 84 Step 2: Substitute the expression for a into the first equation. 163 = 50x + (98x - 24) / 84 163 = 50x + 98x / 84 - 24 / 84 163 = 148x / 84 - 24 / 84 163 = (148x - 24) / 84 13,684 = 148x - 24 13,708 = 148x x = 92.5 Step 3: Substitute the value of x into the expression for a. a = (98x - 24) / 84 a = (98 × 92.5 - 24) / 84 a = (9,065 - 24) / 84 a = 9,041 / 84 a = 107.5 Therefore, the values of x and a are: x = 92.5 a = 107.5 ### Answer: x = 92.5, a = 107.5
Expected answer: {a: 7387/2149, x: 6858/2149}
Correct? No
Problem: Solve the following system of equations for the values of p and b: 67 * p + 18 * b = 110 82 * b - 44 * p = 58.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Step 1: Solve the first equation for p. 67 * p + 18 * b = 110 67 * p = 110 - 18 * b p = (110 - 18 * b) / 67 Step 2: Substitute the expression for p in the second equation. 82 * b - 44 * p = 58 82 * b - 44 * (110 - 18 * b) / 67 = 58 82 * b - 4400 + 792 * b = 3886 874 * b = 8286 b = 9.5 Step 3: Substitute the value of b in the expression for p. p = (110 - 18 * 9.5) / 67 p = (110 - 171) / 67 p = -61 / 67 Therefore, the values of p and b are: p = -61 / 67 b = 9.5 ### Answer: p = -61 / 67, b = 9.5
Expected answer: {b: 4363/3143, p: 3988/3143}
Correct? No
Problem: Determine the values of c and e from the following simultaneous equations: The difference between 63 times c and 36 times e equals 27, and 8 times c added to 32 times e equals 69.
Claude-3 Haiku:
To solve this problem, we need to set up a system of two linear equations with two unknowns, c and e, and then solve for the values of c and e. Given information: - The difference between 63 times c and 36 times e equals 27. - 8 times c added to 32 times e equals 69. Let's represent the equations as follows: Equation 1: 63c - 36e = 27 Equation 2: 8c + 32e = 69 Step 1: Solve for c in terms of e using Equation 2. 8c + 32e = 69 8c = 69 - 32e c = (69 - 32e) / 8 Step 2: Substitute the expression for c into Equation 1 and solve for e. 63c - 36e = 27 63((69 - 32e) / 8) - 36e = 27 (4347 - 2016e) / 8 - 36e = 27 4347 - 2016e - 288e = 216 4347 - 2304e = 216 -2304e = -4131 e = 1.8 Step 3: Substitute the value of e into the expression for c to find the value of c. c = (69 - 32e) / 8 c = (69 - 32(1.8)) / 8 c = (69 - 57.6) / 8 c = 11.4 / 8 c = 1.425 Therefore, the values of c and e are: c = 1.425 e = 1.8 ### Answer: c = 1.425, e = 1.8
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
Claude-3 Haiku:
To solve the given system of linear equations, we can use the elimination method. Step 1: Multiply the first equation by 51 and the second equation by 29 to eliminate the variable p. 51 * (26x + 29p = 107) => 1326x + 1479p = 5457 29 * (61x - 51p = 4) => 1769x - 1479p = 116 Step 2: Subtract the second equation from the first equation to eliminate the variable p. 1326x + 1479p = 5457 - (1769x - 1479p = 116) -443x = 5341 Step 3: Solve for x. -443x = 5341 x = -5341/443 x = -12 Step 4: Substitute the value of x into one of the original equations to find the value of p. Using the first equation: 26x + 29p = 107 26(-12) + 29p = 107 -312 + 29p = 107 29p = 419 p = 419/29 p = 14.4482758620689 Therefore, the values of x and p are: x = -12 p = 14.4482758620689 ### Answer: x = -12, p = 14.4482758620689
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for one of the variables, let's say l. (29 * l) + (17 * g) = 68 29 * l = 68 - 17 * g l = (68 - 17 * g) / 29 Step 2: Substitute the expression for l into the second equation. (31 * g) - (10 * l) = 17 (31 * g) - (10 * (68 - 17 * g) / 29) = 17 (31 * g) - (680 - 170 * g) / 29 = 17 (31 * g - 680 + 170 * g) / 29 = 17 (201 * g - 680) / 29 = 17 201 * g - 680 = 493 201 * g = 1173 g = 5.83 Step 3: Substitute the value of g into the expression for l to find the value of l. l = (68 - 17 * g) / 29 l = (68 - 17 * 5.83) / 29 l = (68 - 99.11) / 29 l = -31.11 / 29 l = -1.07 Therefore, the values for g and l are: g = 5.83 l = -1.07 ### Answer: g = 5.83, l = -1.07
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.
