Qwen2-Math 72B 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
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we'll use the elimination method. First, we'll multiply the first equation by 12 and the second equation by 91 to make the coefficients of q the same: 1. (480 * s) - (1092 * q) = 444 2. (2730 * s) + (1092 * q) = 10991 Next, we'll add the two equations to eliminate q: (480 * s) + (2730 * s) = 444 + 10991 (3210 * s) = 11435 Now, we'll solve for s: s = 11435 / 3210 s = 3.562305295950156 Now that we have the value of s, we can substitute it back into one of the original equations to find q. Let's use the first equation: (40 * 3.562305295950156) - (91 * q) = 37 142.49221183800624 - (91 * q) = 37 Now, we'll solve for q: - (91 * q) = 37 - 142.49221183800624 - (91 * q) = -105.49221183800624 q = -105.49221183800624 / -91 q = 1.1592550751429257 ### Answer: q = 1.1592550751429257, s = 3.562305295950156
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.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations using substitution. First, we can solve the first equation for b: 4 * b = 180 - 44 * m b = (180 - 44 * m) / 4 Now, we can substitute this expression for b into the second equation: 79 * m - 6 * ((180 - 44 * m) / 4) = 48 Simplifying the equation: 79 * m - (6 * 180) / 4 + (6 * 44 * m) / 4 = 48 79 * m - 270 + 66 * m = 48 Combining like terms: 145 * m = 318 m = 318 / 145 m = 2.19310344828 Now, we can substitute the value of m back into the expression for b: b = (180 - 44 * 2.19310344828) / 4 b = (180 - 96.4965476244) / 4 b = 83.5034523756 / 4 b = 20.8758630939 ### Answer: m = 2.19310344828, b = 20.8758630939
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 29 and the second equation by 15 to make the coefficients of g in both equations equal: 29 * (42 * o - 15 * g) = 29 * 28 15 * (47 * o + 29 * g) = 15 * 141 Simplifying these equations, we get: 1218 * o - 435 * g = 812 705 * o + 435 * g = 2115 Now, we can add these two equations to eliminate the g term: 1218 * o - 435 * g + 705 * o + 435 * g = 812 + 2115 Simplifying this equation, we get: 1923 * o = 2927 Dividing both sides by 1923, we get: o = 2927 / 1923 o = 1.522 Now, we can substitute the value of o back into one of the original equations to solve for g. Let's use the first equation: 42 * o - 15 * g = 28 42 * 1.522 - 15 * g = 28 63.924 - 15 * g = 28 Subtracting 63.924 from both sides, we get: -15 * g = -35.924 Dividing both sides by -15, we get: g = 2.395 ### Answer: o = 1.522, g = 2.395
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.
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 45 and the second equation by 6 to make the coefficients of o the same: 45((12 * v) - (6 * o)) = 45 * 21 6((76 * v) - (45 * o)) = 6 * 50 This gives us: 540v - 270o = 945 456v - 270o = 300 Now, we can subtract the second equation from the first to eliminate o: (540v - 270o) - (456v - 270o) = 945 - 300 This simplifies to: 84v = 645 Now, we can solve for v: v = 645 / 84 v = 7.678571428571429 Now that we have the value of v, we can substitute it back into one of the original equations to solve for o. Let's use the first equation: ((12 * 7.678571428571429) - (6 * o)) = 21 This simplifies to: 92.14285714285714 - 6o = 21 Now, we can solve for o: 6o = 92.14285714285714 - 21 6o = 71.14285714285714 o = 71.14285714285714 / 6 o = 11.857142857142858 ### Answer: v = 7.678571428571429, o = 11.857142857142858
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 2 and the second equation by 75 to make the coefficients of f the same: 150 * f + 50 * g = 276 150 * f + 6750 * g = 10725 Next, we can subtract the first equation from the second equation to eliminate f: 6700 * g = 10449 Now, we can solve for g: g = 10449 / 6700 = <<10449/6700=1.56>>1.56 Finally, we can substitute the value of g back into one of the original equations to solve for f: 75 * f + 25 * 1.56 = 138 75 * f + 39 = 138 75 * f = 99 f = 99 / 75 = <<99/75=1.32>>1.32 ### 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.
Qwen2-Math 72B:
Reasoning: First, we can solve the first equation for y: y = (194 - 14 * n) / 4. Substituting this expression for y into the second equation, we get: 7 = 65 * n - 79 * ((194 - 14 * n) / 4). Simplifying this equation, we get: 7 = 65 * n - 194 * 19.75 + 14 * n * 19.75. Combining like terms, we get: 7 = 79 * n - 3825.5. Solving for n, we get: n = (7 + 3825.5) / 79 = <<(7+3825.5)/79=48.5>>48.5. Substituting this value of n back into the expression for y, we get: y = (194 - 14 * 48.5) / 4 = <<(194-14*48.5)/4=1.5>>1.5. ### Answer: y = 1.5, n = 48.5
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.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations using substitution or elimination method. We can use the elimination method by multiplying the first equation by 41 and the second equation by 26 to eliminate l. First equation multiplied by 41: 6026 = 1148 * g + 1066 * l Second equation multiplied by 26: 156 = 1066 * l - 1378 * g Subtracting the second equation from the first equation: 5870 = 2526 * g Solving for g: g = 5870 / 2526 = <<5870/2526=2.32>>2.32 Substituting g = 2.32 into the first equation: 146 = 28 * 2.32 + 26 * l Solving for l: l = (146 - 28 * 2.32) / 26 = <<(146 - 28 * 2.32) / 26=3.2>>3.2 ### Answer: l = 3.2, g = 2.32
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.
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the substitution or elimination method. Let's use the elimination method. First, we can multiply equation 1 by 7 and equation 2 by 56 to make the coefficients of y the same: 1) 140 * p + 392 * y = 630 2) 4984 * p + 392 * y = 5488 Now, we can subtract equation 1 from equation 2 to eliminate y: 4984 * p - 140 * p = 5488 - 630 4844 * p = 4858 p = 4858 / 4844 p = 1.0029 Now that we have the value of p, we can substitute it back into either equation 1 or 2 to find the value of y. Let's use equation 1: 20 * 1.0029 + 56 * y = 90 20.058 + 56 * y = 90 56 * y = 90 - 20.058 56 * y = 69.942 y = 69.942 / 56 y = 1.249 ### Answer: p = 1.0029, y = 1.249
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method. First, we can multiply the first equation by 3 and the second equation by 2 to make the coefficients of d equal: 216*k - 66*d = 51 12*d +138*k = 192 Next, we can add the two equations to eliminate d: 216*k - 66*d + 12*d +138*k = 51 + 192 354*k = 243 k = 243/354 = <<243/354=0.6864406779661017>>0.6864406779661017 Now, we can substitute the value of k back into one of the original equations to find the value of d: 72*0.6864406779661017 - 22*d = 17 49.31355932203389 - 22*d = 17 22*d = 49.31355932203389 - 17 22*d = 32.31355932203389 d = 32.31355932203389/22 = <<32.31355932203389/22=1.4688072419106314>>1.4688072419106314 ### Answer: k = 0.6864406779661017, d = 1.4688072419106314
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
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method. First, we can multiply the second equation by 33/10 to make the coefficients of d in both equations equal: (33/10) * (10 * d + 14 * h) = (33/10) * 91 33 * d + 462 * h = 2997 Now, we can subtract the first equation from this new equation to eliminate d: (33 * d + 462 * h) - (73 * h - 33 * d) = 2997 - 48 33 * d + 462 * h - 73 * h + 33 * d = 2949 66 * d + 389 * h = 2949 Next, we can solve for d in terms of h: 66 * d = 2949 - 389 * h d = (2949 - 389 * h) / 66 Now, we can substitute this expression for d into the second equation: 10 * ((2949 - 389 * h) / 66) + 14 * h = 91 (29490 - 3890 * h) / 66 + 14 * h = 91 29490 - 3890 * h + 924 * h = 6006 -2966 * h = -23484 h = 23484 / 2966 h = 8 Now that we have the value of h, we can substitute it back into the expression for d: d = (2949 - 389 * 8) / 66 d = (2949 - 3112) / 66 d = -163 / 66 d = -2.46969696969697 ### Answer: h = 8, d = -2.46969696969697
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
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we will multiply the first equation by 43 and the second equation by 44 to make the coefficients of e in both equations equal: 1) 43 * (43 * q - 44 * e) = 43 * 29 2) 44 * (30 * q + 43 * e) = 44 * 130 Simplifying the equations, we get: 1) 1849 * q - 1892 * e = 1247 2) 1320 * q + 1892 * e = 5720 Now, we can add the two equations to eliminate the e term: 1849 * q - 1892 * e + 1320 * q + 1892 * e = 1247 + 5720 Simplifying the equation, we get: 3169 * q = 6967 Dividing both sides by 3169, we get: q = 6967 / 3169 q = 2.2 Now, we can substitute the value of q in any of the original equations to find the value of e. Let's substitute it in the first equation: 43 * 2.2 - 44 * e = 29 Simplifying the equation, we get: 94.6 - 44 * e = 29 Subtracting 94.6 from both sides, we get: -44 * e = -65.6 Dividing both sides by -44, we get: e = 65.6 / 44 e = 1.5 Therefore, the values of q and e are 2.2 and 1.5, respectively. ### Answer: q = 2.2, e = 1.5
Expected answer: {e: 4720/3169, q: 6967/3169}
Correct? No
Problem: Find the values of r and a in the following system of equations: 53 = ((89 * r) - (74 * a)) 109 = ((37 * a) + (6 * r))
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the second equation by 89 and the first equation by 37 to make the coefficients of r in both equations equal: (89 * 109) = (89 * ((37 * a) + (6 * r))) (37 * 53) = (37 * ((89 * r) - (74 * a))) Simplifying these equations, we get: 9681 = (3293 * a) + (534 * r) 1961 = (3293 * r) - (2738 * a) Next, we can subtract the second equation from the first equation to eliminate the variable a: (9681 - 1961) = ((3293 * a) + (534 * r)) - ((3293 * r) - (2738 * a)) 7720 = (534 * r) + (2738 * a) - (3293 * r) + (2738 * a) 7720 = (534 * r) - (3293 * r) + (2738 * a) + (2738 * a) 7720 = (-2759 * r) + (5476 * a) Now, we can solve for r by dividing both sides of the equation by -2759: r = (7720 / -2759) + (5476 * a / -2759) r = -2.80 + (-2 * a) Substituting this expression for r into one of the original equations, we can solve for a: 53 = ((89 * (-2.80 + (-2 * a))) - (74 * a)) 53 = (-249.20 + (-178 * a)) - (74 * a) 53 = -249.20 - (252 * a) Now, we can solve for a by dividing both sides of the equation by -252: a = (53 + 249.20) / -252 a = 302.20 / -252 a = -1.20 Finally, we can substitute the value of a back into the expression for r to find the value of r: r = -2.80 + (-2 * (-1.20)) r = -2.80 + 2.40 r = -0.40 Therefore, the values of r and a in the system of equations are r = -0.40 and a = -1.20. ### Answer: r = -0.40, a = -1.20
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.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations step by step. From the first equation: 42g + 12q = 169 From the second equation: 12q - 50g = 19 We can solve this system using substitution or elimination. Let's use elimination for simplicity. Multiply the first equation by 50 and the second equation by 42 to align the coefficients of 'g': 50(42g + 12q) = 50(169) 42(12q - 50g) = 42(19) This gives us: 2100g + 600q = 8450 504q - 2100g = 798 Adding these two equations to eliminate 'g': 600q + 504q = 8450 + 798 1104q = 9248 q = 9248 / 1104 q = 8.375 Substitute q back into one of the original equations to solve for 'g': 42g + 12(8.375) = 169 42g + 100.5 = 169 42g = 68.5 g = 68.5 / 42 g = 1.631 ### Answer: g = 1.631 kg, q = 8.375 kg
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
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 2 to make the coefficients of v the same: 74 * s - 22 * v = 108 Now, we can subtract the second equation from this new equation: (74 * s - 22 * v) - (82 * s - 62 * v) = 108 - 41 This simplifies to: -8 * s + 40 * v = 67 Now, we can solve for s: s = (40 * v - 67) / 8 Next, we can substitute this expression for s into the first equation: 37 * ((40 * v - 67) / 8) - 11 * v = 54 Simplifying this equation, we get: 1480 * v - 2479 - 88 * v = 432 1392 * v = 2911 v = 2911 / 1392 Now, we can substitute this value of v back into the expression for s: s = (40 * (2911 / 1392) - 67) / 8 Simplifying this expression, we get: s = (116440 / 1392 - 67) / 8 s = (83.6 - 67) / 8 s = 16.6 / 8 s = 2.075 Therefore, the values of s and v are: s = 2.075 v = 2911 / 1392 ### Answer: s = 2.075, v = 2911 / 1392
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.
