Claude-3 Sonnet on 8.EE.C.8
Common Core Standard 8.EE.C.8
This standard covers the following:
Analyze and solve pairs of simultaneous linear equations.
Problem: Let's find the values of variables q and s by solving the following system of equations: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121
Claude-3 Sonnet:
To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method. Step 1: Multiply the first equation by 3 and the second equation by 4 to eliminate the variable s. 3(40s - 91q = 37) ⇒ 120s - 273q = 111 4(30s + 12q = 121) ⇒ 120s + 48q = 484 Step 2: Subtract the second equation from the first equation to eliminate s. (120s - 273q = 111) - (120s + 48q = 484) -321q = -373 Step 3: Solve for q. q = -373/-321 q = 1.16 (approximately) Step 4: Substitute the value of q into either of the original equations to find s. Let's use the first equation: (40s) - (91 × 1.16) = 37 40s - 105.56 = 37 40s = 142.56 s = 142.56/40 s = 3.564 (approximately) Therefore, the values of the variables are: q ≈ 1.16 s ≈ 3.564 ### Answer: q ≈ 1.16, s ≈ 3.564
Expected answer: {q: 373/321, s: 2291/642}
Correct? No
Problem: Solve the following system of equations for the values of b and m: 4 * b + 44 * m = 180 and 79 * m - 6 * b = 48.
Claude-3 Sonnet:
To solve the given system of equations for the values of b and m, we can use the elimination method or the substitution method. Using the elimination method: Step 1: Multiply the first equation by 79 and the second equation by 44 to eliminate the variable m. 4 * b * 79 + 44 * m * 79 = 180 * 79 316 * b + 3476 * m = 14220 79 * m * 44 - 6 * b * 44 = 48 * 44 3476 * m - 264 * b = 2112 Step 2: Subtract the second equation from the first equation to eliminate m. 316 * b + 3476 * m - (3476 * m - 264 * b) = 14220 - 2112 316 * b - 264 * b = 12108 52 * b = 12108 b = 12108 / 52 b = 232.8461538461538 Step 3: Substitute the value of b into either of the original equations to find the value of m. Using the first equation: 4 * 232.8461538461538 + 44 * m = 180 931.3846153846154 + 44 * m = 180 44 * m = 180 - 931.3846153846154 44 * m = -751.3846153846154 m = -751.3846153846154 / 44 m = -17.0769666666667 Therefore, the values of b and m are: b = 232.8461538461538 m = -17.0769666666667 ### Answer: b = 232.8461538461538, m = -17.0769666666667
Expected answer: {b: 3027/145, m: 318/145}
Correct? No
Problem: Solve the following system of equations to find the values of o and g: 42 * o - 15 * g = 28 47 * o + 29 * g = 141.
Claude-3 Sonnet:
To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 42 * o - 15 * g = 28 47 * o + 29 * g = 141 Step 1: Multiply the first equation by 29 and the second equation by 15 to eliminate the variable g. 42 * o * 29 - 15 * g * 29 = 28 * 29 1218 * o - 435 * g = 812 47 * o * 15 + 29 * g * 15 = 141 * 15 705 * o + 435 * g = 2115 Step 2: Subtract the second equation from the first equation to eliminate g. (1218 * o - 435 * g) - (705 * o + 435 * g) = 812 - 2115 513 * o = -1303 Step 3: Solve for o. o = -1303 / 513 o = -2.54 Step 4: Substitute the value of o into either of the original equations to find the value of g. Let's use the first equation: 42 * o - 15 * g = 28 42 * (-2.54) - 15 * g = 28 -106.68 - 15 * g = 28 -15 * g = 28 + 106.68 -15 * g = 134.68 g = -8.98 Therefore, the values of o and g are: o = -2.54 g = -8.98 ### Answer: o = -2.54, g = -8.98
Expected answer: {g: 4606/1923, o: 2927/1923}
Correct? No
Problem: Given the following system of equations, determine the values of v and o: The first equation is: ((12 * v) - (6 * o)) = 21 The second equation is: ((76 * v) - (45 * o)) = 50.
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Step 1: Multiply the first equation by (-5) to eliminate the variable v. (-5)((12v - 6o) = 21) -60v + 30o = -105 Step 2: Add the second equation to the modified first equation to eliminate v. (76v - 45o) = 50 -60v + 30o = -105 ----------- 16o = -55 Step 3: Solve for o. o = -55/16 o = -3.4375 Step 4: Substitute the value of o into either of the original equations to find v. Let's use the first equation: (12v - 6(-3.4375)) = 21 12v - (-20.625) = 21 12v = 41.625 v = 41.625/12 v = 3.46875 Therefore, the values of v and o are: v = 3.46875 o = -3.4375 ### Answer: v = 3.46875, o = -3.4375
Expected answer: {o: 83/7, v: 215/28}
Correct? No
Problem: Find the values of f and g in the following system of equations: 75 * f + 25 * g = 138 and 2 * f + 90 * g = 143.
