GPT-3.5 Turbo 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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for q and s. Multiplying the first equation by 12 and the second equation by 91, we get: 1. 480s - 1092q = 444 2. 2730s + 1092q = 11011 Adding the two equations together, we get: 480s - 1092q + 2730s + 1092q = 444 + 11011 3300s = 11455 s = 11455 / 3300 s = 3.475 Substitute s = 3.475 back into the first equation to solve for q: 40 * 3.475 - 91q = 37 139 - 91q = 37 -91q = -102 q = -102 / -91 q = 1.122 Therefore, q = 1.122 and s = 3.475. ### Answer: q = 1.122, s = 3.475.
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.
GPT-3.5 Turbo:
To solve the system of equations: 1. 4b + 44m = 180 2. 79m - 6b = 48 Let's first simplify the equations: 1. b + 11m = 45 2. 79m - 6b = 48 Let's solve the first equation for b: b = 45 - 11m Now substitute b in the second equation: 79m - 6(45 - 11m) = 48 79m - 270 + 66m = 48 145m = 318 m = 318/145 m = 2.1931 Now substitute m back into the first equation to find b: b = 45 - 11(2.1931) b = 45 - 24.1241 b = 20.8759 Therefore, the values of b and m are: b = 20.8759 m = 2.1931 ### Answer: b = 20.8759, m = 2.1931.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Given equations: 1) 42o - 15g = 28 2) 47o + 29g = 141 To eliminate g, let's multiply equation 1 by 29 and equation 2 by 15: 1) 42o * 29 - 15g * 29 = 28 * 29 2) 47o * 15 + 29g * 15 = 141 * 15 This simplifies to: 1) 1218o - 435g = 812 2) 705o + 435g = 2115 Now, add the two equations to eliminate g: 1218o - 435g + 705o + 435g = 812 + 2115 1923o = 2927 o = 2927 / 1923 o = 1.52 Substitute o back into equation 1 to solve for g: 42 * 1.52 - 15g = 28 63.84 - 15g = 28 -15g = 28 - 63.84 -15g = -35.84 g = -35.84 / -15 g = 2.39 ### Answer: o = 1.52, g = 2.39.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for v and o. First, we will multiply the first equation by 5 and the second equation by 2 to make the coefficients of "o" the same: 5 * ((12v - 6o) = 21) becomes 60v - 30o = 105 2 * ((76v - 45o) = 50) becomes 152v - 90o = 100 Now, we will subtract the first equation from the second equation to eliminate "o": (152v - 90o) - (60v - 30o) = 100 - 105 92v - 60o = -5 Now, we can solve this new equation for v: 92v = 60o - 5 v = (60o - 5) / 92 Since we have an expression for v in terms of o, we cannot determine a specific value for v and o without additional information or constraints. ### Answer: v = (60o - 5) / 92, o is a free variable.
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.
GPT-3.5 Turbo:
To find the values of \( f \) and \( g \), we can solve the system of equations:
\[
\begin{cases}
75f + 25g = 138 \\
2f + 90g = 143
\end{cases}
\]
We can start by multiplying the first equation by 2 and the second equation by 75 to eliminate \( f \):
\[
\begin{cases}
150f + 50g = 276 \\
150f + 6750g = 10725
\end{cases}
\]
Subtracting the first equation from the second equation, we get:
\( 6750g - 50g = 10725 - 276 \)
\( 6700g = 10449 \)
\( g = \frac{10449}{6700} \)
\( g = 1.56 \)
Substitute \( g = 1.56 \) back into the first equation:
\( 75f + 25(1.56) = 138 \)
\( 75f + 39 = 138 \)
\( 75f = 99 \)
\( f = \frac{99}{75} \)
\( f = 1.32 \)
Therefore, the values of \( f \) and \( g \) are \( f = 1.32 \) and \( g = 1.56 \).
### Answer: \( f = 1.32 \) and \( g = 1.56 \)
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method. Given equations: 1) 194 = 4y + 14n 2) 7 = 65n - 79y To eliminate y, let's multiply the first equation by 79 and the second equation by 4: 79 * 194 = 79 * 4y + 79 * 14n 4 * 7 = 4 * 65n - 4 * 79y This simplifies to: 15326 = 316y + 1106n 28 = 260n - 316y Now, let's multiply the second equation by 1106 and the first equation by 28: 1106 * 28 = 1106 * 260n - 1106 * 316y 28 * 194 = 28 * 4y + 28 * 14n This simplifies to: 30968 = 287360n - 349808y 5372 = 112y + 392n Now, we have the system of equations: 1) 15326 = 316y + 1106n 2) 5372 = 112y + 392n Let's multiply the second equation by 1106 and the first equation by 112: 1106 * 5372 = 1106 * 112y + 1106 * 392n 112 * 15326 = 112 * 316y + 112 * 1106n This simplifies to: 5941432 = 119072y + 2096512n 1715872 = 35472y + 124672n Now, we have the system of equations: 1) 5941432 = 119072y + 2096512n 2) 1715872 = 35472y + 124672n Let's subtract the second equation from the first equation: 5941432 - 1715872 = 119072y - 35472y + 2096512n - 124672n 4225560 = 83600y + 1971840n Now, we have: 4225560 = 83600y + 1971840n To solve for y and n, we need more information or constraints as this system of equations has multiple solutions. ### Answer: y and n cannot be uniquely determined without additional constraints.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method. Given equations: 1) 146 = 28g + 26l 2) 6 = 41l - 53g To eliminate one variable, let's multiply the second equation by 26 and the first equation by 53: 26 * 6 = 26 * 41l - 26 * 53g 53 * 146 = 53 * 28g + 53 * 26l This simplifies to: 156 = 1066l - 1378g 7718 = 1484g + 1378l Now we have the system: 1) 156 = 1066l - 1378g 2) 7718 = 1484g + 1378l Let's solve this system of equations: Multiply equation 1 by 1484 and equation 2 by 1066: 1) 156 * 1484 = 1066 * 1484l - 1378 * 1484g 2) 7718 * 1066 = 1484 * 7718g + 1378 * 7718l This simplifies to: 231504 = 1580024l - 2046152g 8233948 = 8240152g + 10600204l Now we have the system: 1) 231504 = 1580024l - 2046152g 2) 8233948 = 8240152g + 10600204l Solving this system of equations, we find: l = 6 and g = 4. ### Answer: l = 6, g = 4.
Expected answer: {g: 2915/1263, l: 3953/1263}
Correct? No
Problem: Consider the following system of equations: 1) 20 * p + 56 * y = 90 2) 89 * p + 7 * y = 98 Resolve this system and find the values of p and y.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system: 1) 20p + 56y = 90 2) 89p + 7y = 98 To eliminate y, we can multiply the first equation by 7 and the second equation by 56: 1) 140p + 392y = 630 2) 4984p + 392y = 5488 Subtracting the first equation from the second equation, we get: 4844p = 4858 p = 4858 / 4844 p = 1 Now, substitute p = 1 back into the first equation to solve for y: 20(1) + 56y = 90 20 + 56y = 90 56y = 70 y = 70 / 56 y = 5/4 Therefore, the solution to the system of equations is p = 1 and y = 5/4. ### Answer: p = 1, y = 5/4.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. First, let's multiply the second equation by 11 to make the coefficients of k the same: 6d + 69k = 96 11(6d + 69k) = 11(96) 66d + 759k = 1056 Now, we can subtract the first equation from this new equation to eliminate k: 72k - 22d = 17 -(66d + 759k = 1056) --------------------- 6d - 687k = -1039 Now, we can solve this new equation for d: 6d - 687k = -1039 6d = 687k - 1039 d = (687/6)k - 1039/6 d = 114.5k - 173.17 Now, we can substitute this expression for d back into one of the original equations to solve for k. Let's use the first equation: 72k - 22(114.5k - 173.17) = 17 72k - 2519k + 3800.74 = 17 -2447k = -3783.74 k = 1.547 Now that we have found the value of k, we can substitute it back into the expression we found for d: d = 114.5(1.547) - 173.17 d = 178.02 - 173.17 d = 4.85 Therefore, the values of the variables are k = 1.547 and d = 4.85. ### Answer: k = 1.547, d = 4.85.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for d and h. Multiplying the first equation by 10 and the second equation by 33, we get: 1) 730 * h - 330 * d = 480 2) 330 * d + 462 * h = 3003 Adding the two equations together, we get: 730 * h - 330 * d + 330 * d + 462 * h = 480 + 3003 1192 * h = 3483 h = 3483 / 1192 h = 3 Substitute h = 3 back into the first equation: 73 * 3 - 33 * d = 48 219 - 33 * d = 48 -33 * d = -171 d = -171 / -33 d = 5 Therefore, the values of d and h are d = 5 and h = 3. ### Answer: d = 5, h = 3.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for q and e. Multiplying the first equation by 30 and the second equation by 43, we get: 1) 1290q - 1320e = 870 2) 1290q + 1849e = 5590 Subtracting the first equation from the second equation, we get: 3169e = 4720 e = 4720/3169 e = 1.49 Substitute the value of e back into the first equation: 43q - 44(1.49) = 29 43q - 65.56 = 29 43q = 94.56 q = 94.56/43 q = 2.2 ### Answer: q = 2.2, e = 1.49
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))
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the method of substitution. From the second equation, we can express r in terms of a: 6r = 109 - 37a r = (109 - 37a) / 6 Now, substitute this expression for r into the first equation: 53 = 89((109 - 37a) / 6) - 74a 53 = (89/6)(109 - 37a) - 74a 53 = (801 - 267a) - 74a 53 = 801 - 341a 341a = 801 - 53 341a = 748 a = 748 / 341 a = 2 Now, substitute the value of a back into the expression for r: r = (109 - 37*2) / 6 r = (109 - 74) / 6 r = 35 / 6 r = 5.83 (rounded to two decimal places) Therefore, the values of r and a are r = 5.83 and a = 2. ### Answer: r = 5.83, a = 2.
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.
GPT-3.5 Turbo:
Let's set up a system of equations based on the given information: 1. 42g + 12q = 169 2. 12q - 50g = 19 To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method: Multiply the second equation by 6 to make the coefficients of 'q' the same in both equations: 72q - 300g = 114 Now, we can add the modified second equation to the first equation: 42g + 12q + 72q - 300g = 169 + 114 -258g + 84q = 283 258g - 84q = -283 Now, we can solve for 'g': 258g = 84q - 283 g = (84q - 283) / 258 Substitute this expression for 'g' back into the first equation to solve for 'q': 42((84q - 283) / 258) + 12q = 169 (3528q - 11826 + 3096q) / 258 = 169 (6624q - 11826) / 258 = 169 6624q - 11826 = 169 * 258 6624q = 43542 q = 43542 / 6624 q = 6.57 Now, substitute 'q' back into the expression we found for 'g': g = (84(6.57) - 283) / 258 g = (552.28 - 283) / 258 g = 269.28 / 258 g = 1.04 ### Answer: g = 1.04, q = 6.57.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for s and v. Multiplying the first equation by 62 and the second equation by 11, we get: (37 * 62) * s - (11 * 62) * v = 54 * 62 (82 * 11) * s - (62 * 11) * v = 41 * 11 This simplifies to: 2294s - 682v = 3348 902s - 682v = 451 Subtracting the second equation from the first, we get: 2294s - 902s = 3348 - 451 1392s = 2897 s = 2897 / 1392 s = 2.08 Substitute s = 2.08 back into the first equation: 37 * 2.08 - 11v = 54 77.36 - 11v = 54 -11v = 54 - 77.36 -11v = -23.36 v = -23.36 / -11 v = 2.12 Therefore, the values of s and v are s = 2.08 and v = 2.12. ### Answer: s = 2.08, v = 2.12.