Qwen2-Math 72B:
Reasoning: To solve for c and z, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 48 and the second equation by 41 to make the coefficients of c in both equations equal: 48 * (41 * c + 65 * z) = 48 * 151 41 * (78 * z - 48 * c) = 41 * 48 This gives us: 1968 * c + 3120 * z = 7248 3198 * z - 1968 * c = 1968 Now, we can add the two equations to eliminate c: (1968 * c + 3120 * z) + (3198 * z - 1968 * c) = 7248 + 1968 3120 * z + 3198 * z = 9216 6318 * z = 9216 z = 9216 / 6318 z = 1.458 Now that we have the value of z, we can substitute it back into one of the original equations to solve for c. Let's use the first equation: 41 * c + 65 * 1.458 = 151 41 * c + 94.77 = 151 41 * c = 151 - 94.77 41 * c = 56.23 c = 56.23 / 41 c = 1.371 ### Answer: c = 1.371, z = 1.458
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?
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations using the substitution method. From the first equation, we can express c in terms of i: c = (7 + 91 * i) / 60 Substitute this expression for c into the second equation: 7 * i + 76 * ((7 + 91 * i) / 60) = 148 Simplify and solve for i: i = 1 Substitute i = 1 into the expression for c: c = (7 + 91 * 1) / 60 c = 1.5 ### Answer: c = 1.5, i = 1
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
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 41 and the second equation by 37 to make the coefficients of e in both equations equal: 1) 2911*v - 1517*e = 2829 2) 1517*e + 2812*v = 6808 Now, we can add the two equations to eliminate e: 2911*v + 2812*v = 2829 + 6808 5723*v = 9637 v = 9637/5723 = <<9637/5723=1.6837>>1.6837 Now that we have the value of v, we can substitute it back into one of the original equations to find the value of e. Let's use the first equation: 71*v - 37*e = 69 71*1.6837 - 37*e = 69 119.6427 - 37*e = 69 37*e = 119.6427 - 69 37*e = 50.6427 e = 50.6427/37 = <<50.6427/37=1.3687>>1.3687 ### Answer: e = 1.3687, v = 1.6837
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.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations: 1) 25v + 69d = 113 2) 26d + 42v = 85 To eliminate 'v', we can multiply equation 1 by 42 and equation 2 by 25, then subtract the second equation from the first: 42(25v + 69d) = 42(113) 25(26d + 42v) = 25(85) 1050v + 2898d = 4746 650d + 1050v = 2125 Subtracting the second equation from the first: 2898d - 650d = 4746 - 2125 2248d = 2621 Now, divide both sides by 2248 to find 'd': d = 2621 / 2248 d = 1.165 Substitute 'd' back into equation 1 to find 'v': 25v + 69(1.165) = 113 25v + 80.385 = 113 25v = 113 - 80.385 25v = 32.615 Divide both sides by 25 to find 'v': v = 32.615 / 25 v = 1.3046 ### Answer: d = 1.165, v = 1.3046
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
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 3 and the second equation by 43 to eliminate z: 129 * z - 273 * g = 99 3096 * g - 129 * z = 2924 Add the two equations: 2823 * g = 3023 g = 3023 / 2823 = <<3023/2823=1.0708>>1.0708 Substitute g back into the first equation: 43 * z - 91 * 1.0708 = 33 43 * z = 33 + 91 * 1.0708 = 129.8588 z = 129.8588 / 43 = <<129.8588/43=3.0200>>3.0200 ### Answer: g = 1.0708, z = 3.0200
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 21 and the second equation by 12 to make the coefficients of f the same: 1050c + 252f = 1932 432c - 252f = 408 Next, we can add the two equations together to eliminate f: 1482c = 2340 Now, we can solve for c: c = 2340 / 1482 = 1.57894736842 Since c must be a whole number, we can round it to the nearest whole number, which is 2. Now that we have the value of c, we can substitute it back into one of the original equations to solve for f. Let's use the first equation: 50(2) + 12f = 92 100 + 12f = 92 12f = -8 f = -8 / 12 = -0.66666666667 Since f must also be a whole number, we can round it to the nearest whole number, which is -1. However, since the number of figs cannot be negative, we must have made a mistake in our calculations. Let's go back and check our work. Upon reviewing our work, we see that we made a mistake when we rounded c to the nearest whole number. Since c must be a whole number, we should have rounded it to the nearest whole number that makes sense in the context of the problem. In this case, the only whole number that makes sense for c is 1. Now that we have the correct value of c, we can substitute it back into one of the original equations to solve for f. Let's use the first equation: 50(1) + 12f = 92 50 + 12f = 92 12f = 42 f = 42 / 12 = 3.5 Since f must also be a whole number, we can round it to the nearest whole number, which is 4. Therefore, the quantity of cherries (c) is 1 and the quantity of figs (f) is 4. ### Answer: c = 1, f = 4
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
Qwen2-Math 72B:
Reasoning: Multiply equation 1 by 82 and equation 2 by 87 to eliminate u: 1) 7134u - 6150d = 574 2) 7395d - 7134u = 4698 Add the two equations to eliminate u: 7395d - 6150d = 4698 + 574 1245d = 5272 d = 5272/1245 = <<5272/1245=4.23>>4.23 Substitute d = 4.23 into equation 1: 87u - 75(4.23) = 7 87u - 317.25 = 7 87u = 324.25 u = 324.25/87 = <<324.25/87=3.73>>3.73 ### Answer: u = 3.73, 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?
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations: 75a - 4i = 42 10i + 13a = 160 Multiplying the first equation by 10 and the second by 4, we get: 750a - 40i = 420 40i + 52a = 640 Adding these two equations, we eliminate i: 802a = 1060 a = 1060 / 802 = <<1060/802=1.3216957625937675>>1.3216957625937675 Substituting a back into one of the original equations: 10i + 13(1.3216957625937675) = 160 10i + 17.182044913718977 = 160 10i = 142.81795508628102 i = 14.281795508628102 ### Answer: a = 1.3216957625937675, i = 14.281795508628102
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
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the second equation by 2 to make the coefficients of p in both equations equal: 2 * (27 * c - 18 * p) = 2 * 13 54 * c - 36 * p = 26 Now, we can add the two equations together to eliminate p: (37 * p + 74 * c) + (54 * c - 36 * p) = 183 + 26 37 * p - 36 * p + 74 * c + 54 * c = 209 p + 128 * c = 209 Next, we can solve for p: p = 209 - 128 * c Now, we can substitute this expression for p into one of the original equations to solve for c. Let's use the first equation: 37 * (209 - 128 * c) + 74 * c = 183 7733 - 4736 * c + 74 * c = 183 -4662 * c = -7550 c = 7550 / 4662 c = 1.62 Finally, we can substitute this value of c back into the expression for p to find its value: p = 209 - 128 * 1.62 p = 209 - 207.36 p = 1.64 Therefore, the solution to the system of equations is p = 1.64 and c = 1.62. ### Answer: p = 1.64, c = 1.62
Expected answer: {c: 3775/2331, p: 3979/2331}
Correct? No
Problem: Determine the values of k and p from the following system of equations: (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105.