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Step 1: Multiply the first equation by 2 to eliminate the variable f. 150f + 50g = 276 Step 2: Multiply the second equation by 75 to eliminate the variable f. 150f + 6750g = 10725 Step 3: Subtract the second equation from the first equation to eliminate f. 50g - 6750g = 276 - 10725 -6700g = -10449 g = 1.56 Step 4: Substitute the value of g into either of the original equations to find f. Let's use the first equation: 75f + 25(1.56) = 138 75f + 39 = 138 75f = 99 f = 1.32 Therefore, the values of f and g in the given system of equations are: f = 1.32 g = 1.56 ### Answer: f = 1.32, g = 1.56
Expected answer: {f: 1769/1340, g: 10449/6700}
Correct? No
Problem: Solve the following system of equations for variables y and n: the first equation is 194 = 4 * y + 14 * n and the second equation is 7 = 65 * n - 79 * y.
Claude-3 Sonnet:
To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 194 = 4y + 14n (1) 7 = 65n - 79y (2) Step 1: Multiply equation (1) by 79 and equation (2) by 4 to eliminate the variable y. 79(194) = 79(4y + 14n) 15326 = 316y + 1106n (3) 4(7) = 4(65n - 79y) 28 = 260n - 316y (4) Step 2: Subtract equation (4) from equation (3) to eliminate y. 15326 - 28 = 1106n - 260n 15298 = 846n n = 15298/846 n = 18 Step 3: Substitute the value of n into either equation (1) or (2) to find the value of y. Let's use equation (1): 194 = 4y + 14(18) 194 = 4y + 252 4y = 194 - 252 4y = -58 y = -58/4 y = -14.5 Therefore, the solution to the system of equations is: y = -14.5 n = 18 ### Answer: y = -14.5, n = 18
Expected answer: {n: 7677/683, y: 6256/683}
Correct? No
Problem: Find the values of l and g that satisfy these two equations: The first equation is 146 = 28 * g + 26 * l. The second equation is 6 = 41 * l - 53 * g.
Claude-3 Sonnet:
To solve this system of equations, we need to use the substitution method or the elimination method. Let's use the substitution method. Step 1: Isolate one variable in one of the equations. From the second equation, we can isolate l: 6 = 41l - 53g 41l = 6 + 53g l = (6 + 53g) / 41 Step 2: Substitute the expression for l into the first equation. 146 = 28g + 26((6 + 53g) / 41) 146 = 28g + (156 + 1378g) / 41 146 × 41 = 28g × 41 + 156 + 1378g 5986 = 1148g + 156 + 1378g 5830 = 2526g g = 5830 / 2526 g = 2.31 (approximately) Step 3: Substitute the value of g into either equation to find l. Let's use the second equation: 6 = 41l - 53(2.31) 6 = 41l - 122.43 41l = 128.43 l = 128.43 / 41 l = 3.13 (approximately) Therefore, the values of l and g that satisfy the given equations are: l ≈ 3.13 g ≈ 2.31 ### Answer: l ≈ 3.13, g ≈ 2.31
Expected answer: {g: 2915/1263, l: 3953/1263}
Correct? No
Problem: Consider the following system of equations: 1) 20 * p + 56 * y = 90 2) 89 * p + 7 * y = 98 Resolve this system and find the values of p and y.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 89 and the second equation by -20 to eliminate the variable p. 89(20p + 56y = 90) 1780p + 1984y = 8010 -20(89p + 7y = 98) -1780p - 140y = -1960 Step 2: Add the two equations to eliminate p. 1984y + (-140y) = 8010 + (-1960) 1844y = 6050 y = 6050/1844 y = 3.28 Step 3: Substitute the value of y into either of the original equations to find p. Let's use the first equation: 20p + 56(3.28) = 90 20p + 183.68 = 90 20p = -93.68 p = -4.68 Therefore, the values of p and y are: p = -4.68 y = 3.28 ### Answer: p = -4.68, y = 3.28
Expected answer: {p: 347/346, y: 3025/2422}
Correct? No
Problem: Given the system of equations: 72*k - 22*d = 17 and 6*d +69*k = 96, find the values of variables k and d.