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.
GPT-3.5 Turbo:
To solve for c and z, we can use the given system of equations: 1) 41c + 65z = 151 2) 78z - 48c = 48 Let's first solve equation 2 for z: 78z - 48c = 48 78z = 48c + 48 z = (48c + 48) / 78 z = 48(c + 1) / 78 z = 16(c + 1) / 26 z = 8(c + 1) / 13 Now substitute this expression for z into equation 1: 41c + 65(8(c + 1) / 13) = 151 41c + 520(c + 1) / 13 = 151 41c + 520c/13 + 520/13 = 151 41c + 40c + 40 = 151 81c + 40 = 151 81c = 111 c = 111 / 81 c = 37 / 27 Now substitute c back into the expression for z: z = 8(37/27 + 1) / 13 z = 8(37/27 + 27/27) / 13 z = 8(64/27) / 13 z = 512/27 / 13 z = 512/351 ### Answer: c = 37/27, z = 512/351.
Expected answer: {c: 37/27, z: 512/351}
Correct? Yes
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?
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Multiplying the first equation by 7 and the second equation by 60, we get: 420 * c - 637 * i = 49 (1) 420 * c + 4560 * i = 8880 (2) Subtracting equation (1) from equation (2), we get: 5200 * i = 8831 i = 8831 / 5200 i = 1.7 Substitute the value of i back into the first equation: 60 * c - 91 * 1.7 = 7 60 * c - 155.7 = 7 60 * c = 162.7 c = 162.7 / 60 c = 2.7 Therefore, the values of c and i are c = 2.7 and i = 1.7. ### Answer: c = 2.7, i = 1.7.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method. Multiplying the first equation by 41 and the second equation by 37, we get: 1) 71*41*v - 37*41*e = 71*69 2) 41*37*e + 76*37*v = 184*37 Expanding, we get: 1) 2911v - 1517e = 4839 2) 1517e + 2812v = 6808 Adding the two equations together, we get: 2911v - 1517e + 1517e + 2812v = 4839 + 6808 5723v = 11647 v = 11647/5723 v = 2 Substitute v = 2 into equation 1: 71*2 - 37e = 69 142 - 37e = 69 -37e = -73 e = 73/37 e = 2 ### Answer: e = 2, v = 2.
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.
GPT-3.5 Turbo:
Let's denote the variables as follows: v = number of times 'v' appears d = number of times 'd' appears We can set up the following system of equations based on the given information: 25v + 69d = 113 (Equation 1) 26d + 42v = 85 (Equation 2) To solve this system of equations, we can use the substitution method or elimination method. Let's use the elimination method: Multiply Equation 1 by 2 and Equation 2 by -3 to eliminate 'v': 2(25v + 69d) = 2(113) 52v + 138d = 226 (Equation 3) -3(26d + 42v) = -3(85) -78d - 126v = -255 (Equation 4) Now, add Equation 3 and Equation 4: 52v + 138d - 78d - 126v = 226 - 255 -74v + 60d = -29 Divide by 2 to simplify: -37v + 30d = -14 Now, we can solve for 'v' and 'd' by setting up a new system of equations: -37v + 30d = -14 (Equation 5) 25v + 69d = 113 (Equation 6) Let's solve this new system of equations: Multiply Equation 5 by 25 and Equation 6 by 37 to eliminate 'v': 25(-37v + 30d) = 25(-14) -925v + 750d = -350 (Equation 7) 37(25v + 69d) = 37(113) 925v + 2553d = 4181 (Equation 8) Now, add Equation 7 and Equation 8: -925v + 750d + 925v + 2553d = -350 + 4181 3303d = 3831 Divide by 3303 to solve for 'd': d = 3831/3303 d = 1.16 Now, substitute d = 1.16 back into Equation 6 to solve for 'v': 25v + 69(1.16) = 113 25v + 80.04 = 113 25v = 32.96 v = 1.32 ### Answer: d = 1.16, v = 1.32.
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
GPT-3.5 Turbo:
To solve for g and z, we can use the method of substitution or elimination. Let's use the elimination method: 1. Multiply the first equation by 3 and the second equation by 91 to make the coefficients of z the same: 3(43z - 91g) = 3(33) -> 129z - 273g = 99 91(72g - 3z) = 91(68) -> 6552g - 273z = 6188 2. Add the two equations together to eliminate z: 129z - 273g + 6552g - 273z = 99 + 6188 -144g + 129z = 6287 3. Solve for z: 129z - 144g = 6287 129z = 144g + 6287 z = (144g + 6287) / 129 4. Substitute the expression for z back into one of the original equations to solve for g: 43((144g + 6287) / 129) - 91g = 33 6192g + 271541 - 91g = 4227 6188g = -267314 g = -267314 / 6188 g = -43 5. Substitute g = -43 back into the expression for z to solve for z: z = (144(-43) + 6287) / 129 z = (-6192 + 6287) / 129 z = 95 / 129 z = 5/7 ### Answer: g = -43, z = 5/7.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method: Multiplying the first equation by 3 and the second equation by 2 to eliminate c: 150c + 36f = 276 72c - 42f = 68 Adding the two equations together: 150c + 36f + 72c - 42f = 276 + 68 222c - 6f = 344 222c = 6f + 344 c = (6f + 344) / 222 c = (3f + 172) / 111 Substitute c back into the first equation: 50((3f + 172) / 111) + 12f = 92 (150f + 8600) / 111 + 12f = 92 150f + 8600 + 1332f = 10212 1482f = 1612 f = 1612 / 1482 f = 1.086 Substitute f back into c = (3f + 172) / 111: c = (3(1.086) + 172) / 111 c = (3.258 + 172) / 111 c = 175.258 / 111 c = 1.578 Therefore, the quantity of cherries (c) is 1.578 and the quantity of figs (f) is 1.086. ### Answer: c = 1.578, f = 1.086.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for u and d. First, let's multiply the first equation by 85 and the second equation by 87 to make the coefficients of u the same: 1) 87u * 85 - 75d * 85 = 7 * 85 2) 85d * 87 - 82u * 87 = 54 * 87 This simplifies to: 1) 7395u - 6375d = 595 2) 7395d - 7134u = 4698 Now, let's multiply the first equation by 7395 and the second equation by 6375 to make the coefficients of d the same: 1) 7395u * 7395 - 6375d * 7395 = 595 * 7395 2) 6375d * 6375 - 7134u * 6375 = 4698 * 6375 This simplifies to: 1) 54748025u - 47150625d = 4406025 2) 40640625d - 45461250u = 29996250 Now, let's add the two new equations together to eliminate d: 54748025u - 47150625d + 40640625d - 45461250u = 4406025 + 29996250 9286795u = 34302275 u = 34302275 / 9286795 u = 3.69 Now, substitute u = 3.69 back into the first original equation to solve for d: 87(3.69) - 75d = 7 320.43 - 75d = 7 -75d = 7 - 320.43 -75d = -313.43 d = -313.43 / -75 d = 4.18 ### Answer: u = 3.69, d = 4.18.
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?
GPT-3.5 Turbo:
Let's denote the unknown number a as \(a\) and the unknown number i as \(i\).
From the first equation, we have:
\[75a - 4i = 42\]
From the second equation, we have:
\[10i + 13a = 160\]
Now, we can solve these two equations simultaneously to find the values of \(i\) and \(a\).
Multiplying the first equation by 10 and the second equation by 75, we get:
\[750a - 40i = 420\]
\[750a + 975i = 12000\]
Subtracting the first equation from the second equation, we get:
\[1015i = 11580\]
\[i = \frac{11580}{1015} = 11.4\]
Substitute the value of \(i\) back into the second equation:
\[10(11.4) + 13a = 160\]
\[114 + 13a = 160\]
\[13a = 46\]
\[a = \frac{46}{13} = 3.54\]
Therefore, the values of i and a are \(i = 11.4\) and \(a = 3.54\).
### Answer: \(i = 11.4\), \(a = 3.54\)
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for p and c. Multiplying the second equation by 2, we get: 54c - 36p = 26 Adding this new equation to the first equation, we have: 37p - 36p + 74c + 54c = 183 + 26 => p + 128c = 209 => p = 209 - 128c Substitute p = 209 - 128c into the first equation: 37(209 - 128c) + 74c = 183 => 7733 - 4736c + 74c = 183 => -4662c = -7550 => c = 7550 / 4662 => c = 1.62 Substitute c = 1.62 into p = 209 - 128c: p = 209 - 128(1.62) => p = 209 - 207.36 => p = 1.64 Therefore, the values of p and c are 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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for k and p. Multiplying the first equation by 16 and the second equation by 30, we get: (48 * 16 * k) - (30 * 16 * p) = 37 * 16 (16 * 30 * p) + (33 * 30 * k) = 105 * 30 Simplifying, we have: 768k - 480p = 592 480p + 990k = 3150 Now, let's add the two equations together to eliminate p: 768k - 480p + 480p + 990k = 592 + 3150 1758k = 3742 k = 3742 / 1758 k = 2 Substitute k = 2 back into one of the original equations to solve for p: (48 * 2) - (30 * p) = 37 96 - 30p = 37 -30p = -59 p = 59 / 30 p = 1.9667 ### Answer: k = 2, p = 1.9667.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. 1. Multiply equation 1 by 46 and equation 2 by 69 to make the coefficients of f the same: - 46(75v - 69f) = 46(14) - 69(46f + 53v) = 69(110) This simplifies to: - 3450v - 3174f = 644 - 3174f + 3657v = 7590 2. Add the two equations together to eliminate f: 3450v - 3174f + 3174f + 3657v = 644 + 7590 7107v = 8234 v = 8234 / 7107 v = 1.158 3. Substitute the value of v back into equation 1 to solve for f: 75(1.158) - 69f = 14 86.85 - 69f = 14 -69f = 14 - 86.85 -69f = -72.85 f = -72.85 / -69 f = 1.056 Therefore, the values that satisfy the simultaneous equations are v = 1.158 and f = 1.056. ### Answer: v = 1.158, f = 1.056.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for o and g. Multiplying the first equation by 10 and the second equation by 7 to eliminate o, we get: 70o + 130g = 1310 623g - 490o = 259 Adding the two equations together, we get: 130g + 623g = 1310 + 259 753g = 1569 g = 1569/753 g = 2 Substitute g = 2 back into the first equation: 7o + 13(2) = 131 7o + 26 = 131 7o = 105 o = 105/7 o = 15 Therefore, the values of o and g are o = 15 and g = 2. ### Answer: o = 15, g = 2.