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 16 and the second equation by 30 to make the coefficients of p the same: (48 * k) * 16 - (30 * p) * 16 = 37 * 16 (16 * p) * 30 + (33 * k) * 30 = 105 * 30 This gives us: 768 * k - 480 * p = 592 480 * p + 990 * k = 3150 Next, we can add the two equations together to eliminate p: (768 * k - 480 * p) + (480 * p + 990 * k) = 592 + 3150 This simplifies to: 1758 * k = 3742 Dividing both sides by 1758 gives us: k = 3742 / 1758 = 2.128 Now that we have the value of k, we can substitute it back into one of the original equations to solve for p. Using the first equation: (48 * 2.128) - (30 * p) = 37 Simplifying this gives us: 102.144 - 30 * p = 37 Subtracting 102.144 from both sides gives us: -30 * p = -65.144 Dividing both sides by -30 gives us: p = 65.144 / 30 = 2.171 ### Answer: k = 2.128, p = 2.171
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we'll use elimination. First, we'll multiply the first equation by 46 and the second equation by 69 to make the coefficients of f in both equations equal: 1. 3450v - 3174f = 644 2. 3174f + 3657v = 7690 Now, we'll subtract the first equation from the second equation to eliminate f: 3657v - 3450v = 7690 - 644 207v = 7046 Now, we'll solve for v: v = 7046 / 207 v = <<7046/207=34>>34 Now that we have the value of v, we can substitute it back into one of the original equations to solve for f. Let's use the first equation: 75(34) - 69f = 14 2550 - 69f = 14 -69f = 14 - 2550 -69f = -2536 Now, we'll solve for f: f = -2536 / -69 f = <<-2536/-69=36.75>>36.75 ### Answer: v = 34, f = 36.75
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 70 and the second equation by 13 to get: 490 * o + 910 * g = 9170 1157 * g - 910 * o = 481 Add the two equations to get: 2067 * g = 9651 g = 9651/2067 = <<9651/2067=4.666666666666667>>4.666666666666667 Substitute g = 4.666666666666667 into the first equation to get: 7 * o + 13 * 4.666666666666667 = 131 7 * o = 131 - 13 * 4.666666666666667 = 131 - 60.666666666666664 = 70.33333333333334 o = 70.33333333333334/7 = <<70.33333333333334/7=10.047619047619048>>10.047619047619048 ### Answer: o = 10.047619047619048, g = 4.666666666666667
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the substitution or elimination method. Let's use the elimination method. First, we can multiply the second equation by 3 to make the coefficients of k the same: 99 * k - 33 * d = 138 Now, we can add the two equations to eliminate k: 88 * d - 60 * k + 99 * k - 33 * d = 38 + 138 55 * d + 39 * k = 176 Next, we can solve for d: d = (176 - 39 * k) / 55 Now, we can substitute this expression for d into one of the original equations to solve for k: 88 * ((176 - 39 * k) / 55) - 60 * k = 38 Simplifying this equation, we get: 1568 - 3432 * k / 55 - 60 * k = 38 Multiplying through by 55 to clear the fraction, we get: 86240 - 3432 * k - 3300 * k = 2090 Combining like terms, we get: -6732 * k = -84150 Dividing both sides by -6732, we get: k = 12.5 Now, we can substitute this value of k back into the expression for d to find its value: d = (176 - 39 * 12.5) / 55 d = (176 - 487.5) / 55 d = -211.5 / 55 d = -3.84545454545 ### Answer: d = -3.84545454545, k = 12.5
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the second equation by 12 to make the coefficients of z in both equations equal: 24 * z + 72 * y = 139 24 * z + 984 * y = 1152 Next, we can subtract the first equation from the second equation to eliminate z: (24 * z + 984 * y) - (24 * z + 72 * y) = 1152 - 139 912 * y = 1013 Now, we can solve for y by dividing both sides of the equation by 912: y = 1013 / 912 y = 1.110745862 Next, we can substitute the value of y back into one of the original equations to solve for z. Let's use the first equation: 24 * z + 72 * y = 139 24 * z + 72 * 1.110745862 = 139 24 * z + 79.97343222 = 139 24 * z = 59.02656778 z = 59.02656778 / 24 z = 2.459440324 ### Answer: y = 1.110745862, z = 2.459440324
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method. First, we can multiply the second equation by 29 to make the coefficients of v in both equations equal: (87 * o) - (29 * v) = 1 (232 * v) + (870 * o) = 2639 Next, we can add the two equations to eliminate v: (87 * o) - (29 * v) + (232 * v) + (870 * o) = 1 + 2639 (957 * o) + (203 * v) = 2640 Now, we can solve for o: o = (2640 - 203 * v) / 957 Substituting this expression for o into the first equation, we can solve for v: (87 * ((2640 - 203 * v) / 957)) - (29 * v) = 1 Simplifying this equation, we get: (230880 - 17701 * v) / 957 - 29 * v = 1 Multiplying both sides by 957, we get: 230880 - 17701 * v - 27753 * v = 957 Combining like terms, we get: -45454 * v = -229923 Dividing both sides by -45454, we get: v = 5 Substituting this value of v back into the expression for o, we get: o = (2640 - 203 * 5) / 957 o = 2 ### Answer: v = 5, o = 2
Expected answer: {o: 2647/1566, v: 2629/522}
Correct? No
Problem: Solve the following system of equations for h and p: The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations step by step. We have: (23 * h) - (46 * p) = 58 (11 * p) + (23 * h) = 124 First, we can multiply the second equation by 2 to make the coefficients of h the same in both equations: (23 * h) - (46 * p) = 58 (22 * p) + (46 * h) = 248 Now, we can subtract the first equation from the second equation to eliminate h: (22 * p) + (46 * h) - ((23 * h) - (46 * p)) = 248 - 58 (22 * p) + (46 * h) - (23 * h) + (46 * p) = 190 (68 * p) + (23 * h) = 190 Now, we can substitute the value of (23 * h) from the first equation into the new equation: (68 * p) + 58 = 190 (68 * p) = 132 p = 132 / 68 p = 1.9411764705882353 Now, we can substitute the value of p back into one of the original equations to solve for h. Let's use the second equation: (11 * p) + (23 * h) = 124 (11 * 1.9411764705882353) + (23 * h) = 124 21.352941176470588 + (23 * h) = 124 (23 * h) = 102.64705882352941 h = 102.64705882352941 / 23 h = 4.463 ### Answer: h = 4.463, p = 1.941
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 7 to get 133d + 42f = 840. Add this to the second equation to eliminate f: 133d + 42f + 90d - 42f = 840 + 69, which simplifies to 223d = 909. Solving for d gives d = 909/223 = <<909/223=4>>4. Substitute d = 4 into the first equation to find f: 19(4) + 6f = 120, which simplifies to 76 + 6f = 120. Solving for f gives f = (120 - 76)/6 = <<(120-76)/6=7>>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'?
Qwen2-Math 72B:
Reasoning: To solve this system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we will multiply the first equation by 52 and the second equation by 55 to make the coefficients of 'r' in both equations equal: 1) 42 * m + 55 * r = 127 => 2184 * m + 2860 * r = 6604 2) 66 * m - 52 * r = 44 => 3630 * m - 2860 * r = 2420 Now, we will add the two equations to eliminate 'r': 2184 * m + 2860 * r + 3630 * m - 2860 * r = 6604 + 2420 5814 * m = 9024 m = 9024 / 5814 m = 1.55 Now, we will substitute the value of 'm' in the first equation to find the value of 'r': 42 * 1.55 + 55 * r = 127 65.1 + 55 * r = 127 55 * r = 127 - 65.1 55 * r = 61.9 r = 61.9 / 55 r = 1.125 ### Answer: m = 1.55, r = 1.125
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
Qwen2-Math 72B:
Reasoning: To solve this system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 71 and the second equation by 31 to make the coefficients of n the same: 71(31n + 64w) = 71(120) 31(57w + 71n) = 31(173) Simplifying these equations, we get: 2201n + 4544w = 8520 1767w + 2201n = 5363 Next, we can subtract the second equation from the first equation to eliminate n: (2201n + 4544w) - (1767w + 2201n) = 8520 - 5363 Simplifying this equation, we get: 2777w = 3157 Dividing both sides by 2777, we get: w = 1.136 Now that we have the value of w, we can substitute it back into one of the original equations to solve for n. Let's use the first equation: 31n + 64(1.136) = 120 Simplifying this equation, we get: 31n + 72.96 = 120 Subtracting 72.96 from both sides, we get: 31n = 47.04 Dividing both sides by 31, we get: n = 1.517 Therefore, the values of n and w that satisfy the system of linear equations simultaneously are n = 1.517 and w = 1.136. ### Answer: n = 1.517, w = 1.136
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.
Qwen2-Math 72B:
Reasoning: To solve this system of linear equations, we can use the substitution or elimination method. Here, we will use the elimination method. First, we can multiply equation (1) by 76 and equation (2) by 26 to make the coefficients of u in both equations equal: 76 * (72 * a + 26 * u) = 76 * 112 26 * (76 * u - 77 * a) = 26 * 31 This gives us: 5472 * a + 1976 * u = 8432 1976 * u - 2002 * a = 806 Now, we can subtract the second equation from the first to eliminate u: (5472 * a + 1976 * u) - (1976 * u - 2002 * a) = 8432 - 806 5472 * a + 1976 * u - 1976 * u + 2002 * a = 7626 7474 * a = 7626 Now, we can solve for a: a = 7626 / 7474 a = 1.02 Now that we have the value of a, we can substitute it back into either equation (1) or (2) to solve for u. Let's use equation (1): 72 * 1.02 + 26 * u = 112 73.44 + 26 * u = 112 26 * u = 112 - 73.44 26 * u = 38.56 u = 38.56 / 26 u = 1.48 Therefore, the values of u and a are: u = 1.48 a = 1.02 ### Answer: u = 1.48, a = 1.02
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 3 to get 162n + 54b = 429. Multiply the second equation by 2 to get 174n - 186b = 96. Add the two equations to eliminate b: 336n = 525. Divide both sides by 336 to get n = 525/336 = 1.5625. Substitute n = 1.5625 into the first equation to get 54(1.5625) + 18b = 143. Simplify to get 84.375 + 18b = 143. Subtract 84.375 from both sides to get 18b = 58.625. Divide both sides by 18 to get b = 58.625/18 = 3.2569444444444446. Round b to the nearest hundredth to get b = 3.26. ### Answer: n = 1.5625, b = 3.26
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)?
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we'll use elimination to make the coefficients of p the same in both equations. Multiply the first equation by 88 and the second equation by 7 to align the coefficients of p: (53 * e + 7 * p) * 88 = 103 * 88 (69 * e - 88 * p) * 7 = 6 * 7 This gives us: 4664 * e + 616 * p = 9064 483 * e - 616 * p = 42 Adding these two equations to eliminate p, we get: 5147 * e = 9106 Solving for e, we find: e = 9106 / 5147 = <<9106/5147=1.766>>1.766 (approximately) Substituting e back into one of the original equations to solve for p, we use the first equation: 53 * 1.766 + 7 * p = 103 93.598 + 7 * p = 103 7 * p = 103 - 93.598 7 * p = 9.402 p = 9.402 / 7 = <<9.402/7=1.343>>1.343 (approximately) ### Answer: e ≈ 1.766, p ≈ 1.343
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.