Claude-3 Sonnet:
To solve this system of equations, we can use the substitution method or the elimination method. Using the substitution method: Step 1: Isolate one variable in one of the equations. From the first equation, we can isolate d: 72k - 22d = 17 22d = 72k - 17 d = (72k - 17) / 22 Step 2: Substitute the expression for d into the second equation. 6d + 69k = 96 6((72k - 17) / 22) + 69k = 96 (432k - 102) / 22 + 69k = 96 432k - 102 + 1518k = 2112 1950k = 2214 k = 2214 / 1950 k = 1.135384615 (rounded to 9 decimal places) Step 3: Substitute the value of k into either equation to find d. Using the first equation: 72(1.135384615) - 22d = 17 81.94615 - 22d = 17 22d = 64.94615 d = 64.94615 / 22 d = 2.952098864 (rounded to 9 decimal places) Therefore, the values of the variables are: k = 1.135384615 d = 2.952098864 ### Answer: k = 1.135384615, d = 2.952098864
Expected answer: {d: 1913/650, k: 369/325}
Correct? No
Problem: Solve the following system of equations to find the values of d and h: 73 * h - 33 * d = 48 10 * d + 14 * h = 91
Claude-3 Sonnet:
To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 73h - 33d = 48 10d + 14h = 91 Step 1: Multiply the first equation by 10 and the second equation by 73 to eliminate the variable h. 730h - 330d = 480 730d + 1022h = 6643 Step 2: Subtract the second equation from the first equation to eliminate h. 730h - 330d = 480 -(730d + 1022h = 6643) ----------------- -1060d = -6163 Step 3: Divide both sides by -1060 to find the value of d. d = 6163/1060 d = 5.81 Step 4: Substitute the value of d into either of the original equations to find the value of h. Let's use the first equation: 73h - 33(5.81) = 48 73h - 191.73 = 48 73h = 239.73 h = 239.73/73 h = 3.28 Therefore, the values of d and h are: d = 5.81 h = 3.28 ### Answer: d = 5.81, h = 3.28
Expected answer: {d: 5971/1192, h: 3483/1192}
Correct? No
Problem: Solve the following system of equations for the values of q and e: 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130
Claude-3 Sonnet:
To solve the given system of equations for the values of q and e, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 1) 43q - 44e = 29 2) 30q + 43e = 130 Step 1: Multiply the first equation by 30 and the second equation by (-43) to eliminate the variable e. 30(43q - 44e = 29) 1290q - 1320e = 870 -43(30q + 43e = 130) -1290q - 1849e = -5590 Step 2: Add the two equations to eliminate e. 1290q - 1320e = 870 -1290q - 1849e = -5590 ---------- -529e = -4720 Step 3: Solve for e by dividing both sides by -529. e = 4720 / 529 e = 8.92 (approximately) Step 4: Substitute the value of e into either of the original equations to find q. Let's use the first equation: 43q - 44(8.92) = 29 43q - 393.68 = 29 43q = 422.68 q = 422.68 / 43 q = 9.83 (approximately) Therefore, the values of q and e that satisfy the given system of equations are: q ≈ 9.83 e ≈ 8.92 ### Answer: q ≈ 9.83, e ≈ 8.92
Expected answer: {e: 4720/3169, q: 6967/3169}
Correct? No
Problem: Find the values of r and a in the following system of equations: 53 = ((89 * r) - (74 * a)) 109 = ((37 * a) + (6 * r))
Claude-3 Sonnet:
To solve this system of equations, we need to use the substitution method or the elimination method. Let's use the elimination method. Step 1: Multiply the first equation by 6 and the second equation by 89 to eliminate the variable r. 53 × 6 = 318 = (534 × r) - (444 × a) 109 × 89 = 9701 = (3293 × a) + (534 × r) Step 2: Subtract the second equation from the first equation to eliminate r. 318 - 9701 = -9383 = (534 × r) - (444 × a) - (3293 × a) - (534 × r) -9383 = -3737 × a Step 3: Solve for a. a = -9383 / -3737 a = 2.51 Step 4: Substitute the value of a into either of the original equations to find r. Let's use the second equation: 109 = (37 × a) + (6 × r) 109 = (37 × 2.51) + (6 × r) 109 = 92.87 + 6r 6r = 109 - 92.87 6r = 16.13 r = 16.13 / 6 r = 2.69 Therefore, the values of r and a are: r = 2.69 a = 2.51 ### Answer: r = 2.69, a = 2.51
Expected answer: {a: 9383/3737, r: 271/101}
Correct? No
Problem: Given the following pair of equations where 'g' and 'q' represent the quantities of two different types of grains in a mix, solve for 'g' and 'q': 1. The sum of the weights of 42 units of 'g' and 12 units of 'q' equals 169 kg. 2. The weight of 12 units of 'q' minus the weight of 50 units of 'g' results in 19 kg.