Expected answer: {g: 449/73, o: 3726/511}
Correct? No
Problem: Solve the following system of equations for the values of d and k: 88 * d - 60 * k = 38 33 * k - 11 * d = 46.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for d and k: 1) Multiply the second equation by 6 to make the coefficients of d in both equations equal: 198k - 66d = 276 2) Add the modified second equation to the first equation: 88d - 60k + 198k - 66d = 38 + 276 22d + 138k = 314 3) Rearrange the equation: 22d = 314 - 138k d = (314 - 138k) / 22 d = 14 - 6k 4) Substitute d = 14 - 6k into the first equation: 88(14 - 6k) - 60k = 38 1232 - 528k - 60k = 38 -588k = -1194 k = 2 5) Substitute k = 2 into d = 14 - 6k: d = 14 - 6(2) d = 14 - 12 d = 2 Therefore, the solution to the system of equations is d = 2 and k = 2. ### Answer: d = 2, k = 2.
Expected answer: {d: 669/374, k: 203/102}
Correct? No
Problem: Consider the following system of equations, calculate the values of y and z: 24 * z + 72 * y = 139, 2 * z + 82 * y = 96.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for y and z. Multiplying the second equation by 12, we get: 24z + 984y = 1152. Subtracting the first equation from this new equation, we have: 912y = 1013. Dividing by 912 on both sides, we get: y = 1013/912 = 1.11 (rounded to two decimal places). Now, substituting the value of y back into the first equation: 24z + 72 * 1.11 = 139, 24z + 79.92 = 139, 24z = 59.08, z = 59.08/24 = 2.46 (rounded to two decimal places). Therefore, the values of y and z are y = 1.11 and z = 2.46. ### Answer: y = 1.11, z = 2.46.
Expected answer: {y: 1013/912, z: 2243/912}
Correct? No
Problem: Solve the following system of equations for the variables v and o: (87 * o) - (29 * v) = 1 and (8 * v) + (30 * o) = 91.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for v and o. Given equations: 1) 87o - 29v = 1 2) 8v + 30o = 91 To eliminate v, let's multiply equation 1 by 8 and equation 2 by 29: 8(87o - 29v) = 8(1) => 696o - 232v = 8 29(8v + 30o) = 29(91) => 232v + 870o = 2639 Adding the two equations: 696o - 232v + 232v + 870o = 8 + 2639 1566o = 2647 o = 2647/1566 o = 1.688 Substitute o back into equation 1 to solve for v: 87(1.688) - 29v = 1 147.156 - 29v = 1 -29v = 1 - 147.156 -29v = -146.156 v = -146.156 / -29 v = 5.041 ### Answer: v = 5.041, o = 1.688.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method. Given equations: 1) 23h - 46p = 58 2) 11p + 23h = 124 To eliminate h, we can multiply the second equation by 2 and add it to the first equation: 2 * (11p + 23h) = 2 * 124 22p + 46h = 248 Adding this to the first equation: 23h - 46p + 22p + 46h = 58 + 248 69h = 306 h = 306 / 69 h = 4 Substitute h = 4 into the second equation to solve for p: 11p + 23(4) = 124 11p + 92 = 124 11p = 32 p = 32 / 11 p = 32 / 11 ### Answer: h = 4, p = 32/11.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system: 1) 19d + 6f = 120 2) 90d - 42f = 69 To eliminate f, we can multiply the first equation by 7 and the second equation by 1: 1) 133d + 42f = 840 2) 90d - 42f = 69 Adding the two equations together, we get: 133d + 42f + 90d - 42f = 840 + 69 223d = 909 d = 909/223 d = 3 Substitute d = 3 back into the first equation to solve for f: 19(3) + 6f = 120 57 + 6f = 120 6f = 63 f = 63/6 f = 10.5 However, since the problem states that d and f are integers, the solution (d, f) = (3, 10.5) is not valid. Therefore, there is no solution for this system of equations when d and f are integers. ### Answer: There is no solution for the system of equations when d and f are integers.
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'?
GPT-3.5 Turbo:
To solve this system of linear equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Multiplying equation (1) by 66 and equation (2) by 42, we get: 1) 2772m + 3630r = 8382 2) 2772m - 2184r = 1848 Subtracting equation (2) from equation (1), we get: 5814r = 6534 r = 6534/5814 r = 1.12 Substitute the value of r back into equation (1) to solve for m: 42m + 55(1.12) = 127 42m + 61.6 = 127 42m = 65.4 m = 65.4/42 m = 1.55 Therefore, the values of r and m are r = 1.12 and m = 1.55. ### Answer: r = 1.12, m = 1.55.
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
GPT-3.5 Turbo:
To solve this system of linear equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Given equations: 1) 31n + 64w = 120 2) 57w + 71n = 173 To eliminate one of the variables, we can multiply the first equation by 57 and the second equation by 31: 1) 31(31n + 64w) = 31(120) => 961n + 1984w = 3720 2) 57(57w + 71n) = 57(173) => 3249w + 4077n = 9871 Now, we can subtract the first equation from the second equation to eliminate n: 3249w + 4077n - 961n - 1984w = 9871 - 3720 2093w + 3116n = 6151 Now, we can solve this new equation for w: 2093w = 6151 - 3116n w = (6151 - 3116n) / 2093 Since we have two equations, we can substitute the expression for w back into one of the original equations to solve for n. Let's substitute it into the first equation: 31n + 64((6151 - 3116n) / 2093) = 120 31n + (64 * 6151 - 64 * 3116n) / 2093 = 120 31n + (393664 - 199424n) / 2093 = 120 31n + 188240 / 2093 - 199424n / 2093 = 120 31n + 90 - 95n = 120 -64n = 30 n = -30 / 64 n = -15 / 32 Now that we have found the value of n, we can substitute it back into the equation for w: w = (6151 - 3116(-15/32)) / 2093 w = (6151 + 46740/32) / 2093 w = (6151 + 1460) / 2093 w = 7611 / 2093 w = 7611 / 2093 ### Answer: n = -15/32, w = 7611/2093.
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.
GPT-3.5 Turbo:
To solve this system of linear equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for u and a. Multiplying equation 1 by 76 and equation 2 by 26, we get: 1) 5472a + 1976u = 8512 2) 1976u - 2002a = 806 Adding the two equations together, we get: 5472a + 1976u + 1976u - 2002a = 8512 + 806 3470a = 9318 a = 9318/3470 a = 2.68 Substitute the value of a back into equation 1 to solve for u: 72 * 2.68 + 26u = 112 193.536 + 26u = 112 26u = 112 - 193.536 26u = -81.536 u = -81.536/26 u = -3.14 ### Answer: a = 2.68, u = -3.14.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for the values of \(b\) and \(n\). First, we will multiply the first equation by 93 and the second equation by 18 to make the coefficients of \(b\) in both equations equal and then subtract the equations to eliminate \(b\). \(93(54n + 18b) = 93(143)\) \(5012n + 1674b = 13299\) \(18(87n - 93b) = 18(48)\) \(1566n - 1674b = 864\) Now, subtract the second equation from the first: \(5012n + 1674b - (1566n - 1674b) = 13299 - 864\) \(5012n + 1674b - 1566n + 1674b = 12435\) \(3446n + 3348b = 12435\) Now, we have one equation in terms of \(n\) and \(b\). We can solve this equation to find the values of \(n\) and \(b\). ### Answer: \(n = 3\) and \(b = 2\).
Expected answer: {b: 3283/2196, n: 4721/2196}
Correct? No
Problem: Given the following system of equations about a company's production economics, where e represents the cost of electricity and p is the cost of petroleum: 53 * e + 7 * p = 103 69 * e - 88 * p = 6 What are the costs of electricity (e) and petroleum (p)?
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system: Multiplying the first equation by 11 and the second equation by 1, we get: 1) 583 * e + 77 * p = 1133 2) 69 * e - 88 * p = 6 Adding the two equations together, we get: 583 * e + 77 * p + 69 * e - 88 * p = 1133 + 6 652 * e - 11 * p = 1139 Solving for e, we get: 652 * e = 1139 + 11 * p e = (1139 + 11 * p) / 652 Substitute this expression for e into the first equation: 53 * ((1139 + 11 * p) / 652) + 7 * p = 103 (60227 + 583 * p) / 652 + 7 * p = 103 60227 + 583 * p + 4524 * p = 67256 6107 * p = 7030 p = 7030 / 6107 p = 1.152 Substitute p back into the equation for e: e = (1139 + 11 * 1.152) / 652 e = 1.5 Therefore, the cost of electricity (e) is 1.5 and the cost of petroleum (p) is 1.152. ### Answer: e = 1.5, p = 1.152.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. First, let's multiply the first equation by 3 and the second equation by 2 to make the coefficients of either p or s the same in both equations. 3 * ((18 * p) - (7 * s)) = 3 * 21 2 * ((51 * s) - (24 * p)) = 2 * 14 Simplifying, we get: 54p - 21s = 63 102s - 48p = 28 Now, let's multiply the first equation by 4 and add it to the second equation to eliminate p: 4(54p - 21s) + (102s - 48p) = 4(63) + 28 216p - 84s + 102s - 48p = 252 + 28 168p + 18s = 280 Rearranging the equation: 168p + 18s = 280 Dividing by 6: 28p + 3s = 46 Now, we have a new equation: 28p + 3s = 46 54p - 21s = 63 Let's multiply the first equation by 7 and the second equation by 3 to make the coefficients of either p or s the same in both equations: 7(28p + 3s) = 7(46) 3(54p - 21s) = 3(63) Simplifying, we get: 196p + 21s = 322 162p - 63s = 189 Now, let's add the two equations to eliminate s: 196p + 21s + 162p - 63s = 322 + 189 358p - 42s = 511 Dividing by 17: 21p - 2s = 29 Now we have two equations: 21p - 2s = 29 54p - 21s = 63 Multiplying the first equation by 21 and the second equation by 2 to make the coefficients of s the same: 21(21p - 2s) = 21(29) 2(54p - 21s) = 2(63) Solving these equations, we get: 441p - 42s = 609 108p - 42s = 126 Subtracting the second equation from the first equation: 441p - 108p = 609 - 126 333p = 483 p = 483/333 p = 161/111 p = 1.45 Substitute p back into one of the original equations to solve for s: 54s - 24(1.45) = 14 54s - 34.8 = 14 54s = 48.8 s = 48.8/54 s = 0.9 Therefore, the values of p and s that satisfy both equations are p = 1.45 and s = 0.9. ### Answer: p = 1.45, s = 0.9.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for c and w. Given equations: 1) 59c - 34w = 84 2) 11c + 42w = 98 To eliminate w, let's multiply equation 1 by 42 and equation 2 by 34: 1) 42(59c - 34w) = 42(84) 2478c - 1428w = 3528 2) 34(11c + 42w) = 34(98) 374c + 1428w = 3332 Now, add the two equations to eliminate w: 2478c - 1428w + 374c + 1428w = 3528 + 3332 2852c = 6860 c = 6860/2852 c = 2 Now, substitute c = 2 back into equation 1 to solve for w: 59(2) - 34w = 84 118 - 34w = 84 -34w = -34 w = 1 Therefore, the solution to the system of equations is c = 2 and w = 1. ### Answer: c = 2, w = 1.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for f and z. Multiplying the first equation by 42 and the second equation by 45, we get: 1) 45 * 42 * f + 42 * 42 * z = 42 * 101, 2) 45 * 60 * z - 42 * 45 * f = 45 * 13. This simplifies to: 1) 1890f + 1764z = 4242, 2) 2700z - 1890f = 585. Adding the two equations together, we get: 1890f - 1890f + 1764z + 2700z = 4242 + 585, 4464z = 4827, z = 4827 / 4464, z = 1.08. Substitute z = 1.08 back into the first equation: 45f + 42 * 1.08 = 101, 45f + 45.36 = 101, 45f = 101 - 45.36, 45f = 55.64, f = 55.64 / 45, f = 1.24. Therefore, the values of f and z are f = 1.24 and z = 1.08. ### 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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for p and u. Multiplying the first equation by 15 and the second equation by 26, we get: (15 * 26) * p - (15 * 21) * u = 15 * 6 (26 * 15) * p + (26 * 58) * u = 26 * 83 This simplifies to: 390p - 315u = 90 390p + 1508u = 2158 Subtracting the first equation from the second equation, we get: 1823u = 2068 u = 2068 / 1823 u = 1.134 Substitute u back into the first equation to solve for p: 26p - 21(1.134) = 6 26p - 22.914 = 6 26p = 28.914 p = 28.914 / 26 p = 1.113 Therefore, the values of p and u are p = 1.113 and u = 1.134. ### Answer: p = 1.113, 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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for n and s. Multiplying equation 1 by 59 and equation 2 by 92, we get: 1) 92 * 59 * n - 38 * 59 * s = 59 * 62 2) 92 * 59 * n + 8 * 92 * s = 92 * 75 Subtracting equation 1 from equation 2, we get: (92 * 59 * n + 8 * 92 * s) - (92 * 59 * n - 38 * 59 * s) = 92 * 75 - 59 * 62 Simplifying, we get: 92 * 59 * n + 8 * 92 * s - 92 * 59 * n + 38 * 59 * s = 6908 8 * 92 * s + 38 * 59 * s = 6908 736s + 2242s = 6908 2978s = 6908 s = 6908 / 2978 s = 2 Now, substitute s = 2 back into equation 1 to solve for n: 92n - 38(2) = 62 92n - 76 = 62 92n = 138 n = 138 / 92 n = 3 Therefore, the values of n and s are n = 3 and s = 2. ### Answer: n = 3, s = 2.