Qwen2-Math 72B:
Reasoning: To solve this system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 24 and the second equation by 18 to make the coefficients of p in both equations equal: First equation: 24 * ((18 * p) - (7 * s)) = 24 * 21 Second equation: 18 * ((51 * s) - (24 * p)) = 18 * 14 Simplifying these equations, we get: First equation: 432 * p - 168 * s = 504 Second equation: 918 * s - 432 * p = 252 Now, we can add the two equations to eliminate the variable p: (432 * p - 168 * s) + (-432 * p + 918 * s) = 504 + 252 Simplifying this equation, we get: 750 * s = 756 Dividing both sides by 750, we get: s = 756 / 750 = 1.008 Now that we have the value of s, we can substitute it back into one of the original equations to solve for p. Let's use the first equation: (18 * p) - (7 * 1.008) = 21 Simplifying this equation, we get: 18 * p - 7.056 = 21 Adding 7.056 to both sides, we get: 18 * p = 28.056 Dividing both sides by 18, we get: p = 28.056 / 18 = 1.5586666666666667 Therefore, the values of p and s that satisfy both equations are: p = 1.5586666666666667 s = 1.008 ### Answer: p = 1.5586666666666667, s = 1.008
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.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations: 59c - 34w = 84 11c + 42w = 98 Multiplying the second equation by 34/42 to eliminate w: (11c + 42w) * (34/42) = 98 * (34/42) (11c * 34/42) + (42w * 34/42) = 98 * 34/42 (11c * 34/42) + 34w = 98 * 34/42 Now, subtract the first equation from this new equation: (11c * 34/42) + 34w - (59c - 34w) = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c * 34/42) - 59c = 98 * 34/42 - 84 (11c
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 42 and the second equation by 45 to eliminate f: 42 * (45 * f + 42 * z) = 42 * 101, 45 * (60 * z - 42 * f) = 45 * 13. Simplify the equations: 1890 * f + 1764 * z = 4242, 2700 * z - 1890 * f = 585. Add the two equations to eliminate f: 1764 * z + 2700 * z = 4242 + 585, 4464 * z = 4827. Divide both sides by 4464 to solve for z: z = 4827 / 4464, z = 1.08. Substitute z = 1.08 into the first equation to solve for f: 45 * f + 42 * 1.08 = 101, 45 * f + 45.36 = 101. Subtract 45.36 from both sides: 45 * f = 101 - 45.36, 45 * f = 55.64. Divide both sides by 45 to solve for f: f = 55.64 / 45, f = 1.24. ### Answer: f = 1.24, z = 1.08
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 15 and the second equation by 26 to eliminate p: 15 * (26 * p - 21 * u) = 15 * 6 26 * (15 * p + 58 * u) = 26 * 83 Simplify the equations: 390 * p - 315 * u = 90 390 * p + 1508 * u = 2158 Subtract the first equation from the second equation to eliminate p: (390 * p + 1508 * u) - (390 * p - 315 * u) = 2158 - 90 Simplify the equation: 1823 * u = 2068 Divide both sides by 1823 to solve for u: u = 2068 / 1823 u = 1.134 Substitute the value of u into one of the original equations to solve for p: 26 * p - 21 * 1.134 = 6 Simplify the equation: 26 * p - 23.814 = 6 Add 23.814 to both sides: 26 * p = 29.814 Divide both sides by 26 to solve for p: p = 29.814 / 26 p = 1.147 ### Answer: p = 1.147, u = 1.134
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
Qwen2-Math 72B:
Reasoning: To solve for n and s, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the second equation by 38/8 to make the coefficient of s in both equations the same: (59 * n + 8 * s) * (38/8) = 75 * (38/8) (59 * n * 38/8) + (8 * s * 38/8) = 75 * (38/8) (59 * n * 38/8) + 38 * s = 75 * (38/8) Now, we can subtract the first equation from this new equation to eliminate s: (59 * n * 38/8) + 38 * s - (92 * n - 38 * s) = 75 * (38/8) - 62 (59 * n * 38/8) - 92 * n = 75 * (38/8) - 62 (59 * n * 38/8) - 92 * n = 3450/8 - 62 (59 * n * 38/8) - 92 * n = 3450/8 - 496/8 (59 * n * 38/8) - 92 * n = 2954/8 (59 * n * 38/8) - 92 * n = 369.25 Now, we can solve for n: (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) = 92 * n + 369.25 (59 * n * 38/8) = 92 * n + 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n * 38/8) - 92 * n = 369.25 (59 * n *
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we'll use elimination. First, we'll multiply the first equation by 58 and the second equation by 35 to make the coefficients of e the same: 58 * (87 * z - 35 * e) = 58 * 63 35 * (81 * z + 58 * e) = 35 * 154 This gives us: 5046 * z - 1990 * e = 3654 2835 * z + 2030 * e = 5390 Now, we'll add the two equations to eliminate e: (5046 * z - 1990 * e) + (2835 * z + 2030 * e) = 3654 + 5390 7881 * z = 9044 Now, we'll solve for z: z = 9044 / 7881 z = <<9044/7881=1.1475>>1.1475 Next, we'll substitute the value of z back into one of the original equations to solve for e. Using the first equation: 87 * 1.1475 - 35 * e = 63 99.9375 - 35 * e = 63 Now, we'll solve for e: 35 * e = 99.9375 - 63 35 * e = 36.9375 e = 36.9375 / 35 e = <<36.9375/35=1.0553>>1.0553 ### Answer: z = 1.1475, e = 1.0553
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 36 and the second equation by 44 to make the coefficients of h the same: (36 * 11 * t) + (36 * 44 * h) = 36 * 110 (44 * 98 * t) - (44 * 36 * h) = 44 * 18 This simplifies to: 396 * t + 1584 * h = 3960 4312 * t - 1584 * h = 792 Next, we can add the two equations together to eliminate h: (396 * t) + (4312 * t) = 3960 + 792 4708 * t = 4752 Now, we can solve for t: t = 4752 / 4708 t = 1 Now that we have the value of t, we can substitute it back into one of the original equations to solve for h. Let's use the first equation: (11 * 1) + (44 * h) = 110 11 + 44 * h = 110 44 * h = 99 h = 99 / 44 h = 9 / 4 ### Answer: t = 1, h = 9/4
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.
Qwen2-Math 72B:
Reasoning: ((69 * b) - (50 * o)) = 55 ((41 * o) + (18 * b)) = 85 Solving the system of linear equations, we get: b = 1 o = 2 ### Answer: b = 1, o = 2
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the second equation by 16 to make the coefficients of q in both equations equal: 67 * q - 80 * p = 65, and 96 * q + 80 * p = 2832. Next, we can add the two equations to eliminate p: 163 * q = 2897. Dividing both sides by 163, we get: q = 17.77. Substituting this value of q into the second equation, we get: 6 * 17.77 + 5 * p = 177. Simplifying, we get: 106.62 + 5 * p = 177. Subtracting 106.62 from both sides, we get: 5 * p = 70.38. Dividing both sides by 5, we get: p = 14.076. Therefore, the values of p and q are p = 14.076 and q = 17.77. ### Answer: p = 14.076, 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.
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method. First, we can multiply the first equation by 41 and the second equation by 48 to make the coefficients of f the same: 410 * k + 1968 * f = 6581 1584 * k - 1968 * f = 1488 Now, we can add the two equations to eliminate f: 2094 * k = 8069 k = 8069 / 2094 = <<8069/2094=3.85>>3.85 Now that we have the value of k, we can substitute it back into one of the original equations to find the value of f: 10 * 3.85 + 48 * f = 161 38.5 + 48 * f = 161 48 * f = 122.5 f = 122.5 / 48 = <<122.5/48=2.55>>2.55 ### Answer: k = 3.85, f = 2.55
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
Qwen2-Math 72B:
Reasoning: First, we can solve the first equation for n: n = 58 * l - 31 Then, we can substitute this expression for n into the second equation: 103 = (58 * l - 31) + 73 * l Simplifying the equation, we get: 103 = 131 * l - 31 Adding 31 to both sides, we get: 134 = 131 * l Dividing both sides by 131, we get: l = 134 / 131 = <<134/131=1.023>>1.023 Substituting this value of l back into the expression for n, we get: n = 58 * 1.023 - 31 = <<58*1.023-31=28.934>>28.934 ### Answer: l = 1.023, n = 28.934
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
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 37 and the second equation by 28 to make the coefficients of j equal: 37(29y - 28j) = 37(39) 28(25y + 37j) = 28(116) This gives us: 1073y - 1036j = 1443 700y + 1036j = 3248 Now, we can add the two equations to eliminate j: 1073y + 700y = 1443 + 3248 1773y = 4691 Dividing both sides by 1773, we get: y = 4691 / 1773 y = 2.64 Now that we have the value of y, we can substitute it back into one of the original equations to find the value of j. Let's use the first equation: 29(2.64) - 28j = 39 76.56 - 28j = 39 Subtracting 76.56 from both sides, we get: -28j = 39 - 76.56 -28j = -37.56 Dividing both sides by -28, we get: j = -37.56 / -28 j = 1.34 Therefore, the values of y and j that satisfy the system of equations are y = 2.64 and j = 1.34. ### Answer: y = 2.64, j = 1.34
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 17 and the second equation by 85 to make the coefficients of a in both equations equal: 17*(93u - 85a) = 17*74 85*(96u + 17a) = 85*186 This gives us: 1581u - 1445a = 1258 8160u + 1445a = 15810 Adding these two equations, we get: 9741u = 17068 Dividing both sides by 9741, we get: u = 17068/9741 = 1.75 Substituting this value of u into the first equation, we get: 93*1.75 - 85a = 74 Solving for a, we get: a = (93*1.75 - 74)/85 = 1.00 ### Answer: u = 1.75, a = 1.00
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
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 23 and the second equation by 61 to make the coefficients of g the same: 23 * (97 * u - 61 * g) = 23 * 15 61 * (85 * u + 23 * g) = 61 * 186 This gives us: 2231 * u - 1403 * g = 345 5185 * u + 1403 * g = 11346 Now, we can add these two equations to eliminate g: 2231 * u + 5185 * u = 345 + 11346 7416 * u = 11691 Now, we can solve for u: u = 11691 / 7416 u = 1.576 Now that we have the value of u, we can substitute it back into one of the original equations to find the value of g. Let's use the first equation: 97 * 1.576 - 61 * g = 15 152.872 - 61 * g = 15 -61 * g = 15 - 152.872 -61 * g = -137.872 Now, we can solve for g: g = -137.872 / -61 g = 2.257 ### Answer: u = 1.576, g = 2.257
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.
Qwen2-Math 72B:
Reasoning: Subtracting the first equation from the second equation, we get (85 * r) - (17 * r) = 187 - 103, which simplifies to 68 * r = 84. Dividing both sides by 68, we get r = 84/68 = <<84/68=1.2352941176470588>>1.2352941176470588. Substituting r = 1.2352941176470588 into the first equation, we get (17 * 1.2352941176470588) + (67 * b) = 103, which simplifies to 21 + 67 * b = 103. Subtracting 21 from both sides, we get 67 * b = 82. Dividing both sides by 67, we get b = 82/67 = <<82/67=1.2238805970149254>>1.2238805970149254. ### Answer: r = 1.2352941176470588, b = 1.2238805970149254
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.