Claude-3 Sonnet:
To solve for 'g' and 'q', we need to use the given equations and the substitution method. Given equations: 1. 42g + 12q = 169 2. 12q - 50g = 19 Step 1: Rearrange the second equation to isolate 'q'. 12q - 50g = 19 12q = 50g + 19 q = (50g + 19) / 12 Step 2: Substitute the expression for 'q' into the first equation. 42g + 12((50g + 19) / 12) = 169 42g + 50g + 19 = 169 92g = 150 g = 150 / 92 g = 25 / 18 Step 3: Substitute the value of 'g' into the expression for 'q' to find 'q'. q = (50(25/18) + 19) / 12 q = (1250/18 + 19) / 12 q = (1269/18) / 12 q = 105.75 / 18 q = 5.875 Therefore, the quantities of the two different types of grains in the mix are: g = 25/18 kg q = 5.875 kg ### Answer: g = 25/18 kg, q = 5.875 kg
Expected answer: {g: 75/46, q: 578/69}
Correct? No
Problem: Find the values of s and v for the following system of equations: 37 * s - 11 * v = 54 82 * s - 62 * v = 41
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 62 and the second equation by 11 to eliminate the variable v. 37 * s - 11 * v = 54 (Multiply by 62) 2294 * s - 682 * v = 3348 82 * s - 62 * v = 41 (Multiply by 11) 902 * s - 682 * v = 451 Step 2: Subtract the second equation from the first equation to eliminate v. 2294 * s - 682 * v = 3348 - (902 * s - 682 * v = 451) ------------------------------ 1392 * s = 2897 Step 3: Divide both sides by 1392 to find the value of s. s = 2897 / 1392 s = 2.08 Step 4: Substitute the value of s into either of the original equations to find the value of v. Let's use the first equation: 37 * s - 11 * v = 54 37 * 2.08 - 11 * v = 54 77.96 - 11 * v = 54 -11 * v = -23.96 v = 2.18 Therefore, the values of s and v are: s = 2.08 v = 2.18
Expected answer: {s: 2897/1392, v: 2911/1392}
Correct? No
Problem: Based on the following two equations, find the values for c and z: The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48.
Claude-3 Sonnet:
To solve for the values of c and z, we need to use the system of linear equations given. Let's start by rearranging the equations to get them in the standard form: 41c + 65z = 151 78z - 48c = 48 Now, we can use the elimination method to solve for c and z. Step 1: Multiply the first equation by 48 and the second equation by 41 to eliminate c. 41c × 48 + 65z × 48 = 151 × 48 1968c + 3120z = 7248 78z × 41 - 48c × 41 = 48 × 41 3198z - 1968c = 1968 Step 2: Subtract the second equation from the first equation to eliminate c. (1968c + 3120z) - (3198z - 1968c) = 7248 - 1968 3888c = 5280 c = 5280/3888 c = 1.36 (rounded to two decimal places) Step 3: Substitute the value of c into either of the original equations to find z. Let's use the first equation: 41(1.36) + 65z = 151 56.16 + 65z = 151 65z = 151 - 56.16 65z = 94.84 z = 94.84/65 z = 1.46 (rounded to two decimal places) Therefore, the values of c and z are: c = 1.36 z = 1.46 ### Answer: c = 1.36, z = 1.46
Expected answer: {c: 37/27, z: 512/351}
Correct? No
Problem: Given the following system of equations: 60 * c - 91 * i = 7 7 * i + 76 * c = 148 What are the values of c and i?