Expected answer: {n: 1673/1489, s: 1621/1489}
Correct? No
Problem: Solving the following pairs of simultaneous equations, what would be the values for e and z? The equations are: 87 * z - 35 * e = 63 and 81 * z + 58 * e = 154.
GPT-3.5 Turbo:
To solve the system of equations: 1) 87z - 35e = 63 2) 81z + 58e = 154 We can use the method of substitution or elimination. Let's use the elimination method to solve this system. Multiplying the first equation by 58 and the second equation by 35 to make the coefficients of e the same: 1) 87z * 58 - 35e * 58 = 63 * 58 2) 81z * 35 + 58e * 35 = 154 * 35 This simplifies to: 1) 5046z - 2030e = 3654 2) 2835z + 2030e = 5390 Adding the two equations together to eliminate e: 5046z - 2030e + 2835z + 2030e = 3654 + 5390 7879z = 9044 z = 9044 / 7879 z = 1.147 Substitute z back into the first equation to solve for e: 87 * 1.147 - 35e = 63 100.089 - 35e = 63 -35e = -37.089 e = -37.089 / -35 e = 1.06 Therefore, the values for e and z are e = 1.06 and z = 1.147. ### Answer: e = 1.06, z = 1.147.
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.
GPT-3.5 Turbo:
To solve the system of equations: 1. Multiply the first equation by 9 and the second equation by 1 to make the coefficients of h the same: (99 * t) + (396 * h) = 990 (98 * t) - (36 * h) = 18 2. Add the two equations together to eliminate h: (99t + 98t) + (396h - 36h) = 990 + 18 197t + 360h = 1008 3. Solve for t: 197t = 1008 - 360h t = (1008 - 360h) / 197 4. Substitute t back into the first equation to solve for h: 11((1008 - 360h) / 197) + 44h = 110 (11088 - 3960h) / 197 + 44h = 110 11088 - 3960h + 44h = 110 * 197 11088 - 3916h = 21770 -3916h = 10682 h = -10682 / 3916 h = -2.73 5. Substitute h back into the first equation to solve for t: (11t) + (44 * -2.73) = 110 11t - 119.92 = 110 11t = 229.92 t = 229.92 / 11 t = 20.88 ### Answer: h = -2.73, t = 20.88.
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.
GPT-3.5 Turbo:
To solve the system of linear equations: 1. \(69b - 50o = 55\) 2. \(41o + 18b = 85\) We can use the method of substitution or elimination. Let's use the elimination method to solve this system. Multiplying the first equation by 18 and the second equation by 50 to eliminate \(b\): 1. \(1242b - 900o = 990\) 2. \(2050o + 900b = 4250\) Adding the two equations together: \(1242b - 900o + 2050o + 900b = 990 + 4250\) \(2142b + 1150o = 5240\) Dividing by 2 to simplify: \(1071b + 575o = 2620\) Now, we have a new equation: 3. \(1071b + 575o = 2620\) We can now solve this equation to find the values of \(b\) and \(o\). ### Answer: There is no unique solution to this system of equations.
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.
GPT-3.5 Turbo:
To solve the system of equations: 1. 67q - 80p = 65 2. 6q + 5p = 177 We can use the method of substitution or elimination. Let's use the elimination method by multiplying the second equation by 16 to match the coefficients of p with the first equation. 1. 67q - 80p = 65 2. 96q + 80p = 2832 Adding the two equations together, we get: 163q = 2897 q = 2897/163 q = 17 Substitute q = 17 back into the second equation: 6(17) + 5p = 177 102 + 5p = 177 5p = 75 p = 75/5 p = 15 Therefore, the values of p and q are p = 15 and q = 17. ### Answer: p = 15, q = 17.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Given equations: 1) 10k + 48f = 161 2) 33k - 41f = 31 To eliminate f, we can multiply equation 1 by 41 and equation 2 by 48: 1) 410k + 1968f = 6611 2) 1584k - 1968f = 1488 Adding the two equations together, we get: (410k + 1968f) + (1584k - 1968f) = 6611 + 1488 1994k = 8099 k = 8099 / 1994 k = 4.06 Substitute k back into equation 1 to solve for f: 10(4.06) + 48f = 161 40.6 + 48f = 161 48f = 120.4 f = 120.4 / 48 f = 2.51 ### Answer: k = 4.06, f = 2.51.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the method of substitution. From the second equation, we can express n in terms of l: n = 103 - 73l Now, we substitute this expression for n into the first equation: 31 = 58l - (103 - 73l) 31 = 58l - 103 + 73l 31 = 131l - 103 31 + 103 = 131l 134 = 131l l = 134/131 Now, we substitute the value of l back into the second equation to find n: n = 103 - 73(134/131) n = 103 - 9732/131 n = 103 - 74.3 n = 28.7 Therefore, the values of l and n are: l = 134/131 n = 28.7 ### Answer: l = 134/131, n = 28.7.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. We will multiply the first equation by 37 and the second equation by 28 to make the coefficients of j the same and then subtract the equations to eliminate j. Multiplying the first equation by 37: 37(29y - 28j) = 37(39) 1073y - 1036j = 1443 Multiplying the second equation by 28: 28(25y + 37j) = 28(116) 700y + 1036j = 3248 Subtracting the second equation from the first: 1073y - 1036j - (700y + 1036j) = 1443 - 3248 373y = -1805 y = -1805/373 y = -5 Now, substitute y = -5 back into one of the original equations to solve for j. Let's use the first equation: 29(-5) - 28j = 39 -145 - 28j = 39 -28j = 184 j = -184/28 j = -23/4 Therefore, the values of y and j that satisfy the system of equations are y = -5 and j = -23/4. ### Answer: y = -5, j = -23/4.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of elimination. First, let's multiply the first equation by 96 and the second equation by 93 to make the coefficients of u the same: 1) 93u - 85a = 74 (multiply by 96) 96(93u - 85a) = 96(74) 8928u - 8160a = 7104 2) 96u + 17a = 186 (multiply by 93) 93(96u + 17a) = 93(186) 8928u + 1581a = 17358 Now, subtract the first equation from the second equation to eliminate u: 8928u + 1581a - (8928u - 8160a) = 17358 - 7104 1581a + 8160a = 10254 9741a = 10254 a = 10254/9741 a = 1.05 Substitute the value of a back into the first equation to solve for u: 93u - 85(1.05) = 74 93u - 89.25 = 74 93u = 163.25 u = 163.25/93 u = 1.75 ### Answer: u = 1.75, a = 1.05.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of elimination. First, let's multiply the first equation by 23 and the second equation by 61 to eliminate g: (97 * u - 61 * g) * 23 = 15 * 23 (85 * u + 23 * g) * 61 = 186 * 61 This gives us: 2231u - 1403g = 345 5185u + 1403g = 11346 Now, we can add the two equations to eliminate g: 2231u - 1403g + 5185u + 1403g = 345 + 11346 7416u = 11691 Dividing by 7416 on both sides, we get: u = 11691 / 7416 u = 1.575 Now, substitute u back into one of the original equations to solve for g. Let's use the first equation: 97 * 1.575 - 61g = 15 152.775 - 61g = 15 -61g = 15 - 152.775 -61g = -137.775 g = -137.775 / -61 g = 2.259 ### Answer: u = 1.575, g = 2.259
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Given equations: 1) 17r + 67b = 103 2) 85r + 67b = 187 Subtracting equation 1 from equation 2: (85r + 67b) - (17r + 67b) = 187 - 103 68r = 84 r = 84/68 r = 21/17 Substitute r = 21/17 into equation 1: 17(21/17) + 67b = 103 21 + 67b = 103 67b = 82 b = 82/67 Therefore, the values of r and b that satisfy the system of equations are r = 21/17 and b = 82/67. ### Answer: r = 21/17, b = 82/67.
Expected answer: {b: 82/67, r: 21/17}
Correct? Yes
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for t and n. Multiplying the first equation by 38 and the second equation by 54 to eliminate t: 1. 3230n - 2052t = 1558 2. 2052t + 1620n = 6804 Adding the two equations together to eliminate t: 4850n = 8362 n = 8362 / 4850 n = 1.72577319587 Substitute n back into the first equation to solve for t: 85n - 54t = 41 85(1.72577319587) - 54t = 41 146.875 - 54t = 41 -54t = -105.875 t = 105.875 / 54 t = 1.95925925926 Therefore, the values of t and n are t = 1.95925925926 and n = 1.72577319587. ### Answer: t = 1.95925925926, n = 1.72577319587.
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.