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the substitution or elimination method. Here, we'll use the elimination method. First, we'll multiply the first equation by 30 and the second equation by 54 to make the coefficients of n equal: 1. (30 * 85 * n) - (30 * 54 * t) = 30 * 41 2. (54 * 38 * t) + (54 * 30 * n) = 54 * 126 This simplifies to: 1. 2550n - 1620t = 1230 2. 2052t + 1620n = 6744 Next, we'll subtract the first equation from the second to eliminate n: 2052t + 1620n - (2550n - 1620t) = 6744 - 1230 This simplifies to: 3672t = 5514 Now, we can solve for t: t = 5514 / 3672 t = 1.5 Substitute t = 1.5 into the first original equation to solve for n: (85 * n) - (54 * 1.5) = 41 85n - 81 = 41 85n = 122 n = 122 / 85 n = 1.435 ### Answer: t = 1.5, n = 1.435
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.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations step by step. 1) From the first equation, we have: 75u - 69q = 5 2) From the second equation, we have: 48u + 25q = 140 3) To eliminate q, we can multiply the first equation by 25 and the second equation by 69: (75u - 69q) * 25 = 5 * 25 (48u + 25q) * 69 = 140 * 69 4) This gives us: 1875u - 1725q = 125 3312u + 1725q = 9660 5) Adding these two equations, we get: 5187u = 9785 6) Solving for u, we have: u = 9785 / 5187 = 1.886 7) Substituting u = 1.886 into the first equation, we get: 75 * 1.886 - 69q = 5 8) Solving for q, we have: 69q = 75 * 1.886 - 5 q = (75 * 1.886 - 5) / 69 = 2.000 ### Answer: u = 1.886, q = 2.000
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.
Qwen2-Math 72B:
Reasoning: ((20 * v) + (17 * x)) = 149 and ((24 * x) - (49 * v)) = 22. Solving the system of equations, we get v = 1 and x = 7. ### Answer: v = 1, x = 7
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 91 and the second equation by 41 to make the coefficients of g in both equations equal: 39 * t * 91 + 41 * g * 91 = 147 * 91 94 * t * 41 - 91 * g * 41 = 42 * 41 This simplifies to: 3549 * t + 3731 * g = 13377 3854 * t - 3731 * g = 1722 Now, we can add the two equations to eliminate g: 3549 * t + 3731 * g + 3854 * t - 3731 * g = 13377 + 1722 7403 * t = 15109 Now, we can solve for t: t = 15109 / 7403 t = <<15109/7403=2.04>>2.04 Now that we have the value of t, we can substitute it back into one of the original equations to solve for g. Let's use the first equation: 39 * t + 41 * g = 147 39 * 2.04 + 41 * g = 147 79.56 + 41 * g = 147 Now, we can solve for g: 41 * g = 147 - 79.56 41 * g = 67.44 g = 67.44 / 41 g = <<67.44/41=1.64>>1.64 ### Answer: t = 2.04, g = 1.64
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.
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Let's use elimination for this problem. First, we can multiply the first equation by 55 and the second equation by 6 to make the coefficients of c the same: 55(6c + 55o) = 55(112) 6(55c - 48o) = 6(36) This simplifies to: 330c + 3025o = 6160 330c - 288o = 216 Now, we can subtract the second equation from the first to eliminate c: (330c + 3025o) - (330c - 288o) = 6160 - 216 3025o + 288o = 5944 3313o = 5944 o = 5944 / 3313 o = 1.794 (approximately) Now, we can substitute the value of o back into one of the original equations to solve for c. Let's use the first equation: 6c + 55(1.794) = 112 6c + 98.67 = 112 6c = 112 - 98.67 6c = 13.33 c = 13.33 / 6 c = 2.2217 (approximately) ### Answer: c ≈ 2.2217, o ≈ 1.794
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 94 and the second equation by 68 to eliminate u: 752w + 6392u = 15980, 4828w - 6392u = 536. Add the two equations to eliminate u: 5580w = 16516. Divide both sides by 5580 to solve for w: w = 16516/5580 = <<16516/5580=3>>3. Substitute w = 3 into the first equation to solve for u: 8(3) + 68u = 170, 24 + 68u = 170, 68u = 146, u = 146/68 = <<146/68=2>>2. ### Answer: u = 2, 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
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 38 and the second equation by 68 to make the coefficients of r the same: (68*r + 8*f) * 38 = 92 * 38 (97*f + 38*r) * 68 = 182 * 68 This gives us: 2584*r + 304*f = 3496 6616*f + 2584*r = 12376 Next, we can subtract the first equation from the second equation to eliminate r: (6616*f + 2584*r) - (2584*r + 304*f) = 12376 - 3496 This simplifies to: 6312*f = 8880 Now, we can solve for f: f = 8880 / 6312 f = 1.4 Now that we have the value of f, we can substitute it back into one of the original equations to solve for r. Let's use the first equation: 68*r + 8*1.4 = 92 This simplifies to: 68*r + 11.2 = 92 Now, we can solve for r: 68*r = 92 - 11.2 68*r = 80.8 r = 80.8 / 68 r = 1.19 ### Answer: r = 1.19, f = 1.4
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).
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 39 and the second equation by 67 to make the coefficients of r in both equations equal: 39(67d - 37r) = 39(50) 67(39r - 21d) = 67(16) This gives us: 2613d - 1443r = 1950 2613r - 1407d = 1072 Now, we can add these two equations to eliminate r: (2613d - 1443r) + (2613r - 1407d) = 1950 + 1072 2613d - 1407d = 3022 1206d = 3022 d = 3022/1206 d = 2.5 Now that we have the value of d, we can substitute it back into one of the original equations to find the value of r. Let's use the first equation: 67d - 37r = 50 67(2.5) - 37r = 50 167.5 - 37r = 50 -37r = 50 - 167.5 -37r = -117.5 r = -117.5/-37 r = 3.175675675675676 ### Answer: d = 2.5, r = 3.175675675675676
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.
Qwen2-Math 72B:
Reasoning: To solve the system, we can use the substitution or elimination method. Here, we'll use the elimination method. First, we'll multiply the first equation by 84 to align the coefficients of a: 84 * (163 = 50x + a) => 13752 = 4200x + 84a Now, we'll add this to the second equation to eliminate a: 13752 + 24 = 4200x + 84a + 98x - 84a 13776 = 4298x Solving for x: x = 13776 / 4298 = <<13776/4298=3.2>>3.2 Substitute x back into the first equation to find a: 163 = 50 * 3.2 + a 163 = 160 + a a = 163 - 160 = 3 ### Answer: x = 3.2, a = 3
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 44 and the second equation by 67 to eliminate p: 44 * (67 * p + 18 * b) = 44 * 110 67 * (82 * b - 44 * p) = 67 * 58 Simplify the equations: 2948 * p + 792 * b = 4840 5494 * b - 2948 * p = 3816 Add the two equations to eliminate p: (2948 * p + 792 * b) + (5494 * b - 2948 * p) = 4840 + 3816 Simplify the equation: 6286 * b = 8656 Divide both sides by 6286 to solve for b: b = 8656 / 6286 b = 1.375 Substitute the value of b into one of the original equations to solve for p: 67 * p + 18 * 1.375 = 110 Simplify the equation: 67 * p + 24.75 = 110 Subtract 24.75 from both sides: 67 * p = 85.25 Divide both sides by 67 to solve for p: p = 85.25 / 67 p = 1.2727272727272727 ### Answer: p = 1.2727272727272727, b = 1.375
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.
Qwen2-Math 72B:
Reasoning:
1. Set up the equations:
\[
\begin{align*}
63c - 36e &= 27 \quad \text{(1)} \\
8c + 32e &= 69 \quad \text{(2)}
\end{align*}
\]
2. Simplify equation (2) by dividing all terms by 8:
\[
c + 4e = \frac{69}{8} \quad \text{(3)}
\]
3. Solve equation (3) for c:
\[
c = \frac{69}{8} - 4e \quad \text{(4)}
\]
4. Substitute equation (4) into equation (1):
\[
63\left(\frac{69}{8} - 4e\right) - 36e = 27
\]
5. Simplify and solve for e:
\[
\frac{4293}{8} - 252e - 36e = 27
\]
\[
-288e = 27 - \frac{4293}{8}
\]
\[
-288e = -\frac{3969}{8}
\]
\[
e = \frac{3969}{8 \cdot 288}
\]
\[
e = \frac{3969}{2304}
\]
\[
e = \frac{1323}{768}
\]
\[
e = \frac{441}{256}
\]
6. Substitute e back into equation (4) to find c:
\[
c = \frac{69}{8} - 4\left(\frac{441}{256}\right)
\]
\[
c = \frac{69}{8} - \frac{1764}{256}
\]
\[
c = \frac{552}{256} - \frac{1764}{256}
\]
\[
c = \frac{-1212}{256}
\]
\[
c = -\frac{303}{64}
\]
### Answer: c = -303/64, e = 441/256
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
Qwen2-Math 72B:
Reasoning: To solve the system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 51 and the second equation by 29 to make the coefficients of p the same: 51 * (26 * x + 29 * p) = 51 * 107 29 * (61 * x - 51 * p) = 29 * 4 This gives us: 1326 * x + 1519 * p = 5457 1769 * x - 1479 * p = 116 Next, we can add the two equations together to eliminate p: 1326 * x + 1519 * p + 1769 * x - 1479 * p = 5457 + 116 This simplifies to: 3095 * x = 5573 Dividing both sides by 3095 gives us: x = 5573 / 3095 = <<5573/3095=1.8>>1.8 Now that we have the value of x, we can substitute it back into one of the original equations to solve for p. Using the first equation: 26 * 1.8 + 29 * p = 107 46.8 + 29 * p = 107 Subtracting 46.8 from both sides gives us: 29 * p = 60.2 Dividing both sides by 29 gives us: p = 60.2 / 29 = <<60.2/29=2.1>>2.1 ### Answer: x = 1.8, p = 2.1
Expected answer: {p: 6423/3095, x: 5573/3095}
Correct? No
Problem: Find the values for g and l from the system of equations below. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations using substitution or elimination method. We can use the elimination method by multiplying the first equation by 10 and the second equation by 29 to eliminate l. First equation multiplied by 10: 290l + 170g = 680 Second equation multiplied by 29: 899g - 290l = 493 Adding the two equations: 1069g = 1173 g = 1173/1069 = <<1173/1069=1.1>>1.1 Substituting g = 1.1 in the first equation: 29l + 17(1.1) = 68 29l + 18.7 = 68 29l = 49.3 l = 49.3/29 = <<49.3/29=1.7>>1.7 ### Answer: g = 1.1, l = 1.7
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 24 and the second equation by 33 to eliminate a: 24*(33*a - 54*c) = 24*20 33*(24*a - 26*c) = 33*29 Simplify: 792*a - 1296*c = 480 792*a - 858*c = 957 Subtract the second equation from the first: (792*a - 1296*c) - (792*a - 858*c) = 480 - 957 -438*c = -477 Divide by -438: c = 477/438 = 1.089 Substitute c = 1.089 into the first equation: 33*a - 54*1.089 = 20 33*a - 58.806 = 20 33*a = 78.806 a = 78.806/33 = 2.388 ### Answer: a = 2.388, c = 1.089
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.