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Step 1: Multiply the first equation by 7 and the second equation by 60 to eliminate the variable c. 420c - 637i = 49 420i + 4560c = 8880 Step 2: Subtract the second equation from the first equation to eliminate c. -5197i = -8831 Step 3: Solve for i. i = 8831/5197 i = 1.7 Step 4: Substitute the value of i into either of the original equations to find c. Using the first equation: 60c - 91(1.7) = 7 60c - 154.7 = 7 60c = 161.7 c = 2.695 Therefore, the values of c and i are: c = 2.695 i = 1.7 ### Answer: c = 2.695, 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
Claude-3 Sonnet:
To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 41 and the second equation by 37 to eliminate the variable e. 41(71v - 37e = 69) ⇒ 2911v - 1517e = 2829 37(41e + 76v = 184) ⇒ 1517e + 2812v = 6808 Step 2: Subtract the second equation from the first equation to eliminate e. 2911v - 1517e = 2829 - (1517e + 2812v = 6808) --------------------------- 99v - 3034e = -3979 Since the coefficient of e is -3034, we can set it to 0 by dividing both sides by -3034. 99v - 3034e = -3979 99v = -3979 + 3034(0) 99v = -3979 v = -3979/99 v = -40.19 Step 3: Substitute the value of v into either of the original equations to find e. Let's use the first equation: 71v - 37e = 69 71(-40.19) - 37e = 69 -2853.49 - 37e = 69 -37e = 69 + 2853.49 -37e = 2922.49 e = -2922.49/-37 e = 79.01 Therefore, the values of e and v that satisfy the given simultaneous linear equations are: e = 79.01 v = -40.19 ### Answer: e = 79.01, v = -40.19
Expected answer: {e: 7820/5723, v: 9637/5723}
Correct? No
Problem: What are the values of 'd' and 'v' in the following system of equations? The sum of 25 times 'v' and 69 times 'd' equals 113. The sum of 26 times 'd' and 42 times 'v' equals 85.
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Given equations: 25v + 69d = 113 26d + 42v = 85 Step 1: Multiply the first equation by 26 and the second equation by 25 to eliminate the variable 'v'. 26(25v + 69d) = 2938 25(26d + 42v) = 2125 650v + 1794d = 2938 650d + 1050v = 2125 Subtracting the second equation from the first equation: 744d = 813 d = 813/744 d = 1.092... ≈ 1.09 Step 2: Substitute the value of 'd' into either of the original equations to find the value of 'v'. Substituting d = 1.09 into the first equation: 25v + 69(1.09) = 113 25v + 75.21 = 113 25v = 37.79 v = 37.79/25 v = 1.512 ≈ 1.51 Therefore, the values of 'd' and 'v' in the given system of equations are: d = 1.09 v = 1.51 ### Answer: d = 1.09, v = 1.51
Expected answer: {d: 2621/2248, v: 2927/2248}
Correct? No
Problem: Solve the following equations for g and z: 43 * z - 91 * g = 33 72 * g - 3 * z = 68
Claude-3 Sonnet:
To solve the system of equations for g and z, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 3 and the second equation by 91 to eliminate the variable z. 3(43z - 91g = 33) ⇒ 129z - 273g = 99 91(72g - 3z = 68) ⇒ 6552g - 273z = 6188 Step 2: Add the two equations to eliminate z. 129z - 273g = 99 6552g - 273z = 6188 ----------------- 6279g = 6287 Step 3: Solve for g. 6279g = 6287 g = 6287/6279 g ≈ 1 Step 4: Substitute the value of g into either of the original equations to find z. Let's use the second equation: 72g - 3z = 68 72(1) - 3z = 68 72 - 3z = 68 -3z = -4 z = 4/3 Therefore, the solutions are: g ≈ 1 z = 4/3 ### Answer: g ≈ 1, z = 4/3
Expected answer: {g: 3023/2823, z: 8564/2823}
Correct? No
Problem: Alex's and Brad's baskets contained c cherries and f figs respectively. From an old recipe book, it stated that cherries multiplied by 50 multiplied by the quantity of figs that multiplied by 12, which altogether would weigh about 92 grams. Another recipe book stated that cherries multiplied by 36 minus figs multiplied by 21 should weigh about 34 grams. Can you find the quantity of cherries (c) and figs (f)? Use the following system of equations to solve: 50c + 12f = 92 36c - 21f = 34.