GPT-3.5 Turbo:
Let's denote u as the value of u and q as the value of q. From the first equation: 75u - 69q = 5 From the second equation: 48u + 25q = 140 Now, we can solve this system of equations using either substitution or elimination method. Let's use the elimination method to solve this system of equations: Multiply the first equation by 25 and the second equation by 69 to make the coefficients of q the same: 1875u - 1725q = 125 3312u + 1725q = 9660 Add the two equations together to eliminate q: 5187u = 9785 Divide by 5187 on both sides to solve for u: u = 9785/5187 = 1.885 Substitute u back into the first equation to solve for q: 75(1.885) - 69q = 5 141.375 - 69q = 5 -69q = -136.375 q = 136.375/69 ≈ 1.976 ### Answer: u = 1.885, q = 1.976.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method. Given equations: 1. 20v + 17x = 149 2. 24x - 49v = 22 To eliminate v, let's multiply equation 1 by 49 and equation 2 by 17: 1. 980v + 833x = 7301 2. 408x - 833v = 374 Now, let's add the two equations to eliminate v: 980v + 833x + 408x - 833v = 7301 + 374 147x = 7675 x = 7675 / 147 x = 52.210 Now, substitute x back into equation 1 to find v: 20v + 17(52.210) = 149 20v + 888.77 = 149 20v = -739.77 v = -739.77 / 20 v = -36.989 Therefore, v = -36.989 and x = 52.210. ### Answer: v = -36.989, x = 52.210.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for t and g. Multiplying the first equation by 91 and the second equation by 41, we get: (39 * 91) * t + (41 * 91) * g = 147 * 91 (94 * 41) * t - (91 * 41) * g = 42 * 41 Simplifying, we get: 3549t + 3731g = 13377 3854t - 3731g = 1722 Adding the two equations together, we get: 7403t = 15099 t = 15099 / 7403 t = 2 Substitute t = 2 back into the first equation: 39 * 2 + 41g = 147 78 + 41g = 147 41g = 69 g = 69 / 41 g = 3 Therefore, the values of t and g are t = 2 and g = 3. ### Answer: t = 2, g = 3.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Given equations: 1) 6c + 55o = 112 2) 55c - 48o = 36 To eliminate one variable, let's multiply the first equation by 48 and the second equation by 55: 1) 288c + 2640o = 5376 2) 3025c - 2640o = 1980 Now, add the two equations to eliminate o: (288c + 3025c) + (2640o - 2640o) = 5376 + 1980 3313c = 7356 c = 7356/3313 c = 2 Now, substitute c = 2 into equation 1 to solve for o: 6(2) + 55o = 112 12 + 55o = 112 55o = 100 o = 100/55 o = 20/11 Therefore, the values of c and o that satisfy the system of equations are c = 2 and o = 20/11. ### Answer: c = 2, o = 20/11.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. First, let's multiply the second equation by 8 and the first equation by 94 to make the coefficients of u the same: 8(71w - 94u) = 8(8) 568w - 752u = 64 94(8w + 68u) = 94(170) 752w + 6392u = 15980 Now, we have the system of equations: 568w - 752u = 64, 752w + 6392u = 15980. Adding the two equations together, we get: 1320w + 5640u = 16044. Dividing by 1320, we get: w + 4.27u = 12.2. Now, we can solve for w in terms of u: w = 12.2 - 4.27u. Substitute this expression for w into the first equation: 8(12.2 - 4.27u) + 68u = 170, 97.6 - 34.16u + 68u = 170, 33.84u = 72.4, u = 2.14. Now, substitute u = 2.14 back into w = 12.2 - 4.27u: w = 12.2 - 4.27(2.14), w = 12.2 - 9.15, w = 3.05. Therefore, the values of u and w that satisfy the system of equations are u = 2.14 and w = 3.05. ### Answer: u = 2.14, w = 3.05.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for r and f. Multiplying the first equation by 97 and the second equation by 8, we get: (97)(68*r + 8*f) = 97*92 (8)(97*f + 38*r) = 8*182 Expanding, we get: 6604r + 776f = 8924 776f + 304r = 1456 Subtracting the second equation from the first, we get: 6300r = 7468 r = 7468/6300 r = 118/100 r = 1.18 Substitute r = 1.18 into the first equation: 68*1.18 + 8f = 92 80.24 + 8f = 92 8f = 11.76 f = 11.76/8 f = 1.47 Therefore, the values of r and f are r = 1.18 and f = 1.47. ### Answer: r = 1.18, f = 1.47.
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).
GPT-3.5 Turbo:
To find the values of d and r, we can solve the system of equations: 1) 67d - 37r = 50 2) 39r - 21d = 16 To solve this system, we can use the method of substitution or elimination. Let's use the elimination method: Multiply the first equation by 21 and the second equation by 37 to make the coefficients of d the same: 1) 67d*21 - 37r*21 = 50*21 2) 39r*37 - 21d*37 = 16*37 Simplify the equations: 1) 1407d - 777r = 1050 2) 1443r - 777d = 592 Now, we can add the two equations to eliminate d: 1407d - 777r + 1443r - 777d = 1050 + 592 630d + 666r = 1642 Divide by 6 to simplify: 105d + 111r = 274 Now, we have a new equation: 3) 105d + 111r = 274 We can now solve this equation along with one of the original equations to find the values of d and r. Let's use the first equation: 67d - 37r = 50 Rearrange the equation to solve for d: 67d = 37r + 50 d = (37r + 50) / 67 Now substitute this expression for d into equation 3: 105((37r + 50) / 67) + 111r = 274 (105*37r + 105*50) / 67 + 111r = 274 (3885r + 5250) / 67 + 111r = 274 (3885r + 5250 + 7377r) / 67 = 274 11262r + 5250 = 18358 11262r = 13108 r = 13108 / 11262 r = 1.163 Now substitute r back into the equation to find d: 67d = 37(1.163) + 50 67d = 43.031 + 50 67d = 93.031 d = 93.031 / 67 d = 1.388 Therefore, the cost of each apple (d) is $1.39 and the cost of each ragdoll (r) is $1.16. ### Answer: d = 1.39, r = 1.16.
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.
GPT-3.5 Turbo:
Let's solve the system of equations: 1) 163 = 50x + a 2) 24 = 98x - 84a To solve this system, we can use the method of substitution or elimination. Let's use the elimination method: Multiply the first equation by 84 and the second equation by a to eliminate the variable a: 1) 163*84 = 50*84x + 84a 2) 24a = 98a*x - 84a^2 Now, we have: 1) 13692 = 4200x + 84a 2) 24a = 98x - 84a Now, let's multiply the second equation by 84: 3) 2016 = 8232x - 7056a Now, we have the following system of equations: 1) 13692 = 4200x + 84a 2) 2016 = 8232x - 7056a Let's solve this system of equations: From equation 1: 84a = 13692 - 4200x a = (13692 - 4200x) / 84 a = 163 - 50x Substitute a = 163 - 50x into equation 2: 2016 = 8232x - 7056(163 - 50x) 2016 = 8232x - 114048 + 352800x 2016 = 360032x - 114048 360032x = 2016 + 114048 360032x = 116064 x = 116064 / 360032 x = 0.3229 Now, substitute x = 0.3229 into a = 163 - 50x: a = 163 - 50(0.3229) a = 163 - 16.145 a = 146.855 Therefore, the values of x and a are x = 0.3229 and a = 146.855. ### Answer: x = 0.3229, a = 146.855.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for p and b. Multiplying the first equation by 44 and the second equation by 18, we get: 44 * (67 * p + 18 * b) = 44 * 110 18 * (82 * b - 44 * p) = 18 * 58 Simplifying, we get: 2948p + 792b = 4840 1476b - 792p = 1044 Now, let's add the two equations together to eliminate p: 2948p + 792b + 1476b - 792p = 4840 + 1044 2156b = 5884 b = 5884 / 2156 b = 2.73 Now, substitute b back into one of the original equations to solve for p: 67p + 18(2.73) = 110 67p + 49.14 = 110 67p = 60.86 p = 60.86 / 67 p = 0.91 Therefore, the values of p and b are p = 0.91 and b = 2.73. ### Answer: p = 0.91, b = 2.73.
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.
GPT-3.5 Turbo:
Let's denote the two equations as follows: 1) 63c - 36e = 27 2) 8c + 32e = 69 To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method by multiplying the first equation by 2 to make the coefficients of c the same: 1) 126c - 72e = 54 2) 8c + 32e = 69 Now, we can add the two equations together to eliminate c: 126c - 72e + 8c + 32e = 54 + 69 134c - 40e = 123 Now, we can solve for c: 134c - 40e = 123 134c = 40e + 123 c = (40e + 123) / 134 Now, we can substitute this expression for c into one of the original equations to solve for e. Let's use the first equation: 63c - 36e = 27 63((40e + 123) / 134) - 36e = 27 (2520e + 7629) / 134 - 36e = 27 (2520e + 7629) - 36e * 134 = 27 * 134 2520e + 7629 - 4824e = 3618 -2304e = -4011 e = -4011 / -2304 e = 69/128 Now that we have found the value of e, we can substitute it back into the expression for c to find c: c = (40 * (69/128) + 123) / 134 c = (2760/128 + 123) / 134 c = (21.5625 + 123) / 134 c = 144.5625 / 134 c = 1.0784 Therefore, the values of c and e are c = 1.0784 and e = 69/128. ### Answer: c = 1.0784, e = 69/128. ###
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
GPT-3.5 Turbo:
To solve this system of linear equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Multiplying the first equation by 51 and the second equation by 29, we get: 1) 26 * 51 * x + 29 * 51 * p = 107 * 51 2) 61 * 29 * x - 51 * 29 * p = 4 * 29 This simplifies to: 1) 1326x + 1479p = 5457 2) 1769x - 1479p = 116 Adding the two equations together, we get: 3095x = 5573 x = 5573 / 3095 x = 1.8 Substitute x = 1.8 into the first equation: 26 * 1.8 + 29p = 107 46.8 + 29p = 107 29p = 60.2 p = 60.2 / 29 p ≈ 2.07 Therefore, the values of x and p are x = 1.8 and p ≈ 2.07. ### Answer: x = 1.8, p ≈ 2.07.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method. First, let's multiply the first equation by 10 and the second equation by 17 to eliminate the variable "l". Equation 1: 290l + 170g = 680 Equation 2: 527g - 170l = 289 Now, let's add the two equations together to eliminate "l": 290l + 170g + 527g - 170l = 680 + 289 120g + 120l = 969 g + l = 969/120 g + l = 8.075 Now, we have g + l = 8.075. Since g and l are integers, the closest integers that add up to 8.075 are 8 and 1. Therefore, g = 8 and l = 1. ### Answer: g = 8, l = 1.
Expected answer: {g: 1173/1069, l: 1819/1069}
Correct? No
Problem: Given the following system of equations, what are the values for a and c? 33*a - 54*c = 20 24*a - 26*c = 29.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for a and c. Multiplying the first equation by 24 and the second equation by 33, we get: 792a - 1296c = 480 792a - 858c = 957 Subtracting the second equation from the first equation, we get: -438c = -477 c = 477/438 c = 39/36 c = 13/12 Substitute c = 13/12 back into the first equation: 33a - 54(13/12) = 20 33a - 702/12 = 20 33a - 58.5 = 20 33a = 78.5 a = 78.5/33 a = 2.5 Therefore, the values for a and c are a = 2.5 and c = 13/12. ### Answer: a = 2.5, c = 13/12.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. First, let's multiply the first equation by 43 and the second equation by 71 to make the coefficients of d the same: (71d + 9j) * 43: 43(71d) + 43(9j) = 43(132) 3053d + 387j = 5676 (43j - 84d) * 71: 71(43j) - 71(84d) = 71(91) 3053j - 5976d = 6461 Now, we have the system of equations: 3053d + 387j = 5676 -5976d + 3053j = 6461 Let's add these two equations to eliminate d: 3053d - 5976d + 387j + 3053j = 5676 + 6461 -2923d + 3440j = 12137 Now, we can solve for j: 3440j = 12137 + 2923d j = (12137 + 2923d) / 3440 Now, we can substitute this expression for j back into one of the original equations to solve for d. Let's substitute it into the first equation: 71d + 9((12137 + 2923d) / 3440) = 132 71d + (109233 + 26207d) / 3440 = 132 71d + 109233 + 26207d = 132 * 3440 71d + 26207d = 450720 - 109233 26278d = 341487 d = 341487 / 26278 d = 13 Now, substitute d = 13 back into the first equation to solve for j: 71(13) + 9j = 132 923 + 9j = 132 9j = 132 - 923 9j = -791 j = -791 / 9 j = -87.888... Therefore, the solution to the system of equations is d = 13 and j = -87.888.... ### Answer: d = 13, j = -87.888...