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method. First, we can multiply the first equation by 84 and the second equation by 71 to make the coefficients of d the same: 84(71d + 9j) = 84(132) 71(43j - 84d) = 71(91) This gives us: 5964d + 756j = 11088 3053j - 5964d = 6461 Now, we can add these two equations to eliminate d: 5964d + 756j + 3053j - 5964d = 11088 + 6461 This simplifies to: 3809j = 17549 Now, we can solve for j: j = 17549 / 3809 j = 4.608 Next, we can substitute the value of j back into one of the original equations to solve for d. Let's use the first equation: 71d + 9(4.608) = 132 This simplifies to: 71d + 41.472 = 132 Now, we can solve for d: 71d = 132 - 41.472 71d = 90.528 d = 90.528 / 71 d = 1.275 Therefore, the solution to the system of equations is d = 1.275 and j = 4.608. ### Answer: d = 1.275, j = 4.608
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.
Qwen2-Math 72B:
Reasoning: Let's solve this system of equations using the substitution method. First, we can solve the first equation for o: o = (163 - 26n) / 81 Next, we can substitute this expression for o into the second equation: 87n + 23((163 - 26n) / 81) = 143 Multiplying both sides by 81 to eliminate the fraction, we get: 7047n + 3749 - 598n = 11583 Combining like terms, we get: 6449n = 7834 Dividing both sides by 6449, we get: n = 7834 / 6449 = 1.214 Substituting this value of n back into the expression for o, we get: o = (163 - 26(1.214)) / 81 = 1.506 Therefore, the values of o and n are approximately 1.506 and 1.214, respectively. ### Answer: o = 1.506, n = 1.214
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 61 and the second equation by 46 to eliminate s: 61 * (46 * s - 61 * w) = 61 * 2 46 * (61 * s + 67 * w) = 46 * 178 Simplify: 2796 * s - 3721 * w = 122 2806 * s + 3082 * w = 8188 Subtract the first equation from the second equation to eliminate s: (2806 * s + 3082 * w) - (2796 * s - 3721 * w) = 8188 - 122 Simplify: 10 * s + 6803 * w = 8066 Divide by 10: s + 680.3 * w = 806.6 Substitute s = 806.6 - 680.3 * w into the first equation: 46 * (806.6 - 680.3 * w) - 61 * w = 2 Simplify: 37097.6 - 31293.8 * w - 61 * w = 2 Combine like terms: -31354.8 * w = -37095.6 Divide by -31354.8: w = 1.183 Substitute w = 1.183 into the equation s = 806.6 - 680.3 * w: s = 806.6 - 680.3 * 1.183 Simplify: s = 806.6 - 806.5 s = 0.1 ### Answer: w = 1.183, s = 0.1
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.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations using the substitution method. From the first equation, we can express k in terms of i: k = (86 + 8*i) / 51. Substituting this expression into the second equation, we get: 39*i + 8*((86 + 8*i) / 51) = 193. Simplifying this equation, we get: 39*i + (688 + 64*i) / 51 = 193. Multiplying both sides by 51, we get: 1989*i + 688 + 64*i = 9843. Combining like terms, we get: 2053*i = 9155. Dividing both sides by 2053, we get: i = 4.46. Substituting this value of i back into the expression for k, we get: k = (86 + 8*4.46) / 51 = 2.04. ### Answer: i = 4.46, k = 2.04
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
Qwen2-Math 72B:
Reasoning: Multiply the second equation by 17 to get 1581m + 17l = 2142. Subtract the first equation from this to eliminate l: 1581m + 17l - (26m + 17l) = 2142 - 146, which simplifies to 1555m = 1996. Dividing both sides by 1555 gives m = 1996/1555 = 1.2836. Substitute this value of m into the second equation to get 93(1.2836) + l = 126, which simplifies to 119.2368 + l = 126. Subtracting 119.2368 from both sides gives l = 126 - 119.2368 = 6.7632. ### Answer: l = 6.7632, m = 1.2836
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
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we'll use elimination. First, we'll multiply equation 1 by 29 and equation 2 by 8 to make the coefficients of 'f' the same: 1) (29 * 8 * f) + (29 * 39 * h) = 29 * 66 2) (8 * 29 * f) + (8 * 46 * h) = 8 * 119 Simplifying, we get: 1) 232f + 1131h = 1914 2) 232f + 368h = 952 Now, we'll subtract equation 2 from equation 1 to eliminate 'f': (232f + 1131h) - (232f + 368h) = 1914 - 952 Simplifying, we get: 763h = 962 Dividing both sides by 763, we get: h = 962 / 763 = 1.26 Now, we can substitute the value of 'h' back into one of the original equations to solve for 'f'. Let's use equation 1: (8 * f) + (39 * 1.26) = 66 Simplifying, we get: 8f + 49.14 = 66 Subtracting 49.14 from both sides, we get: 8f = 16.86 Dividing both sides by 8, we get: f = 16.86 / 8 = 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.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations step by step. First, we can multiply the first equation by 3 and the second equation by 1 to make the coefficients of b in both equations equal. This gives us: 66 * b + 69 * k = 327 and 62 * k - 69 * b = 14. Now, we can add the two equations together to eliminate b: (66 * b + 69 * k) + (62 * k - 69 * b) = 327 + 14, which simplifies to 131 * k = 341. Dividing both sides by 131 gives us k = 341 / 131 = 2.6. Substituting k = 2.6 into the first equation gives us 22 * b + 23 * 2.6 = 109, which simplifies to 22 * b + 59.8 = 109. Subtracting 59.8 from both sides gives us 22 * b = 49.2, and dividing both sides by 22 gives us b = 49.2 / 22 = 2.2. Therefore, the values of k and b that satisfy the system of equations are k = 2.6 and b = 2.2. ### Answer: k = 2.6, b = 2.2
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.
Qwen2-Math 72B:
Reasoning: To solve the system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 2 and the second equation by 55 to make the coefficients of s the same: 1. 168m - 110s = 2 2. 3465m - 110s = 5390 Next, we can subtract the first equation from the second equation to eliminate s: 3465m - 168m = 5390 - 2 3297m = 5388 Now, we can solve for m: m = 5388 / 3297 m = 1.63 Next, we can substitute the value of m back into one of the original equations to solve for s. Let's use the first equation: 84(1.63) - 55s = 1 136.92 - 55s = 1 -55s = 1 - 136.92 -55s = -135.92 Now, we can solve for s: s = -135.92 / -55 s = 2.47 Therefore, the solution to the system of linear equations is m = 1.63 and s = 2.47. ### Answer: m = 1.63, s = 2.47
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.
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the substitution or elimination method. Here, we'll use the elimination method. First, we can multiply the first equation by 51 and the second equation by 6 to make the coefficients of b the same: 1. (51 * 6 * b) + (51 * 5 * k) = 51 * 105 2. (6 * 95 * k) - (6 * 51 * b) = 6 * 67 This simplifies to: 1. 306b + 255k = 5355 2. 570k - 306b = 402 Now, we can add these two equations to eliminate b: 306b + 255k + 570k - 306b = 5355 + 402 This simplifies to: 825k = 5757 Now, we can solve for k: k = 5757 / 825 k = 7 Next, we can substitute the value of k back into one of the original equations to solve for b. Let's use the first equation: (6 * b) + (5 * 7) = 105 This simplifies to: 6b + 35 = 105 Now, we can solve for b: 6b = 105 - 35 6b = 70 b = 70 / 6 b = 11.666... Since b must be a whole number (as it represents the number of items), we can round b to the nearest whole number, which is 12. ### Answer: b = 12, 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.
Qwen2-Math 72B:
Reasoning: First, we can solve for d in terms of u using the first equation: 33d = 114 - 62u d = (114 - 62u) / 33 Next, we can substitute this expression for d into the second equation: 32u + 40((114 - 62u) / 33) = 83 Simplifying this equation, we get: 32u + (40 * 114) / 33 - (40 * 62u) / 33 = 83 32u + 1360 / 33 - 2480u / 33 = 83 32u - 2480u / 33 = 83 - 1360 / 33 (32u * 33) - 2480u = (83 * 33) - 1360 1056u - 2480u = 2739 - 1360 -1424u = 1379 u = -1379 / 1424 Now that we have the value of u, we can substitute it back into the expression for d: d = (114 - 62(-1379 / 1424)) / 33 d = (114 + 85618 / 1424) / 33 d = (114 * 1424 + 85618) / (33 * 1424) d = (162456 + 85618) / 46992 d = 248074 / 46992 d = 5.28 Therefore, the values of d and u are d = 5.28 and u = -1379 / 1424. ### Answer: d = 5.28, u = -1379 / 1424
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
Qwen2-Math 72B:
Reasoning: Let's solve this system of equations using the substitution method. First, we can solve the first equation for c: 5 * c + 28 * p = 102 5 * c = 102 - 28 * p c = (102 - 28 * p) / 5 Now, we can substitute this expression for c into the second equation: 20 * c - 67 * p = 13 20 * ((102 - 28 * p) / 5) - 67 * p = 13 4 * (102 - 28 * p) - 67 * p = 13 408 - 112 * p - 67 * p = 13 408 - 179 * p = 13 -179 * p = 13 - 408 -179 * p = -395 p = -395 / -179 p = 2.2 Now that we have the value of p, we can substitute it back into the expression for c: c = (102 - 28 * p) / 5 c = (102 - 28 * 2.2) / 5 c = (102 - 61.6) / 5 c = 40.4 / 5 c = 8.08 ### Answer: c = 8.08, p = 2.2
Expected answer: {c: 7198/895, p: 395/179}
Correct? No
Problem: Given the following system of linear equations, determine the value of r and q. The system of equations is: 58 * r + 12 * q = 147 97 * q - 76 * r = 76.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations using the substitution method. First, we can solve the first equation for r: 58 * r = 147 - 12 * q r = (147 - 12 * q) / 58 Now, we can substitute this expression for r into the second equation: 97 * q - 76 * ((147 - 12 * q) / 58) = 76 Simplifying this equation, we get: 97 * q - (147 * 76 / 58) + (12 * 76 * q / 58) = 76 97 * q - 186 + 152 * q / 7 = 76 Multiplying both sides by 7 to eliminate the fraction, we get: 679 * q - 1302 + 152 * q = 532 Combining like terms, we get: 831 * q = 1834 Dividing both sides by 831, we get: q = 1834 / 831 q = 2.206 Now, we can substitute this value of q back into the expression for r: r = (147 - 12 * 2.206) / 58 r = (147 - 26.472) / 58 r = 120.528 / 58 r = 2.078 ### Answer: r = 2.078, q = 2.206
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 2 and the second equation by 3 to eliminate k: 94 * v - 6 * k = 20 219 * v + 6 * k = 384 Add the two equations to eliminate k: 313 * v = 404 v = 404 / 313 = <<404/313=1.2907348242805112>>1.2907348242805112 Substitute v into the first equation to solve for k: 47 * 1.2907348242805112 - 3 * k = 10 60.66453674018392 - 3 * k = 10 3 * k = 50.66453674018392 k = 50.66453674018392 / 3 = <<50.66453674018392/3=16.88817891339464>>16.88817891339464 ### Answer: v = 1.2907348242805112, k = 16.88817891339464
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?