Claude-3 Haiku:
To solve for the values of a and c, we can use the method of substitution or elimination. Let's use the method of elimination. First, we will multiply the first equation by 26 and the second equation by 54 to eliminate the variable c. Multiplying the first equation by 26: 26(33*a - 54*c = 20) 858*a - 1404*c = 520 Multiplying the second equation by 54: 54(24*a - 26*c = 29) 1296*a - 1404*c = 1566 Now, we subtract the two equations to eliminate the variable c: 1296*a - 1404*c - 858*a + 1404*c = 1566 - 520 438*a = 1046 a = 1046/438 a = 2.39 Now, we can substitute the value of a into either of the original equations to solve for c. Using the first equation: 33*2.39 - 54*c = 20 78.87 - 54*c = 20 -54*c = -58.87 c = 1.09 Therefore, the values for a and c are: a = 2.39 c = 1.09 ### Answer: a = 2.39, c = 1.09
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.
Claude-3 Haiku:
Okay, let's solve this system of equations with two variables, d and j. To solve this system, we can use the elimination method. First, let's multiply the first equation by 84 and the second equation by 71 to eliminate one of the variables. Multiplying the first equation by 84: 84(71d + 9j) = 84(132) 5964d + 756j = 11088 Multiplying the second equation by 71: 71(43j - 84d) = 71(91) 3053j - 5964d = 6461 Now, we can subtract the second equation from the first equation to eliminate the variable d. 5964d + 756j = 11088 -3053j + 5964d = -6461 -------------- 756j = 17549 Solving for j: j = 17549 / 756 j = 23.2 Now, we can substitute the value of j into either of the original equations to solve for d. Let's use the first equation: 71d + 9(23.2) = 132 71d + 208.8 = 132 71d = -76.8 d = -1.08 Therefore, the solution to the system of equations is: d = -1.08 j = 23.2 ### Answer: d = -1.08, j = 23.2
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Given the system of equations: (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143 Step 1: Solve for o in the first equation. (81 * o) + (26 * n) = 163 81 * o = 163 - 26 * n o = (163 - 26 * n) / 81 Step 2: Substitute the expression for o in the second equation. (87 * n) + (23 * o) = 143 (87 * n) + (23 * (163 - 26 * n) / 81) = 143 (87 * n) + (3739 - 598 * n) / 81 = 143 (87 * n) + 3739 - 598 * n = 143 * 81 (87 * n) - 598 * n + 3739 = 11583 -511 * n + 3739 = 11583 -511 * n = 7844 n = -7844 / 511 n = -15.35 Step 3: Substitute the value of n in the expression for o. o = (163 - 26 * n) / 81 o = (163 - 26 * (-15.35)) / 81 o = (163 + 398.1) / 81 o = 561.1 / 81 o = 6.93 Therefore, the values of o and n are: o = 6.93 n = -15.35 ### Answer: o = 6.93, n = -15.35
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Step 1: Solve the first equation for s. 46 * s - 61 * w = 2 46 * s = 61 * w + 2 s = (61 * w + 2) / 46 Step 2: Substitute the expression for s in the second equation. 61 * s + 67 * w = 178 61 * ((61 * w + 2) / 46) + 67 * w = 178 (61^2 * w + 122) / 46 + 67 * w = 178 (3721 * w + 122) / 46 + 67 * w = 178 3721 * w + 122 + 2082 * w = 8228 5803 * w = 8106 w = 1.4 Step 3: Substitute the value of w in the first equation to find the value of s. 46 * s - 61 * 1.4 = 2 46 * s - 85.4 = 2 46 * s = 87.4 s = 1.9 Therefore, the solution to the system of equations is: w = 1.4 s = 1.9 ### Answer: w = 1.4, s = 1.9
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve for i in the first equation. 51*k - 8*i = 86 8*i = 51*k - 86 i = (51*k - 86)/8 Step 2: Substitute the expression for i in the second equation. 39*i + 8*k = 193 39*((51*k - 86)/8) + 8*k = 193 (39*51*k - 39*86)/8 + 8*k = 193 (1989*k - 3354)/8 + 8*k = 193 1989*k - 3354 + 64*k = 1544 2053*k = 4898 k = 2.39 Step 3: Substitute the value of k in the expression for i. i = (51*k - 86)/8 i = (51*2.39 - 86)/8 i = (122.89 - 86)/8 i = 36.89/8 i = 4.61 ### Answer: i = 4.61, k = 2.39
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