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method. Given equations: 1) 81o + 26n = 163 2) 87n + 23o = 143 To eliminate o, let's multiply equation 1 by 23 and equation 2 by 26: 1) 1863o + 598n = 3749 2) 2262n + 598o = 3718 Now, subtract equation 2 from equation 1: -399n = 31 n = -31/399 = -0.0777 Substitute n back into equation 1: 81o + 26(-0.0777) = 163 81o - 2.0202 = 163 81o = 165.0202 o = 165.0202/81 ≈ 2.037 ### Answer: o ≈ 2.037, n ≈ -0.0777.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for w and s. Multiplying the first equation by 67 and the second equation by 61, we get: (46 * 67) * s - (61 * 67) * w = 2 * 67 (61 * 61) * s + (67 * 61) * w = 178 * 61 Simplifying, we get: 3082s - 4077w = 134 3721s + 4087w = 10858 Adding the two equations together, we get: 3082s - 4077w + 3721s + 4087w = 134 + 10858 6803s = 10992 s = 10992 / 6803 s ≈ 1.615 Substitute s back into one of the original equations to solve for w: 61 * 1.615 + 67w = 178 98.315 + 67w = 178 67w = 178 - 98.315 67w ≈ 79.685 w ≈ 1.189 Therefore, the solution to the system of equations is w ≈ 1.189 and s ≈ 1.615. ### Answer: w ≈ 1.189, s ≈ 1.615.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for i and k. Given equations: 1) 51*k - 8*i = 86 2) 39*i + 8*k = 193 To eliminate k, let's multiply equation 1 by 8 and equation 2 by -8: 1) 408*k - 64*i = 688 2) -312*i - 64*k = -1544 Adding the two equations together: 408*k - 64*i - 312*i - 64*k = 688 - 1544 344*k - 376*i = -856 Now, let's solve for k in terms of i: 344*k = 376*i - 856 k = (376*i - 856) / 344 k = (94*i - 214) / 86 Substitute this expression for k back into equation 1: 51*((94*i - 214) / 86) - 8*i = 86 (4794*i - 10914) / 86 - 8*i = 86 4794*i - 10914 - 688*i = 7396 4106*i = 18310 i = 18310 / 4106 i = 4.46 Now, substitute i back into the expression for k: k = (94*4.46 - 214) / 86 k = (419.24 - 214) / 86 k = 205.24 / 86 k = 2.39 ### Answer: i = 4.46, k = 2.39.
Expected answer: {i: 9155/2053, k: 4898/2053}
Correct? No
Problem: Solve the following system of equations for the values of l and m: 26m + 17l = 146 93m + l = 126
GPT-3.5 Turbo:
To solve this system of equations, we can use the substitution method. From the second equation, we can express l in terms of m: l = 126 - 93m Now we substitute this expression for l into the first equation: 26m + 17(126 - 93m) = 146 26m + 2142 - 1581m = 146 -1555m = -1996 m = 1996/1555 m = 1.285 Now we substitute m back into the expression for l: l = 126 - 93(1.285) l = 126 - 119.205 l = 6.795 Therefore, the values of l and m are l = 6.795 and m = 1.285. ### Answer: l = 6.795, m = 1.285.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method: 1) (8 * f) + (39 * h) = 66 2) (29 * f) + (46 * h) = 119 To eliminate 'f', we can multiply the first equation by 29 and the second equation by 8: 1) 29(8 * f) + 29(39 * h) = 29(66) 2) 8(29 * f) + 8(46 * h) = 8(119) Simplify the equations: 1) 232f + 1131h = 1914 2) 232f + 368h = 952 Now, subtract the second equation from the first equation to eliminate 'f': 232f - 232f + 1131h - 368h = 1914 - 952 763h = 962 h = 962 / 763 h = 2 Now, substitute h = 2 back into the first equation to solve for 'f': 8f + 39(2) = 66 8f + 78 = 66 8f = 66 - 78 8f = -12 f = -12 / 8 f = -1.5 Therefore, the values of 'h' and 'f' are h = 2 and f = -1.5. ### Answer: h = 2, f = -1.5.
Expected answer: {f: 1605/763, h: 962/763}
Correct? No
Problem: Find the values of k and b that satisfy the following system of equations: 22 * b + 23 * k = 109 and 62 * k - 69 * b = 14.
GPT-3.5 Turbo:
To find the values of \( k \) and \( b \), we can solve the system of equations:
\[
\begin{cases}
22b + 23k = 109 \\
62k - 69b = 14
\end{cases}
\]
We can solve this system by multiplying the first equation by 3 and the second equation by 1 to eliminate \( b \):
\[
\begin{cases}
66b + 69k = 327 \\
62k - 69b = 14
\end{cases}
\]
Adding the two equations together, we get:
\( 66b + 69k + 62k - 69b = 327 + 14 \)
\( -3b + 131k = 341 \)
Solving for \( b \), we get:
\( -3b = 341 - 131k \)
\( b = \frac{131k - 341}{3} \)
Since \( b \) and \( k \) are integers, we need to find a value of \( k \) that makes \( b \) an integer. Let's try some values of \( k \) to find the corresponding \( b \):
For \( k = 3 \), \( b = \frac{131*3 - 341}{3} = \frac{393 - 341}{3} = \frac{52}{3} \) (not an integer)
For \( k = 4 \), \( b = \frac{131*4 - 341}{3} = \frac{524 - 341}{3} = \frac{183}{3} = 61 \) (an integer)
Therefore, the values of \( k \) and \( b \) that satisfy the system of equations are \( k = 4 \) and \( b = 61 \).
### Answer: \( k = 4 \) and \( b = 61 \)
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.
GPT-3.5 Turbo:
To solve this system of linear equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for m and s. 1. Multiply the second equation by 55 and the first equation by 2 to make the coefficients of s the same: - 110(63m - 2s) = 110(98) - 6930m - 220s = 10780 - 168m - 110s = 2 2. Subtract the second modified equation from the first modified equation: - (84m - 55s) - (168m - 110s) = 1 - 2 - -84m + 55s - 168m + 110s = -1 - -252m + 165s = -1 3. Solve for m: - -252m + 165s = -1 - 252m - 165s = 1 - 0m = 0 - There are infinitely many solutions for m. 4. Substitute m back into one of the original equations to solve for s: - 84m - 55s = 1 - 84(0) - 55s = 1 - -55s = 1 - s = -1/55 ### Answer: There are infinitely many solutions for m, and s = -1/55.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method:
1. \(6b + 5k = 105\)
2. \(95k - 51b = 67\)
To eliminate \(b\), we can multiply the first equation by 51 and the second equation by 5:
1. \(51(6b) + 51(5k) = 51(105)\) which simplifies to \(306b + 255k = 5355\)
2. \(5(95k) - 5(51b) = 5(67)\) which simplifies to \(475k - 255b = 335\)
Now, we can subtract the second equation from the first:
\((306b + 255k) - (475k - 255b) = 5355 - 335\)
This simplifies to \(306b + 255k - 475k + 255b = 5020\)
Combining like terms, we get \(561b - 220k = 5020\)
Now, we can solve for \(b\) in terms of \(k\):
\(561b = 220k + 5020\)
\(b = \frac{220k + 5020}{561}\)
Now, we substitute this expression for \(b\) back into the first equation:
\(6\left(\frac{220k + 5020}{561}\right) + 5k = 105\)
Solving this equation will give us the value of \(k\), and then we can find the value of \(b\).
After solving the equations, we find that \(k = 9\) and \(b = 4\).
### Answer: \(k = 9\) and \(b = 4\)
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for d and u. Multiplying the first equation by 32 and the second equation by -33, we get: (32)(33d) + (32)(62u) = (32)(114) -33(32u) - 33(40d) = -33(83) This simplifies to: 1056d + 1984u = 3648 -1056u - 1320d = -2739 Adding the two equations together, we get: 664u = 909 u = 909/664 u = 1.37 Substitute u = 1.37 back into the first equation: 33d + 62(1.37) = 114 33d + 85.94 = 114 33d = 28.06 d = 28.06/33 d = 0.85 Therefore, the values of d and u are d = 0.85 and u = 1.37. ### Answer: d = 0.85, u = 1.37.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system of equations. Multiplying the first equation by 20 and the second equation by 5, we get: 100c + 560p = 2040 (Equation 1) 100c - 335p = 65 (Equation 2) Subtracting Equation 2 from Equation 1, we get: (100c + 560p) - (100c - 335p) = 2040 - 65 100c + 560p - 100c + 335p = 1975 895p = 1975 p = 1975 / 895 p = 35/16 Substitute p = 35/16 into the first equation: 5c + 28(35/16) = 102 5c + 980/16 = 102 5c + 61.25 = 102 5c = 40.75 c = 40.75 / 5 c = 8.15 Therefore, the solution to the system of equations is c = 8.15 and p = 35/16. ### Answer: c = 8.15, p = 35/16.
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.
GPT-3.5 Turbo:
To solve this system of linear equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for r and q. Multiplying the first equation by 97 and the second equation by 58 to eliminate r: (58 * r + 12 * q) * 97 = 147 * 97 (97 * q - 76 * r) * 58 = 76 * 58 Expanding the equations: 5626r + 1164q = 14259 5626q - 4408r = 4408 Now, we can solve this system of equations. Let's multiply the second equation by 1164 and add it to the first equation to eliminate r: 5626r + 1164q + 6546424q - 5099712r = 14259 + 5119712 6546424q - 5094086r = 5119971 Rearranging the equation: 6546424q - 5099712r = 5119971 Now, we have a new system of equations: 5626r + 1164q = 14259 6546424q - 5099712r = 5119971 Solving this new system of equations, we find: r = 3 q = 2 ### Answer: r = 3, q = 2.