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations step by step. First, we can multiply the first equation by 27 and the second equation by 89 to eliminate p: 27*(89p - 41q) = 27*60 89*(27p + 91q) = 89*124 This gives us: 2393p - 1107q = 1620 2403p + 8099q = 11036 Subtracting the first equation from the second gives: 10p + 9206q = 9416 Solving for p gives: p = (9416 - 9206q)/10 Substituting this into the first equation gives: 89*((9416 - 9206q)/10) - 41q = 60 Simplifying this gives: 84744 - 81954q - 410q = 600 Solving for q gives: q = (84744 - 600)/(81954 + 410) = 1 Substituting q = 1 into the equation for p gives: p = (9416 - 9206*1)/10 = 21 ### Answer: p = 21, q = 1
Expected answer: {p: 5272/4603, q: 4708/4603}
Correct? No
Problem: Find the values of u and d that make these equations true: The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations step by step. First, we can multiply the second equation by 3 to make the coefficients of d in both equations the same: (81 * u) + (108 * d) = 213 Now, we can subtract the first equation from this new equation to eliminate d: (81 * u) + (108 * d) - ((88 * u) - (96 * d)) = 213 - 12 Simplifying this, we get: (81 * u) + (108 * d) - (88 * u) + (96 * d) = 201 Combining like terms, we get: (81 * u - 88 * u) + (108 * d + 96 * d) = 201 -7 * u + 204 * d = 201 Now, we can solve for u: u = (201 - 204 * d) / -7 Substituting this expression for u into the first equation, we get: (88 * ((201 - 204 * d) / -7)) - (96 * d) = 12 Simplifying this, we get: (88 * (201 - 204 * d)) / -7 - 96 * d = 12 Multiplying both sides by -7 to get rid of the fraction, we get: 88 * (201 - 204 * d) - 672 * d = -84 Expanding this, we get: 17728 - 17952 * d - 672 * d = -84 Combining like terms, we get: 17728 - 18624 * d = -84 Now, we can solve for d: d = (17728 + 84) / 18624 d = 17812 / 18624 d = 0.956 Substituting this value of d back into the expression for u, we get: u = (201 - 204 * 0.956) / -7 u = (201 - 194.904) / -7 u = 6.096 / -7 u = -0.871 ### Answer: u = -0.871, d = 0.956
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.
Qwen2-Math 72B:
Reasoning: Multiply the second equation by 9 to get 27 * n - 18 * f = 630. Subtract the first equation from this to get 7 * n = 621. Divide both sides by 7 to get n = 88.71428571428571. Substitute this value of n into the first equation to get 20 * 88.71428571428571 - 18 * f = 9. Solve for f to get f = 97.5. ### Answer: n = 88.71428571428571, f = 97.5
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 22 and the second equation by 97 to eliminate j: 154 * l + 2134 * j = 2926 2134 * j + 3599 * l = 7372 Subtract the first equation from the second equation to eliminate j: 3599 * l - 154 * l = 7372 - 2926 3445 * l = 4446 l = 4446 / 3445 = 1.29 Substitute l = 1.29 into the first equation to solve for j: 7 * 1.29 + 97 * j = 133 9.03 + 97 * j = 133 97 * j = 123.97 j = 123.97 / 97 = 1.28 ### Answer: j = 1.28, l = 1.29
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 21 and the second equation by 46 to eliminate f: 441d - 966f = 1113 368d + 3864f = 6118 Add the two equations: 809d = 7231 d = 7231/809 = <<7231/809=9>>9 Substitute d = 9 into the first equation: 21(9) - 46f = 53 189 - 46f = 53 -46f = 53 - 189 -46f = -136 f = -136/-46 = <<-136/-46=3>>3 ### Answer: d = 9, f = 3
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.
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 68 and the second equation by 11 to make the coefficients of w equal: 68(74u + 11w) = 68(116) 11(66u - 68w) = 11(21) Simplifying these equations, we get: 5032u + 748w = 7888 726u - 748w = 231 Next, we can add these two equations to eliminate w: (5032u + 748w) + (726u - 748w) = 7888 + 231 5758u = 8119 Dividing both sides by 5758, we get: u = 8119/5758 Simplifying this fraction, we get: u = 1.41 Now that we have the value of u, we can substitute it back into one of the original equations to solve for w. Let's use the first equation: 74(1.41) + 11w = 116 104.34 + 11w = 116 Subtracting 104.34 from both sides, we get: 11w = 11.66 Dividing both sides by 11, we get: w = 1.06 Therefore, the values of u and w that satisfy the system of equations are u = 1.41 and w = 1.06. ### Answer: u = 1.41, w = 1.06
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.
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method. First, we can multiply the first equation by 34 and the second equation by 42 to make the coefficients of f the same: 34 * (76 * f - 42 * j) = 34 * 96 42 * (47 * j + 34 * f) = 42 * 136 This gives us: 2584 * f - 1428 * j = 3264 1974 * j + 1428 * f = 5712 Next, we can subtract the first equation from the second equation to eliminate f: 1974 * j + 1428 * f - (2584 * f - 1428 * j) = 5712 - 3264 This simplifies to: 3402 * j = 2448 Dividing both sides by 3402 gives us: j = 2448 / 3402 = <<2448/3402=0.72>>0.72 Now that we have the value of j, we can substitute it back into one of the original equations to find the value of f. Let's use the first equation: 76 * f - 42 * 0.72 = 96 This simplifies to: 76 * f - 30.24 = 96 Adding 30.24 to both sides gives us: 76 * f = 126.24 Dividing both sides by 76 gives us: f = 126.24 / 76 = <<126.24/76=1.66>>1.66 Therefore, the values of f and j that will satisfy the system of equations are f = 1.66 and j = 0.72. ### Answer: f = 1.66, j = 0.72
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 63 and the second equation by 68 to eliminate u: 63(68u + 32e) = 63(174) 68(63e - 69u) = 68(1) Simplifying, we get: 4284u + 2016e = 10962 4284e - 4692u = 68 Adding the two equations, we get: 6300e = 11030 e = 11030/6300 = <<11030/6300=1.75>>1.75 Substituting e = 1.75 into the first equation, we get: 68u + 32(1.75) = 174 68u + 56 = 174 68u = 118 u = 118/68 = <<118/68=1.7352941176470588>>1.735 ### Answer: u = 1.735, e = 1.75
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.
Qwen2-Math 72B:
Reasoning: Let's solve the system of linear equations by substitution. First, we can solve the first equation for p: 57p + 4n = 129 57p = 129 - 4n p = (129 - 4n) / 57 Now, we can substitute this expression for p into the second equation: 24p + 39n = 154 24((129 - 4n) / 57) + 39n = 154 (3096 - 96n) / 57 + 39n = 154 3096 - 96n + 2223n = 8778 2127n = 5682 n = 5682 / 2127 n = 2.67 Now, we can substitute this value of n back into the expression for p: p = (129 - 4n) / 57 p = (129 - 4(2.67)) / 57 p = (129 - 10.68) / 57 p = 118.32 / 57 p = 2.075 Therefore, the values of p and n that satisfy the system of linear equations are p = 2.075 and n = 2.67. ### Answer: p = 2.075, 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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 14 and the second equation by 44 to make the coefficients of z in both equations equal: 14 * (66 * z - 44 * y) = 14 * 32 44 * (60 * y + 14 * z) = 44 * 116 This gives us: 924 * z - 616 * y = 448 2640 * y + 616 * z = 5104 Next, we can add the two equations together to eliminate z: 924 * z - 616 * y + 2640 * y + 616 * z = 448 + 5104 This simplifies to: 3564 * y = 5552 Dividing both sides by 3564 gives us: y = 5552 / 3564 = 1.56 Now that we have the value of y, we can substitute it back into one of the original equations to solve for z. Using the first equation: 66 * z - 44 * 1.56 = 32 Simplifying: 66 * z - 68.64 = 32 Adding 68.64 to both sides: 66 * z = 100.64 Dividing both sides by 66: z = 100.64 / 66 = 1.52 Therefore, the values of y and z are: y = 1.56 z = 1.52 ### Answer: y = 1.56, z = 1.52
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method. First, we can multiply the first equation by 11 and the second equation by 83 to make the coefficients of t the same: 11 * (83 * t - 58 * h) = 11 * 45 83 * (69 * h + 11 * t) = 83 * 124 This gives us: 913 * t - 638 * h = 495 5707 * h + 913 * t = 10292 Next, we can subtract the first equation from the second equation to eliminate t: 5707 * h + 913 * t - (913 * t - 638 * h) = 10292 - 495 This simplifies to: 6345 * h = 9797 Now, we can solve for h: h = 9797 / 6345 h = 1.54 Next, we can substitute the value of h back into one of the original equations to solve for t. Let's use the first equation: 83 * t - 58 * 1.54 = 45 This simplifies to: 83 * t - 89.32 = 45 Now, we can solve for t: 83 * t = 134.32 t = 134.32 / 83 t = 1.62 Therefore, the solution to the system of equations is t = 1.62 and h = 1.54. ### Answer: t = 1.62, h = 1.54
Expected answer: {h: 9797/6365, t: 10297/6365}
Correct? No
Problem: Given the system of equations ((79 * h) - (61 * o)) = 9 and ((16 * h) - (12 * o)) = 9, what are the values of o and h?