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Step 1: Solve the second equation for l. 93m + l = 126 l = 126 - 93m Step 2: Substitute the expression for l in the first equation. 26m + 17(126 - 93m) = 146 26m + 2142 - 1581m = 146 -1555m + 2142 = 146 -1555m = -1996 m = 1.28 Step 3: Substitute the value of m in the second equation to find the value of l. l = 126 - 93(1.28) l = 126 - 119.04 l = 6.96 Therefore, the values of l and m are: l = 6.96 m = 1.28 ### Answer: l = 6.96, m = 1.28
Expected answer: {l: 10302/1555, m: 1996/1555}
Correct? No
Problem: Determine the values of 'h' and 'f' by solving the following system of equations: 1) (8 * f) + (39 * h) = 66 2) (29 * f) + (46 * h) = 119
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve for 'f' in the first equation. (8 * f) + (39 * h) = 66 8 * f = 66 - (39 * h) f = (66 - 39 * h) / 8 Step 2: Substitute the expression for 'f' into the second equation. (29 * f) + (46 * h) = 119 (29 * (66 - 39 * h) / 8) + (46 * h) = 119 (1914 - 1131 * h) / 8 + 46 * h = 119 1914 - 1131 * h + 368 * h = 952 -763 * h = -962 h = 1.26 Step 3: Substitute the value of 'h' into the expression for 'f' to find its value. f = (66 - 39 * h) / 8 f = (66 - 39 * 1.26) / 8 f = (66 - 49.14) / 8 f = 16.86 / 8 f = 2.11 Therefore, the values of 'h' and 'f' are: h = 1.26 f = 2.11 ### Answer: h = 1.26, f = 2.11
Expected answer: {f: 1605/763, h: 962/763}
Correct? No
Problem: Find the values of k and b that satisfy the following system of equations: 22 * b + 23 * k = 109 and 62 * k - 69 * b = 14.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. First, let's solve the first equation for b: 22 * b + 23 * k = 109 22 * b = 109 - 23 * k b = (109 - 23 * k) / 22 Now, let's substitute this expression for b into the second equation: 62 * k - 69 * b = 14 62 * k - 69 * ((109 - 23 * k) / 22) = 14 62 * k - 69 * (109 / 22 - 23 * k / 22) = 14 62 * k - 69 * (4.95 - 1.045 * k) = 14 62 * k - 341.55 + 72.105 * k = 14 134.105 * k = 355.55 k = 2.65 Now, let's substitute the value of k back into the first equation to find the value of b: 22 * b + 23 * 2.65 = 109 22 * b + 61.45 = 109 22 * b = 47.55 b = 2.16 Therefore, the values of k and b that satisfy the system of equations are: k = 2.65 b = 2.16 ### Answer: k = 2.65, b = 2.16
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.
Claude-3 Haiku:
To solve the given system of linear equations, we can use the substitution method. Given equations: 1. 84m - 55s = 1 2. 63m - 2s = 98 Step 1: Solve for s in the second equation. 63m - 2s = 98 2s = 63m - 98 s = (63m - 98) / 2 Step 2: Substitute the expression for s in the first equation. 84m - 55[(63m - 98) / 2] = 1 84m - (55/2)(63m - 98) = 1 84m - (27.5m - 53.9) = 1 84m - 27.5m + 53.9 = 1 56.5m = -52.9 m = -52.9 / 56.5 m = -0.936 Step 3: Substitute the value of m in the expression for s to find the value of s. s = (63m - 98) / 2 s = (63(-0.936) - 98) / 2 s = (-58.968 - 98) / 2 s = -156.968 / 2 s = -78.484 Therefore, the values of m and s are: m = -0.936 s = -78.484 ### Answer: m = -0.936, s = -78.484
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for b. (6 * b) + (5 * k) = 105 6b + 5k = 105 b = (105 - 5k) / 6 Step 2: Substitute the expression for b into the second equation. (95 * k) - (51 * b) = 67 (95 * k) - (51 * (105 - 5k) / 6) = 67 (95 * k) - (5355 - 255k) / 6 = 67 (95 * k) - 892.5 + 42.5k = 67 137.5k = 960 k = 7 Step 3: Substitute the value of k into the first equation to find the value of b. b = (105 - 5k) / 6 b = (105 - 5 * 7) / 6 b = (105 - 35) / 6 b = 70 / 6 b = 11.67 (rounded to two decimal places) Therefore, the values of b and k are: b = 11.67 k = 7 ### Answer: b = 11.67, k = 7
Expected answer: {b: 1928/165, k: 1919/275}
Correct? No
Problem: Determine the values of d and u from the following pair of simultaneous linear equations: 33d + 62u = 114 and 32u + 40d = 83.