Claude-3 Sonnet:
To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method. Step 1: Multiply the first equation by 3 to eliminate the variable c. 150c + 36f = 276 Step 2: Multiply the second equation by 2 to eliminate the variable c. 72c - 42f = 68 Step 3: Subtract the second equation from the first equation to eliminate c. 150c + 36f = 276 - (72c - 42f = 68) ------------------ 78f = 208 Step 4: Solve for f. f = 208/78 f = 2.67 (approximately) Step 5: Substitute the value of f into either equation to find c. Let's use the first equation: 50c + 12(2.67) = 92 50c + 32.04 = 92 50c = 92 - 32.04 50c = 59.96 c = 59.96/50 c = 1.199 (approximately) Therefore, the quantity of cherries (c) is approximately 1.199, and the quantity of figs (f) is approximately 2.67. ### Answer: c = 1.199, f = 2.67
Expected answer: {c: 30/19, f: 62/57}
Correct? No
Problem: Here are the equations you'll need to solve (remember, the goal is to find the values of u and d): 1) 87u - 75d = 7 2) 85d - 82u = 54
Claude-3 Sonnet:
To solve for the values of u and d, we need to use the given system of linear equations: 1) 87u - 75d = 7 2) 85d - 82u = 54 Step 1: Multiply the first equation by 82 and the second equation by 87 to eliminate u. (87u - 75d) × 82 = 7 × 82 7134u - 6150d = 574 (85d - 82u) × 87 = 54 × 87 7395d - 7134u = 4698 Step 2: Add the two equations to eliminate u. 7134u - 6150d = 574 7395d - 7134u = 4698 ---------- 245d = 5272 Step 3: Solve for d. d = 5272/245 d = 21.5 Step 4: Substitute the value of d into either of the original equations to find u. Using the first equation: 87u - 75(21.5) = 7 87u - 1612.5 = 7 87u = 1619.5 u = 18.6 Therefore, the values of u and d are: u = 18.6 d = 21.5 ### Answer: u = 18.6, d = 21.5
Expected answer: {d: 5272/1245, u: 929/249}
Correct? No
Problem: Bill is trying to decipher a secret code. He manages to translate two sentences into the following mathematical expressions: (75 * his unknown number a) - (4 * his unknown number i) equals 42, and (10 * i) + (13 * a) equals 160. Using these equations, can you figure out the values of i and a for him?
Claude-3 Sonnet:
To solve this problem, we need to use the system of two linear equations with two unknowns (a and i) and find their values. Given equations: (75 * a) - (4 * i) = 42 (10 * i) + (13 * a) = 160 Step 1: Rearrange the first equation to get i in terms of a. (75 * a) - (4 * i) = 42 -4i = 42 - (75 * a) i = (75a - 42) / 4 Step 2: Substitute the expression for i into the second equation. (10 * i) + (13 * a) = 160 10 * ((75a - 42) / 4) + 13a = 160 (187.5a - 105) + 13a = 160 200.5a = 265 a = 265 / 200.5 a = 1.32 Step 3: Substitute the value of a into the expression for i to find the value of i. i = (75a - 42) / 4 i = (75 * 1.32 - 42) / 4 i = (99 - 42) / 4 i = 57 / 4 i = 14.25 Therefore, the values of a and i are: a = 1.32 i = 14.25 ### Answer: a = 1.32, i = 14.25
Expected answer: {a: 530/401, i: 5727/401}
Correct? No
Problem: Solve the following system of equations for the values of p and c: 37 * p + 74 * c = 183 27 * c - 18 * p = 13
Claude-3 Sonnet:
To solve the given system of equations for the values of p and c, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 37p + 74c = 183 27c - 18p = 13 Step 1: Multiply the first equation by 18 and the second equation by 37 to eliminate p. 18(37p + 74c = 183) 666p + 1332c = 3294 37(27c - 18p = 13) 999c - 666p = 481 Step 2: Subtract the second equation from the first equation to eliminate p. (1332c - 999c) = (3294 - 481) 333c = 2813 Step 3: Solve for c. c = 2813 / 333 c = 8.45 (approximately) Step 4: Substitute the value of c into either of the original equations to find p. Let's use the second equation: 27c - 18p = 13 27(8.45) - 18p = 13 227.15 - 18p = 13 -18p = -214.15 p = 11.9 (approximately) Therefore, the solution to the system of equations is: p ≈ 11.9 c ≈ 8.45 ### Answer: p ≈ 11.9, c ≈ 8.45
Expected answer: {c: 3775/2331, p: 3979/2331}
Correct? No
Problem: Determine the values of k and p from the following system of equations: (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105.
Claude-3 Sonnet:
To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method. Step 1: Multiply the first equation by 2 to eliminate the variable k. 2(48k - 30p = 37) 96k - 60p = 74 Step 2: Multiply the second equation by -3 to eliminate the variable k. -3(16p + 33k = 105) -48p - 99k = -315 Step 3: Add the two equations to eliminate k. 96k - 60p = 74 -48p - 99k = -315 -------------- -108p = -241 Step 4: Solve for p. p = -241/-108 p = 2.23 (approximately) Step 5: Substitute the value of p into either of the original equations to find k. Let's use the second equation: 16p + 33k = 105 16(2.23) + 33k = 105 35.68 + 33k = 105 33k = 69.32 k = 2.1 (approximately) Therefore, the values of k and p are: k ≈ 2.1 p ≈ 2.23 ### Answer: k ≈ 2.1, p ≈ 2.23
Expected answer: {k: 1871/879, p: 1273/586}
Correct? No
Problem: Consider the following pair of equations. 1. 75v - 69f = 14 2. 46f + 53v = 110 Determine the values for v and f that satisfy these simultaneous equations.