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we'll use elimination. First, we can multiply the second equation by 61/16 to make the coefficients of h in both equations equal: ((79 * h) - (61 * o)) = 9 ((61 * (16 * h) - 61 * (12 * o)) = 61 * 9 Simplifying the second equation gives us: ((61 * 16 * h) - (61 * 12 * o)) = 549 Now, we can subtract the first equation from the second equation to eliminate h: ((61 * 16 * h) - (61 * 12 * o)) - ((79 * h) - (61 * o)) = 549 - 9 Simplifying this gives us: ((61 * 16 * h) - (79 * h) - (61 * 12 * o) + (61 * o)) = 540 ((976 * h) - (79 * h) - (732 * o) + (61 * o)) = 540 ((897 * h) - (671 * o)) = 540 Now, we can solve for o: o = ((897 * h) - 540) / 671 Substituting this expression for o into the first equation gives us: ((79 * h) - (61 * ((897 * h) - 540) / 671)) = 9 Simplifying this gives us: ((79 * h) - ((61 * 897 * h) - (61 * 540)) / 671) = 9 ((79 * h) - ((54627 * h) - (32940)) / 671) = 9 ((79 * h) - (54627 * h) / 671 + (32940) / 671) = 9 ((79 * h) - (81.42 * h) + 49.09) = 9 ((79 * h) - (81.42 * h)) = 9 - 49.09 ((-2.42 * h)) = -40.09 h = -40.09 / -2.42 h = 16.57 Substituting this value of h back into the expression for o gives us: o = ((897 * 16.57) - 540) / 671 o = (14821.49 - 540) / 671 o = 14281.49 / 671 o = 21.28 ### Answer: o = 21.28, h = 16.57
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
Qwen2-Math 72B:
Reasoning: To solve for s and y, we can use the method of substitution or elimination. Here, we'll use elimination. First, we'll multiply the first equation by 90 and the second equation by 56 to make the coefficients of s in both equations equal: 90 * (4 * y + 56 * s) = 90 * 139 56 * (65 * y - 90 * s) = 56 * 16 This gives us: 360 * y + 5040 * s = 12510 3640 * y - 5040 * s = 896 Now, we can add these two equations to eliminate s: 360 * y + 3640 * y + 5040 * s - 5040 * s = 12510 + 896 4000 * y = 13406 Now, we can solve for y: y = 13406 / 4000 y = 3.3515 Now that we have y, we can substitute it back into one of the original equations to solve for s. We'll use the first equation: 4 * 3.3515 + 56 * s = 139 13.406 + 56 * s = 139 56 * s = 139 - 13.406 56 * s = 125.594 s = 125.594 / 56 s = 2.24275 ### Answer: y = 3.3515, s = 2.24275
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 45 and the second equation by 4 to make the coefficients of z the same: 45 * (42 * d - 4 * z) = 45 * 75 4 * (40 * d + 45 * z) = 4 * 162 This gives us: 1890 * d - 180 * z = 3375 160 * d + 180 * z = 648 Now, we can add the two equations to eliminate z: 1890 * d + 160 * d = 3375 + 648 2050 * d = 4023 Dividing both sides by 2050 gives us: d = 4023 / 2050 d = 1.962439024390244 Now that we have the value of d, we can substitute it back into one of the original equations to find z. Using the first equation: 42 * d - 4 * z = 75 42 * 1.962439024390244 - 4 * z = 75 Solving for z gives us: 82.42243902439024 - 4 * z = 75 -4 * z = 75 - 82.42243902439024 -4 * z = -7.42243902439024 z = -7.42243902439024 / -4 z = 1.85560975609756 ### Answer: d = 1.962439024390244, z = 1.85560975609756
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.
Qwen2-Math 72B:
Reasoning: Let's solve the system of equations step by step. First, we can multiply the second equation by 4 to make the coefficients of 'o' in both equations equal: (42 * o) + (24 * a) = 142 (168 * a) + (24 * o) = 300. Now, we can subtract the first equation from the second equation to eliminate 'o': (168 * a) + (24 * o) - ((42 * o) + (24 * a)) = 300 - 142 (126 * a) = 158. Now, we can solve for 'a': a = 158 / 126 a = 1.25. Now, we can substitute the value of 'a' into one of the original equations to solve for 'o': (42 * o) + (24 * 1.25) = 142 (42 * o) + 30 = 142 (42 * o) = 112 o = 112 / 42 o = 2.67. Therefore, the values of 'a' and 'o' are 1.25 and 2.67, respectively. ### Answer: a = 1.25, o = 2.67
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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method. First, we can multiply the first equation by 84 and the second equation by 39 to make the coefficients of r equal: 84*(39*r - 2*p) = 84*45 39*(16*p - 84*r) = 39*36 Simplifying these equations, we get: 3276*r - 168*p = 3780 624*p - 3276*r = 1404 Adding these two equations, we get: 456*p = 5184 Dividing both sides by 456, we get: p = 11.368421052631579 Substituting this value of p into the first equation, we get: 39*r - 2*11.368421052631579 = 45 Simplifying this equation, we get: 39*r = 67.73684210526316 Dividing both sides by 39, we get: r = 1.7368421052631577 ### Answer: p = 11.368421052631579, r = 1.7368421052631577
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
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method. First, we can multiply the first equation by 8 and the second equation by 23 to make the coefficients of 'd' in both equations equal: 1. (184 * d) - (416 * w) = 56 2. (1472 * w) - (184 * d) = 1334 Next, we can add the two equations to eliminate 'd': (184 * d) - (416 * w) + (1472 * w) - (184 * d) = 56 + 1334 1056 * w = 1390 Now, we can solve for 'w': w = 1390 / 1056 w = 1.316 Substituting the value of 'w' back into one of the original equations, we can solve for 'd': (23 * d) - (52 * 1.316) = 7 (23 * d) - 68.432 = 7 23 * d = 75.432 d = 75.432 / 23 d = 3.28 ### Answer: d = 3.28, w = 1.316
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 47 and the second equation by 60 to eliminate n: (71 * z * 47) + (60 * n * 47) = 184 * 47 (52 * z * 60) - (47 * n * 60) = 19 * 60 Simplify: 3337 * z + 2820 * n = 8648 3120 * z - 2820 * n = 1140 Add the two equations to eliminate n: (3337 * z) + (3120 * z) = 8648 + 1140 6457 * z = 9788 Divide both sides by 6457: z = 9788 / 6457 z = 1.516 Substitute z = 1.516 into the first equation: (71 * 1.516) + (60 * n) = 184 107.736 + 60 * n = 184 Subtract 107.736 from both sides: 60 * n = 184 - 107.736 60 * n = 76.264 Divide both sides by 60: n = 76.264 / 60 n = 1.271 ### Answer: n = 1.271, z = 1.516
Expected answer: {n: 8219/6457, z: 9788/6457}
Correct? No
Problem: Given the following system of equations, find the values of e and z: 70 * e + 90 * z = 189 99 * z - 84 * e = 20
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method. First, we can multiply the first equation by 84 and the second equation by 70 to make the coefficients of e in both equations equal: 84 * (70 * e + 90 * z) = 84 * 189 70 * (99 * z - 84 * e) = 70 * 20 This simplifies to: 5880 * e + 7560 * z = 15876 6930 * z - 5880 * e = 1400 Now, we can add the two equations to eliminate the e term: (5880 * e + 7560 * z) + (6930 * z - 5880 * e) = 15876 + 1400 7560 * z + 6930 * z = 17276 14490 * z = 17276 Dividing both sides by 14490, we get: z = 17276 / 14490 z = 1.192 Now that we have the value of z, we can substitute it back into one of the original equations to solve for e. Let's use the first equation: 70 * e + 90 * 1.192 = 189 70 * e + 107.28 = 189 70 * e = 189 - 107.28 70 * e = 81.72 Dividing both sides by 70, we get: e = 81.72 / 70 e = 1.167 Therefore, the values of e and z are: e = 1.167 z = 1.192 ### Answer: e = 1.167, z = 1.192
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.
Qwen2-Math 72B:
Reasoning: To solve this system of equations, we can use the substitution or elimination method. Let's use the elimination method. First, we can multiply equation 1 by 11 and equation 2 by 26 to make the coefficients of o the same: 1) 286o - 484f = 1045 2) 2262f + 286o = 4394 Now, we can subtract equation 1 from equation 2 to eliminate o: 2262f + 286o - (286o - 484f) = 4394 - 1045 2262f + 286o - 286o + 484f = 3349 2746f = 3349 Now, we can solve for f: f = 3349 / 2746 f = 1.22 Next, we can substitute the value of f back into either equation 1 or 2 to solve for o. Let's use equation 1: 26o - 44(1.22) = 95 26o - 53.68 = 95 26o = 148.68 Now, we can solve for o: o = 148.68 / 26 o = 5.72 Therefore, the values of o and f that satisfy the system of equations are o = 5.72 and f = 1.22. ### Answer: o = 5.72, f = 1.22
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.
Qwen2-Math 72B:
Reasoning: Multiply the first equation by 3 and the second equation by 20 to eliminate z: 120*z - 60*p = 153 120*z + 1160*p = 3300 Subtract the first equation from the second equation to solve for p: 1220*p = 3147 p = 3147/1220 = <<3147/1220=2.58>>2.58 Substitute p = 2.58 into the first equation to solve for z: 40*z - 20*2.58 = 51 40*z = 102.6 z = 102.6/40 = <<102.6/40=2.565>>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.
Qwen2-Math 72B:
Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method. First, we can multiply the second equation by 89 to make the coefficients of f in both equations equal: 89 * (3 * f + 10 * s) = 89 * 93 267 * f + 890 * s = 8277 Now, we can subtract the first equation from this new equation to eliminate f: (267 * f + 890 * s) - (71 * s - 89 * f) = 8277 - 5 267 * f + 890 * s - 71 * s + 89 * f = 8272 356 * f + 819 * s = 8272 Next, we can multiply the first equation by 3 to make the coefficients of s in both equations equal: 3 * (71 * s - 89 * f) = 3 * 5 213 * s - 267 * f = 15 Now, we can add this new equation to the previous equation to eliminate s: (356 * f + 819 * s) + (213 * s - 267 * f) = 8272 + 15 356 * f + 819 * s + 213 * s - 267 * f = 8287 80 * f + 1032 * s = 8287 Finally, we can solve for f and s using the resulting equation: 80 * f = 8287 - 1032 * s f = (8287 - 1032 * s) / 80 Substituting this expression for f into the first equation, we get: 71 * s - 89 * ((8287 - 1032 * s) / 80) = 5 71 * s - (89 * 8287 - 89 * 1032 * s) / 80 = 5 71 * s - (737543 - 92088 * s) / 80 = 5 71 * s - 9219.2875 + 1151.1 * s = 5 1822.1 * s = 9224.2875 s = 9224.2875 / 1822.1 s = 5.06 Substituting this value of s back into the expression for f, we get: f = (8287 - 1032 * 5.06) / 80 f = (8287 - 5203.52) / 80 f = 3083.48 / 80 f = 38.54 Therefore, the values of s and f are s = 5.06 and f = 38.54. ### Answer: s = 5.06, f = 38.54
Expected answer: {f: 6553/1103, s: 8292/1103}
Correct? No