Claude-3 Haiku:
To solve this system of simultaneous linear equations, we can use the substitution method. Step 1: Solve for d in the first equation. 33d + 62u = 114 33d = 114 - 62u d = (114 - 62u) / 33 Step 2: Substitute the expression for d into the second equation. 32u + 40d = 83 32u + 40((114 - 62u) / 33) = 83 32u + (4560 - 2480u) / 33 = 83 32u + 2080 / 33 - 75.15u = 83 -43.15u = 83 - 2080 / 33 -43.15u = 83 - 63.03 -43.15u = 19.97 u = -19.97 / 43.15 u = -0.46 Step 3: Substitute the value of u into the expression for d to find the value of d. d = (114 - 62u) / 33 d = (114 - 62(-0.46)) / 33 d = (114 + 28.52) / 33 d = 142.52 / 33 d = 4.32 Therefore, the values of d and u are: d = 4.32 u = -0.46 ### Answer: d = 4.32, u = -0.46
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
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for c. 5 * c + 28 * p = 102 5 * c = 102 - 28 * p c = (102 - 28 * p) / 5 Step 2: Substitute the expression for c into the second equation. 20 * c - 67 * p = 13 20 * ((102 - 28 * p) / 5) - 67 * p = 13 (2040 - 560 * p) / 5 - 67 * p = 13 2040 - 560 * p - 335 * p = 65 2040 - 895 * p = 65 -895 * p = -1975 p = 2.21 Step 3: Substitute the value of p back into the first equation to find the value of c. c = (102 - 28 * 2.21) / 5 c = (102 - 61.88) / 5 c = 40.12 / 5 c = 8.024 ### Answer: c = 8.024, p = 2.21
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.
Claude-3 Haiku:
To solve this system of linear equations, we can use the substitution method. Step 1: Solve for r in the first equation. 58 * r + 12 * q = 147 58 * r = 147 - 12 * q r = (147 - 12 * q) / 58 Step 2: Substitute the expression for r in the second equation. 97 * q - 76 * r = 76 97 * q - 76 * (147 - 12 * q) / 58 = 76 97 * q - 11172 + 912 * q = 4408 1009 * q = 15580 q = 15580 / 1009 q = 15.45 Step 3: Substitute the value of q back into the expression for r. r = (147 - 12 * 15.45) / 58 r = (147 - 185.4) / 58 r = -38.4 / 58 r = -0.66 Therefore, the values of r and q are: r = -0.66 q = 15.45 ### Answer: r = -0.66, q = 15.45
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.
Claude-3 Haiku:
To solve the system of simultaneous linear equations, we can use the elimination method. Given equations: 47 * v - 3 * k = 10 73 * v + 2 * k = 128 Step 1: Multiply the first equation by 2 and the second equation by 3 to eliminate the variable k. 94 * v - 6 * k = 20 219 * v + 6 * k = 384 Step 2: Subtract the first equation from the second equation to eliminate the variable k. 125 * v = 364 Step 3: Solve for the variable v. v = 364 / 125 v = 2.912 Step 4: Substitute the value of v into one of the original equations to solve for the variable k. Let's use the first equation: 47 * v - 3 * k = 10 47 * 2.912 - 3 * k = 10 137.064 - 3 * k = 10 -3 * k = -127.064 k = 42.355 Therefore, the solution to the system of equations is: v = 2.912 k = 42.355 ### Answer: v = 2.912, k = 42.355
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?
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Let's start by solving the first equation for p: 89p - 41q = 60 p = (60 + 41q) / 89 Now, we can substitute this expression for p into the second equation: 27(60 + 41q) / 89 + 91q = 124 1620 + 1107q + 2457q = 11,036 3564q = 9,416 q = 2.64 Now, we can substitute the value of q back into the first equation to find the value of p: p = (60 + 41(2.64)) / 89 p = (60 + 108.24) / 89 p = 168.24 / 89 p = 1.89 ### Answer: p = 1.89, q = 2.64
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
Claude-3 Haiku:
To find the values of u and d that make the given equations true, we need to solve the system of linear equations. Let's start with the first equation: (88 * u) - (96 * d) = 12 Rearranging the terms, we get: 88u - 96d = 12 Now, let's look at the second equation: (27 * u) + (36 * d) = 71 Solving the system of equations: From the first equation, we can express d in terms of u: d = (88u - 12) / 96 Substituting this expression for d in the second equation, we get: 27u + 36((88u - 12) / 96) = 71 27u + 33.6u - 12.6 = 71 60.6u = 83.6 u = 83.6 / 60.6 u = 1.38 Now, we can substitute the value of u back into the first equation to find the value of d: d = (88 * 1.38 - 12) / 96 d = (121.44 - 12) / 96 d = 109.44 / 96 d = 1.14 Therefore, the values of u and d that make the equations true are: u = 1.38 d = 1.14 ### Answer: u = 1.38, d = 1.14