Claude-3 Sonnet:
To solve this pair of simultaneous equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 1. 75v - 69f = 14 2. 46f + 53v = 110 Step 1: Multiply the first equation by 46 and the second equation by 69 to eliminate the variable f. 46 × (75v - 69f = 14) ⇒ 3450v - 3174f = 644 69 × (46f + 53v = 110) ⇒ 3174f + 3657v = 7590 Step 2: Subtract the second equation from the first equation to eliminate f. (3450v - 3174f = 644) - (3174f + 3657v = 7590) -207v = -6946 v = 33.5 Step 3: Substitute the value of v into either of the original equations to find the value of f. Let's use the first equation: 75v - 69f = 14 75(33.5) - 69f = 14 2512.5 - 69f = 14 -69f = 14 - 2512.5 -69f = -2498.5 f = 36.2 Therefore, the values that satisfy the given simultaneous equations are: v = 33.5 f = 36.2 ### Answer: v = 33.5, f = 36.2
Expected answer: {f: 7508/7107, v: 358/309}
Correct? No
Problem: Determine the values of o and g from the following system of equations: 7 * o + 13 * g = 131 89 * g - 70 * o = 37.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 7o + 13g = 131 89g - 70o = 37 Step 1: Multiply the first equation by 70 and the second equation by 7 to eliminate the variable o. 490o + 910g = 9,170 623g - 490o = 259 Step 2: Subtract the second equation from the first equation to eliminate o. 490o + 910g = 9,170 -(623g - 490o = 259) -------------- 287g = 8,911 Step 3: Divide both sides by 287 to find the value of g. g = 8,911 / 287 g = 31 Step 4: Substitute the value of g into either of the original equations to find the value of o. Let's use the first equation: 7o + 13(31) = 131 7o + 403 = 131 7o = -272 o = -272 / 7 o = -39 Therefore, the values of o and g are: o = -39 g = 31 ### Answer: o = -39, g = 31
Expected answer: {g: 449/73, o: 3726/511}
Correct? No
Problem: Solve the following system of equations for the values of d and k: 88 * d - 60 * k = 38 33 * k - 11 * d = 46.
Claude-3 Sonnet:
To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 88 * d - 60 * k = 38 33 * k - 11 * d = 46 Step 1: Multiply the first equation by 11 and the second equation by 88 to eliminate the variable d. 968 * d - 660 * k = 418 2904 * k - 968 * d = 4048 Step 2: Subtract the second equation from the first equation to eliminate d. 968 * d - 660 * k = 418 - (2904 * k - 968 * d = 4048) ----------------------------------- 3564 * k = -3630 Step 3: Solve for k. k = -3630 / 3564 k = -1.02 Step 4: Substitute the value of k into either of the original equations to find d. Let's use the first equation: 88 * d - 60 * (-1.02) = 38 88 * d + 61.2 = 38 88 * d = 38 - 61.2 88 * d = -23.2 d = -23.2 / 88 d = -0.264 Therefore, the values of d and k are: d = -0.264 k = -1.02 ### Answer: d = -0.264, k = -1.02
Expected answer: {d: 669/374, k: 203/102}
Correct? No
Problem: Consider the following system of equations, calculate the values of y and z: 24 * z + 72 * y = 139, 2 * z + 82 * y = 96.
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Step 1: Multiply the first equation by 2 and the second equation by 24 to eliminate the variable z. 48z + 144y = 278 (First equation × 2) 48z + 1968y = 2304 (Second equation × 24) Step 2: Subtract the second equation from the first equation to eliminate z. 48z + 144y = 278 - (48z + 1968y = 2304) -------------- -1824y = -2026 Step 3: Solve for y. -1824y = -2026 y = -2026/-1824 y = 1.11 Step 4: Substitute the value of y into either of the original equations to find z. Let's use the second equation: 2z + 82y = 96 Substitute y = 1.11 2z + 82(1.11) = 96 2z + 91.02 = 96 2z = 96 - 91.02 2z = 4.98 z = 4.98/2 z = 2.49 Therefore, the values of y and z are: y = 1.11 z = 2.49 ### Answer: y = 1.11, z = 2.49
Expected answer: {y: 1013/912, z: 2243/912}
Correct? No
Problem: Solve the following system of equations for the variables v and o: (87 * o) - (29 * v) = 1 and (8 * v) + (30 * o) = 91.