Expected answer: {q: 7790/3269, r: 13347/6538}
Correct? No
Problem: Resolve the provided system of simultaneous linear equations for v and k: 47 * v - 3 * k = 10 73 * v + 2 * k = 128.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of elimination. First, let's multiply the first equation by 2 and the second equation by 3 to eliminate k: 2(47v - 3k) = 2(10) 3(73v + 2k) = 3(128) This simplifies to: 94v - 6k = 20 219v + 6k = 384 Now, we can add the two equations together to eliminate k: 94v - 6k + 219v + 6k = 20 + 384 313v = 404 v = 404/313 Therefore, v = 404/313. Now, we can substitute the value of v back into one of the original equations to solve for k. Let's use the first equation: 47v - 3k = 10 47(404/313) - 3k = 10 18988/313 - 3k = 10 -3k = 10 - 18988/313 -3k = 3130/313 - 18988/313 -3k = -15858/313 k = (-15858/313) / -3 k = 5286/313 Therefore, k = 5286/313. ### Answer: v = 404/313, k = 5286/313.
Expected answer: {k: 5286/313, v: 404/313}
Correct? Yes
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?
GPT-3.5 Turbo:
Let's denote the first equation as: 89p - 41q = 60 And the second equation as: 27p + 91q = 124 To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. First, let's multiply the first equation by 27 and the second equation by 89 to make the coefficients of p the same: 27(89p - 41q) = 27(60) 89(27p + 91q) = 89(124) This simplifies to: 2403p - 1147q = 1620 2403p + 8099q = 11036 Subtracting the first equation from the second equation, we get: 9246q = 9424 q = 9424 / 9246 q = 1.02 Now, substitute q back into the first equation to solve for p: 89p - 41(1.02) = 60 89p - 41.82 = 60 89p = 101.82 p = 101.82 / 89 p = 1.14 Therefore, the values of p and q are p = 1.14 and q = 1.02. ### Answer: p = 1.14, q = 1.02.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. First, let's multiply the first equation by 3 and the second equation by 2 to make the coefficients of u the same: 3 * (88u - 96d) = 3 * 12 264u - 288d = 36 2 * (27u + 36d) = 2 * 71 54u + 72d = 142 Now, let's subtract the second equation from the first equation to eliminate u: (264u - 288d) - (54u + 72d) = 36 - 142 210u - 360d = -106 Now, we can solve this new equation for u: 210u = 360d - 106 u = (360d - 106) / 210 u = 1.7143d - 0.5048 Now, we can substitute this expression for u back into one of the original equations to solve for d. Let's use the first equation: 88(1.7143d - 0.5048) - 96d = 12 151.4294d - 44.3744 - 96d = 12 55.4294d = 56.3744 d = 1.0182 Now that we have found d, we can substitute it back into the expression we found for u: u = 1.7143(1.0182) - 0.5048 u = 1.7429 - 0.5048 u = 1.2381 Therefore, the values of u and d that make the equations true are u = 1.2381 and d = 1.0182. ### Answer: u = 1.2381, d = 1.0182.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Given equations: 1) 20n - 18f = 9 2) 3n - 2f = 70 To eliminate f, let's multiply equation 2 by 9 and equation 1 by 2: 2) 27n - 18f = 630 1) 40n - 36f = 18 Subtracting equation 1 from equation 2: 27n - 18f - 40n + 36f = 630 - 18 -13n + 18f = 612 13n - 18f = -612 Adding this new equation to the original equation 1: 33n = -594 n = -594 / 33 n = -18 Substitute n = -18 into equation 1: 20(-18) - 18f = 9 -360 - 18f = 9 -18f = 369 f = -369 / 18 f = -20.5 ### Answer: n = -18, f = -20.5.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for j and l. Multiplying the first equation by 22 and the second equation by 7, we get: 154 * l + 2134 * j = 2926 154 * j + 259 * l = 532 Subtracting the second equation from the first equation, we get: (154 * l + 2134 * j) - (154 * j + 259 * l) = 2926 - 532 154l + 2134j - 154j - 259l = 2394 -105l + 1980j = 2394 105l - 1980j = -2394 Now, let's solve this new equation: 105l - 1980j = -2394 Dividing by 15: 7l - 132j = -159.6 Now, we can solve for j: 7l - 132j = -159.6 -132j = -7l - 159.6 132j = 7l + 159.6 j = (7l + 159.6) / 132 Now, we substitute this expression for j back into one of the original equations to solve for l and then find j. Let's substitute j back into the first equation: 7l + 97 * ((7l + 159.6) / 132) = 133 7l + (679l + 159.6*97) / 132 = 133 7l + (679l + 15463.2) / 132 = 133 7l + 5.1705l + 117.2 = 133 12.1705l = 15.8 l = 15.8 / 12.1705 l ≈ 1.3 Now, substitute l back into j = (7l + 159.6) / 132 to find j: j = (7*1.3 + 159.6) / 132 j = (9.1 + 159.6) / 132 j = 168.7 / 132 j ≈ 1.28 Therefore, the values of l and j are approximately l ≈ 1.3 and j ≈ 1.28. ### Answer: l ≈ 1.3, j ≈ 1.28.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for d and f. Multiplying the first equation by 4 and the second equation by 23, we get: 1) 84d - 184f = 212 2) 184d + 1932f = 3069 Adding the two equations together, we get: 268d + 1748f = 3281 Dividing by 4, we get: 67d + 437f = 820.25 Now, we can solve this new equation along with the first equation to find the values of d and f. 21d - 46f = 53 67d + 437f = 820.25 Multiplying the first equation by 67 and the second equation by 21, we get: 1) 1407d - 3082f = 3541 2) 1407d + 9267f = 17245.25 Subtracting the first equation from the second equation, we get: 12349f = 13704.25 f = 13704.25 / 12349 f = 1.11 Substitute the value of f back into the first equation: 21d - 46(1.11) = 53 21d - 51.06 = 53 21d = 104.06 d = 104.06 / 21 d = 4.96 Therefore, the values of d and f are d = 4.96 and f = 1.11. ### Answer: d = 4.96, f = 1.11.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. First, let's multiply the second equation by 11 to make the coefficients of w in both equations the same: (66u - 68w) * 11 = 21 * 11 726u - 748w = 231 Now, we have the system of equations: 74u + 11w = 116 726u - 748w = 231 Adding the two equations together, we get: 800u = 347 u = 347/800 u = 0.43375 Now, substitute u back into the first equation to solve for w: 74(0.43375) + 11w = 116 32 + 11w = 116 11w = 84 w = 84/11 w = 7.63636 ### Answer: u = 0.43375, w = 7.63636.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Given equations: 1) 76f - 42j = 96 2) 47j + 34f = 136 To eliminate one variable, let's multiply the first equation by 47 and the second equation by 76: 1) 47 * (76f - 42j) = 47 * 96 3572f - 1974j = 4512 2) 76 * (47j + 34f) = 76 * 136 3572j + 2584f = 10336 Now, we have the equations: 1) 3572f - 1974j = 4512 2) 3572j + 2584f = 10336 Let's add these two equations to eliminate f: 3572f - 1974j + 3572j + 2584f = 4512 + 10336 6156f + 1598j = 14848 Now, we can solve for f: 6156f = 14848 - 1598j f = (14848 - 1598j) / 6156 f = 2.4 - 0.26j Now, we can substitute this value of f back into one of the original equations to solve for j. Let's substitute it into the first equation: 76 * (2.4 - 0.26j) - 42j = 96 182.4 - 19.76j - 42j = 96 182.4 - 61.76j = 96 -61.76j = -86.4 j = -86.4 / -61.76 j = 1.4 Now that we have found j, we can substitute it back into the equation f = 2.4 - 0.26j to find f: f = 2.4 - 0.26 * 1.4 f = 2.4 - 0.364 f = 2.036 Therefore, the values of f and j that satisfy the system of equations are f = 2.036 and j = 1.4. ### Answer: f = 2.036, j = 1.4.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. First, let's multiply the second equation by 32 to make the coefficients of e in both equations the same: 32(63e - 69u) = 32(1) 2016e - 2208u = 32 Now, we can add this modified second equation to the first equation to eliminate u: 68u + 32e + 2016e - 2208u = 174 + 32 -2140u + 2048e = 206 Now, we can solve for e: 2048e = 2140u + 206 e = (2140u + 206) / 2048 e = 107u/1024 + 103/1024 Now, we substitute this expression for e back into the first equation: 68u + 32(107u/1024 + 103/1024) = 174 68u + 3424u/1024 + 3296/1024 = 174 68u + 3.34765625u + 3.21875 = 174 71.34765625u = 170.78125 u = 170.78125 / 71.34765625 u ≈ 2.39 Now that we have found u, we can substitute it back into the second equation to find e: 63e - 69(2.39) = 1 63e - 167.91 = 1 63e = 168.91 e ≈ 2.68 Therefore, the values of u and e are approximately u ≈ 2.39 and e ≈ 2.68. ### Answer: u ≈ 2.39, e ≈ 2.68.
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.
GPT-3.5 Turbo:
To solve this system of linear equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Multiplying the first equation by 24 and the second equation by 57, we get: (24 * 57p) + (24 * 4n) = 24 * 129 (57 * 24p) + (57 * 39n) = 57 * 154 This simplifies to: 1368p + 96n = 3096 1368p + 2223n = 8778 Subtracting the first equation from the second equation, we get: 2223n - 96n = 8778 - 3096 2127n = 5682 n = 5682 / 2127 n = 2.67 Substitute n = 2.67 back into the first equation: 57p + 4(2.67) = 129 57p + 10.68 = 129 57p = 118.32 p = 118.32 / 57 p = 2.08 Therefore, the values of p and n that satisfy the system of equations are p = 2.08 and n = 2.67. ### Answer: p = 2.08, n = 2.67.
Expected answer: {n: 1894/709, p: 4415/2127}
Correct? No
Problem: Determine the values of y and z from the given system of equations: 66 * z - 44 * y = 32 60 * y + 14 * z = 116.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Multiplying the first equation by 3 and the second equation by 2, we get: 1) 198z - 132y = 96 2) 120y + 28z = 232 Now, let's multiply the second equation by 3 and subtract it from the first equation: 198z - 132y - 360y - 84z = 96 - 696 114z - 492y = -600 Dividing by 6: 19z - 82y = -100 Now, we have a new equation: 19z - 82y = -100 Now, we can solve this equation along with one of the original equations to find the values of y and z. Let's use the first equation: 66z - 44y = 32 Rearranging: 66z = 44y + 32 z = (44y + 32) / 66 z = (22y + 16) / 33 Substitute z = (22y + 16) / 33 into 19z - 82y = -100: 19((22y + 16) / 33) - 82y = -100 (418y + 304) / 33 - 82y = -100 418y + 304 - 33 * 82y = -3300 418y + 304 - 2706y = -3300 -2288y = -3604 y = -3604 / -2288 y = 157/251 Now, substitute y = 157/251 back into z = (22y + 16) / 33: z = (22(157/251) + 16) / 33 z = (3454/251 + 16) / 33 z = (3454/251 + 16 * 251 / 251) / 33 z = (3454 + 4016) / 251 / 33 z = 7470 / 251 / 33 z = 30 / 251 Therefore, the values of y and z are y = 157/251 and z = 30/251. ### Answer: y = 157/251, z = 30/251.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method. Multiplying the first equation by 11 and the second equation by 58, we get: (1) 11 * (83t - 58h) = 11 * 45 => 913t - 638h = 495 (2) 58 * (69h + 11t) = 58 * 124 => 4002h + 638t = 7192 Now, we can add equations (1) and (2) to eliminate t: (913t - 638h) + (638t + 4002h) = 495 + 7192 => 1551t + 3364h = 7687 Now, we can solve this new equation for t: 1551t = 7687 - 3364h t = (7687 - 3364h) / 1551 Now, we can substitute this expression for t back into one of the original equations to solve for h. Let's substitute it into the second equation: 69h + 11 * ((7687 - 3364h) / 1551) = 124 69h + (84557 - 36904h) / 1551 = 124 69h + 54.5 - 23.8h = 124 45.2h = 69.5 h = 69.5 / 45.2 h = 1.536946 Now, we can substitute this value of h back into the expression we found for t: t = (7687 - 3364 * 1.536946) / 1551 t = (7687 - 5169.6) / 1551 t = 2517.4 / 1551 t = 1.623 ### Answer: t = 1.623, h = 1.537.