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Given equations: 20 * n - 18 * f = 9 3 * n - 2 * f = 70 Step 1: Solve for f in the second equation. 3 * n - 2 * f = 70 2 * f = 3 * n - 70 f = (3 * n - 70) / 2 Step 2: Substitute the expression for f in the first equation. 20 * n - 18 * ((3 * n - 70) / 2) = 9 20 * n - 27 * n + 630 = 9 -7 * n + 630 = 9 -7 * n = -621 n = 89 Step 3: Substitute the value of n in the expression for f to find the value of f. f = (3 * 89 - 70) / 2 f = (267 - 70) / 2 f = 197 / 2 f = 98.5 Therefore, the values of f and n are: f = 98.5 n = 89 ### Answer: f = 98.5, n = 89
Expected answer: {f: 1373/14, n: 621/7}
Correct? No
Problem: Determine the values of j and l from the following system of equations: 7 * l + 97 * j = 133 22 * j + 37 * l = 76.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. First, let's solve for j in the second equation: 22 * j + 37 * l = 76 22 * j = 76 - 37 * l j = (76 - 37 * l) / 22 Now, let's substitute this expression for j into the first equation: 7 * l + 97 * ((76 - 37 * l) / 22) = 133 7 * l + 97 * (76 / 22 - 37 * l / 22) = 133 7 * l + 97 * (3.45 - 1.68 * l) = 133 7 * l + 335.65 - 163.16 * l = 133 -156.16 * l = -202.65 l = 1.3 Now, let's substitute the value of l into the expression for j: j = (76 - 37 * 1.3) / 22 j = (76 - 48.1) / 22 j = 27.9 / 22 j = 1.27 ### Answer: j = 1.27, l = 1.3
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.
Claude-3 Haiku:
To solve this system of equations, we can use the elimination method. Step 1: Multiply the first equation by 8 and the second equation by 21 to eliminate the variable d. 8(21d - 46f = 53) 168d - 368f = 424 21(8d + 84f = 133) 168d + 1764f = 2793 Step 2: Subtract the first equation from the second equation to eliminate the variable d. 1764f - 368f = 2793 - 424 1396f = 2369 Step 3: Solve for the variable f. f = 2369 / 1396 f = 1.7 Step 4: Substitute the value of f into one of the original equations to solve for the variable d. Let's use the first equation: 21d - 46f = 53 Substituting f = 1.7, we get: 21d - 46(1.7) = 53 21d - 78.2 = 53 21d = 131.2 d = 6.25 Therefore, the values of d and f are: d = 6.25 f = 1.7 ### Answer: d = 6.25, f = 1.7
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.
Claude-3 Haiku:
To solve this system of linear equations, we can use the elimination method. Step 1: Multiply the first equation by 66 and the second equation by 74 to eliminate one of the variables. 66(74u + 11w = 116) 4,884u + 726w = 7,656 74(66u - 68w = 21) 4,884u - 5,032w = 1,554 Step 2: Subtract the second equation from the first equation to eliminate the u variable. 4,884u + 726w = 7,656 4,884u - 5,032w = 1,554 --------------------------------------------------------------------------- 1,758w = 6,102 Step 3: Solve for w. w = 6,102 / 1,758 w = 3.47 Step 4: Substitute the value of w into one of the original equations to solve for u. Let's use the first equation: 74u + 11w = 116 74u + 11(3.47) = 116 74u + 38.17 = 116 74u = 77.83 u = 1.05 Therefore, the values of u and w that satisfy the given system of equations are: u = 1.05 w = 3.47 ### Answer: u = 1.05, w = 3.47
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for f. 76 * f - 42 * j = 96 76 * f = 96 + 42 * j f = (96 + 42 * j) / 76 Step 2: Substitute the 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 + 42 * j / 76) = 136 47 * j + 3264 / 76 + 1428 * j / 76 = 136 47 * j + 4692 / 76 = 136 47 * j + 61.7 = 136 47 * j = 74.3 j = 1.58 Step 3: Substitute the value of j into the expression for f. f = (96 + 42 * j) / 76 f = (96 + 42 * 1.58) / 76 f = (96 + 66.36) / 76 f = 162.36 / 76 f = 2.14 Therefore, the values of f and j that satisfy the system of equations are: f = 2.14 j = 1.58 ### Answer: f = 2.14, j = 1.58
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve for u in the first equation. 68u + 32e = 174 68u = 174 - 32e u = (174 - 32e) / 68 Step 2: Substitute the expression for u into the second equation. 63e - 69u = 1 63e - 69((174 - 32e) / 68) = 1 63e - (174 - 32e) = 68 63e - 174 + 32e = 68 95e = 242 e = 242 / 95 e = <<242/95>>2.54 Step 3: Substitute the value of e into the expression for u to find the value of u. u = (174 - 32e) / 68 u = (174 - 32(2.54)) / 68 u = (174 - 81.28) / 68 u = 92.72 / 68 u = <<92.72/68>>1.36 Therefore, the values of u and e are: u = 1.36 e = 2.54 ### Answer: u = 1.36, e = 2.54
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.
Claude-3 Haiku:
To solve this system of linear equations, we can use the elimination method. Step 1: Multiply the first equation by 24 and the second equation by 57 to eliminate one of the variables. 24(57p + 4n = 129) 1368p + 96n = 3096 57(24p + 39n = 154) 1368p + 2223n = 8778 Step 2: Subtract the second equation from the first equation to eliminate the p variable. (1368p + 96n = 3096) - (1368p + 2223n = 8778) -2127n = -5682 n = 2.67 Step 3: Substitute the value of n into one of the original equations to find the value of p. Using the first equation: 57p + 4(2.67) = 129 57p + 10.68 = 129 57p = 118.32 p = 2.08 Therefore, the values of p and n that satisfy the given system of linear equations are: p = 2.08 n = 2.67 ### Answer: p = 2.08, n = 2.67
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.
Claude-3 Haiku:
To solve for the values of y and z, we can use the given system of linear equations. Given equations: 66 * z - 44 * y = 32 60 * y + 14 * z = 116 Step 1: Solve for y in the second equation. 60 * y + 14 * z = 116 60 * y = 116 - 14 * z y = (116 - 14 * z) / 60 Step 2: Substitute the expression for y into the first equation. 66 * z - 44 * ((116 - 14 * z) / 60) = 32 66 * z - 44 * (116 / 60 - 14 * z / 60) = 32 66 * z - 44 * (1.933 - 0.233 * z) = 32 66 * z - 85.052 + 10.252 * z = 32 76.252 * z - 85.052 = 32 76.252 * z = 117.052 z = 117.052 / 76.252 z = 1.536 Step 3: Substitute the value of z into the expression for y to find the value of y. y = (116 - 14 * 1.536) / 60 y = (116 - 21.504) / 60 y = 94.496 / 60 y = 1.575 Therefore, the values of y and z are: y = 1.575 z = 1.536 ### Answer: y = 1.575, z = 1.536
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. First, let's solve the first equation for t: 83 * t - 58 * h = 45 83 * t = 45 + 58 * h t = (45 + 58 * h) / 83 Now, let's substitute this expression for t into the second equation: 69 * h + 11 * (45 + 58 * h) / 83 = 124 69 * h + 11 * (45 / 83 + 58 * h / 83) = 124 69 * h + 11 * (45 / 83 + 58 * h / 83) = 124 69 * h + 11 * (45 / 83) + 11 * (58 * h / 83) = 124 69 * h + 11 * (45 / 83) + 638 * h / 83 = 124 69 * h + 11 * (45 / 83) + 638 * h / 83 - 124 = 0 (69 + 638 / 83) * h + 11 * (45 / 83) - 124 = 0 707 * h / 83 + 495 / 83 - 124 = 0 707 * h / 83 = 124 - 495 / 83 707 * h / 83 = -371 / 83 h = -371 / (707 / 83) h = -371 / 8.5 h = -43.6 Now, let's substitute the value of h into the first equation to find the value of t: 83 * t - 58 * (-43.6) = 45 83 * t + 2528.8 = 45 83 * t = 45 - 2528.8 t = (45 - 2528.8) / 83 t = -29.9 ### Answer: t = -29.9, h = -43.6
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?
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Let's start by solving for h in the second equation: (16 * h) - (12 * o) = 9 16h - 12o = 9 h = (9 + 12o) / 16 Now, let's substitute this expression for h into the first equation: (79 * ((9 + 12o) / 16)) - (61 * o) = 9 (79 * (9 + 12o) / 16) - (61 * o) = 9 (711 + 948o) / 16 - 61o = 9 711 + 948o - 976o = 144 948o - 976o = -567 -28o = -567 o = 20.25 Now, let's substitute the value of o back into the expression for h: h = (9 + 12o) / 16 h = (9 + 12 * 20.25) / 16 h = (9 + 243) / 16 h = 252 / 16 h = 15.75 ### Answer: o = 20.25, h = 15.75
Expected answer: {h: 63/4, o: 81/4}
Correct? No
Problem: Solve for variables s and y when the following system of equations is satisfied: 4 * y + 56 * s = 139 65 * y - 90 * s = 16
Claude-3 Haiku:
To solve for the variables s and y, we need to use the given system of equations and solve for them. Given equations: 4 * y + 56 * s = 139 65 * y - 90 * s = 16 Step 1: Solve for s in the first equation. 4 * y + 56 * s = 139 56 * s = 139 - 4 * y s = (139 - 4 * y) / 56 Step 2: Substitute the expression for s in the second equation. 65 * y - 90 * s = 16 65 * y - 90 * (139 - 4 * y) / 56 = 16 65 * y - 90 * (139 / 56 - 4 * y / 56) = 16 65 * y - 90 * (2.482142857 - 0.0714285714 * y) = 16 65 * y - 223.3928571 + 6.428571429 * y = 16 71.42857143 * y = 239.3928571 y = 3.35 Step 3: Substitute the value of y in the expression for s. s = (139 - 4 * y) / 56 s = (139 - 4 * 3.35) / 56 s = (139 - 13.4) / 56 s = 125.6 / 56 s = 2.25 Therefore, the values of the variables s and y are: s = 2.25 y = 3.35 ### Answer: s = 2.25, y = 3.35
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.