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?
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for h and o. Multiplying the first equation by 4 and the second equation by 61, we get: 1) 316h - 244o = 36 2) 976h - 732o = 549 Now, subtracting the first equation from the second equation: 976h - 732o - 316h + 244o = 549 - 36 660h - 488o = 513 660h = 488o + 513 h = (488o + 513) / 660 Substitute the expression for h into the first equation: 79 * ((488o + 513) / 660) - 61o = 9 (79 * 488o + 79 * 513 - 61 * 660o) / 660 = 9 (38692o + 40527 - 40260o) / 660 = 9 (38692o - 40260o + 40527) / 660 = 9 -1568o + 40527 = 5940 -1568o = 18813 o = -18813 / 1568 o = -12 Substitute the value of o back into the expression for h: h = (488(-12) + 513) / 660 h = (-5856 + 513) / 660 h = -5343 / 660 h = -81 Therefore, the values of o and h are o = -12 and h = -81. ### Answer: o = -12, h = -81.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for variables s and y. Multiplying the first equation by 65 and the second equation by 4, we get: 1) 260y + 3640s = 9015 2) 260y - 360s = 64 Subtracting equation 2 from equation 1, we get: 4000s = 8951 s = 8951 / 4000 s = 2.23775 Substitute s = 2.23775 into the first equation: 4y + 56(2.23775) = 139 4y + 125.22 = 139 4y = 13.78 y = 13.78 / 4 y = 3.445 ### Answer: s = 2.23775, y = 3.445.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. First, let's multiply the first equation by 10 and the second equation by 2 to make the coefficients of z the same: 1) 420d - 40z = 750 2) 80d + 90z = 324 Now, let's multiply the first equation by 9 and the second equation by 4 to make the coefficients of d the same: 1) 378d - 36z = 6750 2) 320d + 360z = 1296 Now, let's subtract the second equation from the first equation to eliminate d: 58d - 396z = 5454 Now, we can solve for z: 58d - 396z = 5454 -396z = 5454 z = -5454 / 396 z = -13.77 Now, substitute the value of z back into one of the original equations to solve for d. Let's use the first equation: 42d - 4z = 75 42d - 4(-13.77) = 75 42d + 55.08 = 75 42d = 19.92 d = 19.92 / 42 d = 0.47 Therefore, the values of d and z are d = 0.47 and z = -13.77. ### Answer: d = 0.47, z = -13.77.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for 'a' and 'o'. Multiplying the second equation by 4, we get: (42 * a) + (6 * o) = 75 Multiplying by 4 gives: 4(42 * a) + 4(6 * o) = 4 * 75 168a + 24o = 300 Now, we can subtract the first equation from this new equation: (168a + 24o) - (42 * o + 24 * a) = 300 - 142 168a + 24o - 42o - 24a = 158 144a - 18o = 158 Dividing by 6: 24a - 3o = 26 Now, we have a system of equations: 42o + 24a = 142 24a - 3o = 26 Multiplying the second equation by 8: 8(24a - 3o) = 8 * 26 192a - 24o = 208 Adding this new equation to the first equation: (42o + 24a) + (192a - 24o) = 142 + 208 168a = 350 a = 350 / 168 a = 2.0833 Substitute 'a' back into the second equation: 24(2.0833) - 3o = 26 50 - 3o = 26 -3o = -24 o = 8 Therefore, the values of 'a' and 'o' are: a = 2.0833 o = 8 ### Answer: a = 2.0833, o = 8.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for p and r. Given equations: 1) 39r - 2p = 45 2) 16p - 84r = 36 To eliminate p, let's multiply equation 1 by 8 and equation 2 by 1: 1) 312r - 16p = 360 2) 16p - 84r = 36 Adding the two equations together, we get: 312r - 16p + 16p - 84r = 360 + 36 228r = 396 r = 396/228 r = 11/6 Substitute r = 11/6 into equation 1: 39(11/6) - 2p = 45 429/6 - 2p = 45 2p = 429/6 - 45 2p = 429/6 - 270/6 2p = 159/6 p = 159/12 p = 53/4 Therefore, the values of p and r are p = 53/4 and r = 11/6. ### Answer: p = 53/4, r = 11/6.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method: 1. Multiply the first equation by 8 and the second equation by 52 to make the coefficients of 'd' the same: - 8(23d - 52w) = 8(7) => 184d - 416w = 56 - 52(64w - 8d) = 52(58) => 3328w - 416d = 3016 2. Add the two equations together to eliminate 'd': 184d - 416w + 3328w - 416d = 56 + 3016 -232d - 84w = 3072 3. Solve for 'd': -232d = 84w + 3072 d = (84w + 3072) / -232 4. Substitute the expression for 'd' back into the first equation to solve for 'w': 23((84w + 3072) / -232) - 52w = 7 (1932w + 70944) / -232 - 52w = 7 1932w + 70944 + 11984w = -1624 13916w = -72068 w = -72068 / 13916 w = -5 5. Substitute 'w = -5' back into the expression for 'd' to solve for 'd': d = (84(-5) + 3072) / -232 d = (-420 + 3072) / -232 d = 2652 / -232 d = -11 Therefore, the values of 'd' and 'w' are d = -11 and w = -5. ### Answer: d = -11, w = -5.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for n and z. Given equations: 1) 71z + 60n = 184 2) 52z - 47n = 19 To eliminate n, let's multiply equation 1 by 47 and equation 2 by 60: 1) 47(71z) + 47(60n) = 47(184) => 3337z + 2820n = 8648 2) 60(52z) - 60(47n) = 60(19) => 3120z - 2820n = 1140 Now, let's add the two equations to eliminate n: 3337z + 2820n + 3120z - 2820n = 8648 + 1140 6457z = 9788 z = 9788 / 6457 z = 1.515 Now, substitute z back into equation 1 to solve for n: 71(1.515) + 60n = 184 107.565 + 60n = 184 60n = 76.435 n = 76.435 / 60 n = 1.273 ### Answer: n = 1.273, z = 1.515.
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
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for e and z.
Multiplying the second equation by 10 to make the coefficients of e the same:
\( 70e + 90z = 189 \)
\( 990z - 840e = 200 \)
Now, we can subtract the first equation from the second equation:
\( 990z - 840e - (70e + 90z) = 200 - 189 \)
\( 950z - 910e = 11 \)
Now, we can solve for e in terms of z:
\( 910e = 950z - 11 \)
\( e = \frac{950z - 11}{910} \)
Substitute this expression for e back into the first equation:
\( 70 * \left( \frac{950z - 11}{910} \right) + 90z = 189 \)
\( \frac{66500z - 770}{910} + 90z = 189 \)
\( 66500z - 770 + 81900z = 189 * 910 \)
\( 148400z = 171990 \)
\( z = \frac{171990}{148400} \)
\( z = 1.159 \)
Now, substitute z back into the expression for e:
\( e = \frac{950(1.159) - 11}{910} \)
\( e = \frac{1101.05 - 11}{910} \)
\( e = \frac{1090.05}{910} \)
\( e = 1.2 \)
Therefore, the values of e and z are e = 1.2 and z = 1.159.
### Answer: e = 1.2, z = 1.159.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. 1) 26o - 44f = 95 2) 87f + 11o = 169 Let's multiply the first equation by 11 and the second equation by 2 to make the coefficients of o the same: 1) 286o - 484f = 1045 2) 174f + 22o = 338 Now, let's add the two equations together to eliminate o: 286o - 484f + 174f + 22o = 1045 + 338 308o - 310f = 1383 Now, let's solve for o: 308o - 310f = 1383 308o = 310f + 1383 o = (310f + 1383) / 308 Now, we substitute this expression for o back into one of the original equations to solve for f. Let's use the first equation: 26((310f + 1383) / 308) - 44f = 95 (26 * 310f + 26 * 1383) / 308 - 44f = 95 (8060f + 35958) / 308 - 44f = 95 (8060f + 35958 - 13552f) / 308 = 95 (6708f + 35958) / 308 = 95 6708f + 35958 = 308 * 95 6708f + 35958 = 29840 6708f = 29840 - 35958 6708f = -6128 f = -6128 / 6708 f = -0.912 Now that we have found the value of f, we can substitute it back into the expression we found for o to find the value of o: o = (310(-0.912) + 1383) / 308 o = (-284.52 + 1383) / 308 o = 1098.48 / 308 o = 3.566 Therefore, the values of o and f that satisfy the system of equations are o = 3.566 and f = -0.912. ### Answer: o = 3.566, f = -0.912.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for z and p. Multiplying the first equation by 3, we get: 120z - 60p = 153 (1) Now, we will subtract the second equation from this modified first equation: 120z - 60p - (6z + 58p) = 153 - 165 114z - 118p = -12 Solving for z, we get: 114z = 118p - 12 z = (118p - 12) / 114 z = (59p - 6) / 57 Now, we substitute this expression for z back into the first equation: 40*((59p - 6) / 57) - 20p = 51 (40*59p - 40*6 - 20*57p) / 57 = 51 (2360p - 240 - 1140p) / 57 = 51 1220p - 240 = 57*51 1220p = 2897 p = 2897 / 1220 p = 2.375 Now, we substitute p back into the expression we found for z: z = (59*2.375 - 6) / 57 z = (140.125 - 6) / 57 z = 134.125 / 57 z = 2.354 Therefore, the values of z and p are z = 2.354 and p = 2.375. ### Answer: z = 2.354, p = 2.375.
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.
GPT-3.5 Turbo:
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to solve this system. Multiplying the first equation by 3 and the second equation by 71, we get: 3*(71s) - 3*(89f) = 3*5 71*(3f) + 71*(10s) = 71*93 This simplifies to: 213s - 267f = 15 213f + 710s = 6603 Now, let's multiply the first equation by 213 and the second equation by 267: 213*(213s) - 213*(267f) = 213*15 267*(213f) + 267*(710s) = 267*6603 This simplifies to: 45429s - 56971f = 3195 56871f + 189870s = 1766801 Adding the two new equations together, we get: 45429s - 56971f + 56871f + 189870s = 3195 + 1766801 235299s = 1769996 s = 1769996 / 235299 s = 7.52 Substitute s = 7.52 into the first original equation: 71*7.52 - 89f = 5 533.92 - 89f = 5 -89f = 5 - 533.92 -89f = -528.92 f = -528.92 / -89 f = 5.94 Therefore, the values of s and f are s = 7.52 and f = 5.94. ### Answer: s = 7.52, f = 5.94.
Expected answer: {f: 6553/1103, s: 8292/1103}
Correct? No