Claude-3 Haiku:
To solve the given pair of simultaneous equations, we can use the elimination method. Step 1: Multiply the first equation by 40 and the second equation by 42 to eliminate the variable d. 40 * (42 * d - 4 * z = 75) 1680 * d - 160 * z = 3000 42 * (40 * d + 45 * z = 162) 1680 * d + 1890 * z = 6804 Step 2: Subtract the first equation from the second equation to eliminate the variable d. 1890 * z - (-160 * z) = 6804 - 3000 2050 * z = 3804 z = 3804 / 2050 z = 1.86 Step 3: Substitute the value of z into one of the original equations to find the value of d. Let's use the first equation: 42 * d - 4 * 1.86 = 75 42 * d - 7.44 = 75 42 * d = 82.44 d = 82.44 / 42 d = 1.96 Therefore, the values of d and z are: d = 1.96 z = 1.86 ### Answer: d = 1.96, z = 1.86
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.
Claude-3 Haiku:
To solve this system of equations, we need to find the values of 'a' and 'o'. Given equations: (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 Step 1: Solve for 'o' in the first equation. (42 * o) + (24 * a) = 142 42 * o = 142 - 24 * a o = (142 - 24 * a) / 42 Step 2: Substitute the expression for 'o' into the second equation. (42 * a) + (6 * ((142 - 24 * a) / 42)) = 75 (42 * a) + (6 * (142 / 42 - 24 * a / 42)) = 75 (42 * a) + (6 * (3.38 - 0.57 * a)) = 75 (42 * a) + (20.28 - 3.42 * a) = 75 38.58 * a = 54.72 a = 54.72 / 38.58 a = 1.42 Step 3: Substitute the value of 'a' into the expression for 'o' to find its value. o = (142 - 24 * a) / 42 o = (142 - 24 * 1.42) / 42 o = (142 - 34.08) / 42 o = 107.92 / 42 o = 2.57 Therefore, the values of 'a' and 'o' are: a = 1.42 o = 2.57 ### Answer: a = 1.42, o = 2.57
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. First, let's solve the first equation for p: 39*r - 2*p = 45 2*p = 39*r - 45 p = (39*r - 45)/2 Now, let's substitute this expression for p into the second equation: 16*p - 84*r = 36 16*((39*r - 45)/2) - 84*r = 36 (16*39*r - 16*45)/2 - 84*r = 36 (624*r - 720)/2 - 84*r = 36 312*r - 360 - 168*r = 72 144*r = 432 r = 3 Now, let's substitute the value of r back into the first equation to find the value of p: 39*3 - 2*p = 45 117 - 2*p = 45 -2*p = -72 p = 36 Therefore, the values of p and r are: p = 36 r = 3 ### Answer: p = 36, r = 3
Expected answer: {p: 216/19, r: 33/19}
Correct? No
Problem: Given the following system of equations, solve for the values of 'd' and 'w': 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58
Claude-3 Haiku:
To solve for the values of 'd' and 'w', we can use the substitution method. Step 1: Rearrange the second equation to solve for 'd' in terms of 'w'. (64 * w) - (8 * d) = 58 8 * d = (64 * w) - 58 d = ((64 * w) - 58) / 8 Step 2: Substitute the expression for 'd' into the first equation. (23 * ((64 * w) - 58) / 8) - (52 * w) = 7 (23 * (64 * w - 58)) / 8 - (52 * w) = 7 (1472 * w - 1334) / 8 - (52 * w) = 7 (1472 * w - 1334 - 416 * w) / 8 = 7 (1056 * w - 1334) / 8 = 7 1056 * w - 1334 = 56 1056 * w = 1390 w = 1.32 Step 3: Substitute the value of 'w' into the expression for 'd' to find the value of 'd'. d = ((64 * 1.32) - 58) / 8 d = (84.48 - 58) / 8 d = 26.48 / 8 d = 3.31 Therefore, the values of 'd' and 'w' are: d = 3.31 w = 1.32 ### Answer: d = 3.31, w = 1.32
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.
Claude-3 Haiku:
To solve the given system of equations, we can use the substitution method. Given equations: (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19 Step 1: Solve for n in the second equation. (52 * z) - (47 * n) = 19 -47 * n = (52 * z) - 19 n = ((52 * z) - 19) / -47 n = ((52 * z) - 19) / 47 Step 2: Substitute the expression for n in the first equation. (71 * z) + (60 * ((52 * z) - 19) / 47) = 184 (71 * z) + (3120 * z - 1140) / 47 = 184 (71 * z) + (3120 * z - 1140) / 47 = 184 (71 * z) + (66.38 * z - 24.26) = 184 137.38 * z - 24.26 = 184 137.38 * z = 208.26 z = 208.26 / 137.38 z = 1.52 Step 3: Substitute the value of z in the expression for n. n = ((52 * 1.52) - 19) / 47 n = (79.04 - 19) / 47 n = 60.04 / 47 n = 1.28 Therefore, the values of n and z are: n = 1.28 z = 1.52 ### Answer: n = 1.28, z = 1.52
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
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for e. 70 * e + 90 * z = 189 70 * e = 189 - 90 * z e = (189 - 90 * z) / 70 Step 2: Substitute the expression for e into the second equation. 99 * z - 84 * e = 20 99 * z - 84 * ((189 - 90 * z) / 70) = 20 99 * z - (2646 - 7560 * z) / 70 = 20 99 * z - 37.8571 * (189 - 90 * z) = 20 99 * z - 7143.4286 + 3402.8571 * z = 20 4.5714 * z - 7143.4286 = 20 4.5714 * z = 7163.4286 z = 1565 Step 3: Substitute the value of z into the first equation to find the value of e. 70 * e + 90 * 1565 = 189 70 * e + 141,850 = 189 70 * e = 189 - 141,850 e = -2,023.5714 Therefore, the values of e and z are: e = -2,023.5714 z = 1565 ### Answer: e = -2,023.5714, z = 1565
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for o. 26o - 44f = 95 26o = 95 + 44f o = (95 + 44f) / 26 Step 2: Substitute the expression for o into the second equation. 87f + 11o = 169 87f + 11((95 + 44f) / 26) = 169 87f + 1045 + 484f = 4394 571f = 3349 f = 5.86 Step 3: Substitute the value of f into the first equation to find the value of o. o = (95 + 44f) / 26 o = (95 + 44(5.86)) / 26 o = (95 + 257.84) / 26 o = 352.84 / 26 o = 13.57 Therefore, the values of o and f that satisfy the system of equations are: o = 13.57 f = 5.86 ### Answer: o = 13.57, f = 5.86
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.
Claude-3 Haiku:
To solve this system of equations, we can use the substitution method. First, let's solve for z in the first equation: 40*z - 20*p = 51 40*z = 51 + 20*p z = (51 + 20*p) / 40 Now, let's substitute this expression for z in the second equation: 6*z + 58*p = 165 6*((51 + 20*p) / 40) + 58*p = 165 (306 + 120*p) / 40 + 58*p = 165 306 + 120*p + 2320*p = 6600 2440*p = 6294 p = 6294 / 2440 p = 2.58 Now, let's substitute the value of p back into the first equation to find the value of z: z = (51 + 20*p) / 40 z = (51 + 20*2.58) / 40 z = (51 + 51.6) / 40 z = 102.6 / 40 z = 2.565 ### Answer: z = 2.565, p = 2.58
Expected answer: {p: 3147/1220, z: 3129/1220}
Correct? No
Problem: Find the values of s and f from the given system of equations: (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93.
Claude-3 Haiku:
To find the values of s and f, we need to solve the given system of linear equations. Given equations: (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93 Step 1: Solve for s in the second equation. (3 * f) + (10 * s) = 93 10 * s = 93 - (3 * f) s = (93 - 3 * f) / 10 Step 2: Substitute the expression for s in the first equation. (71 * s) - (89 * f) = 5 (71 * (93 - 3 * f) / 10) - (89 * f) = 5 (6603 - 213 * f) / 10 - (89 * f) = 5 6603 - 213 * f - 890 * f = 50 6603 - 1103 * f = 50 -1103 * f = -6553 f = 5.94 Step 3: Substitute the value of f in the expression for s. s = (93 - 3 * f) / 10 s = (93 - 3 * 5.94) / 10 s = (93 - 17.82) / 10 s = 7.52 Therefore, the values of s and f are: s = 7.52 f = 5.94 ### Answer: s = 7.52, f = 5.94
Expected answer: {f: 6553/1103, s: 8292/1103}
Correct? No