Claude-3 Sonnet:
To solve the given system of equations for v and o, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: (87 * o) - (29 * v) = 1 (8 * v) + (30 * o) = 91 Step 1: Multiply the first equation by 8 and the second equation by 29 to eliminate the variable v. 8 × [(87 * o) - (29 * v) = 1] 696o - 232v = 8 29 × [(8 * v) + (30 * o) = 91] 232v + 870o = 2639 Step 2: Subtract the second equation from the first equation to eliminate v. 696o - 232v = 8 -(232v + 870o = 2639) -------------- -174o = -2631 Step 3: Solve for o. -174o = -2631 o = 2631/174 o = 15 Step 4: Substitute the value of o into either of the original equations to find v. Let's use the first equation: (87 * o) - (29 * v) = 1 (87 * 15) - (29 * v) = 1 1305 - 29v = 1 29v = 1304 v = 1304/29 v = 45 Therefore, the solution to the system of equations is: v = 45 o = 15 ### Answer: v = 45, o = 15
Expected answer: {o: 2647/1566, v: 2629/522}
Correct? No
Problem: Solve the following system of equations for h and p: The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124.
Claude-3 Sonnet:
To solve the given system of equations for h and p, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: (23 × h) - (46 × p) = 58 ... (1) (11 × p) + (23 × h) = 124 ... (2) Step 1: Multiply equation (1) by 11 and equation (2) by 46 to eliminate the variable h. 11 × [(23 × h) - (46 × p) = 58] ⇒ 253h - 506p = 638 ... (3) 46 × [(11 × p) + (23 × h) = 124] ⇒ 506p + 1058h = 5704 ... (4) Step 2: Subtract equation (3) from equation (4) to eliminate h. (506p + 1058h = 5704) - (253h - 506p = 638) ⇒ 1564p = 6342 ⇒ p = 4 Step 3: Substitute the value of p = 4 into either equation (1) or (2) to find the value of h. Substituting p = 4 in equation (1): (23 × h) - (46 × 4) = 58 23h - 184 = 58 23h = 242 h = 242 / 23 h = 10.52 Therefore, the solution to the system of equations is: h = 10.52 p = 4 ### Answer: h = 10.52, p = 4
Expected answer: {h: 2114/437, p: 22/19}
Correct? No
Problem: If the following system of equations represents the relationship between integers d and f, determine their values: 1) 19d + 6f = 120, 2) 90d - 42f = 69.
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Step 1: Multiply the first equation by 6 and the second equation by 1 to eliminate the variable f. 6(19d + 6f = 120) ⇒ 114d + 36f = 720 1(90d - 42f = 69) ⇒ 90d - 42f = 69 Step 2: Subtract the second equation from the first equation to eliminate f. (114d + 36f = 720) - (90d - 42f = 69) 24d + 78f = 651 Step 3: Divide both sides of the equation by 78 to find the value of f. f = 651/78 f = 8.346... ≈ 8 (since we are dealing with integers) Step 4: Substitute the value of f into either of the original equations to find the value of d. Let's use the first equation: 19d + 6(8) = 120 19d + 48 = 120 19d = 72 d = 72/19 d = 3.789... ≈ 4 (since we are dealing with integers) Therefore, the values of d and f that satisfy the given system of equations are: d = 4 f = 8 ### Answer: d = 4, f = 8
Expected answer: {d: 909/223, f: 3163/446}
Correct? No
Problem: Here's a problem: A system of linear equations is given by the following formulas: 1) 42 * m + 55 * r = 127 2) 66 * m - 52 * r = 44 Can you solve this system to find the values of 'r' and 'm'?
Claude-3 Sonnet:
To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by (-66) and the second equation by 42 to eliminate the variable 'm'. (-66)(42m + 55r = 127) ⇒ -2772m - 3630r = -8382 (42)(66m - 52r = 44) ⇒ 2772m - 2184r = 1848 Step 2: Add the two equations to eliminate 'm'. -3630r = -8382 -2184r = 1848 ----------- -5814r = -6534 Step 3: Divide both sides by -5814 to find the value of 'r'. r = -6534/-5814 r = 1.124 (approximately) Step 4: Substitute the value of 'r' into either of the original equations to find the value of 'm'. Let's use the first equation: 42m + 55(1.124) = 127 42m + 61.82 = 127 42m = 65.18 m = 65.18/42 m = 1.552 (approximately) Therefore, the solution to the system of linear equations is: r = 1.124 m = 1.552 ### Answer: r = 1.124, m = 1.552
Expected answer: {m: 1504/969, r: 363/323}
Correct? No
Problem: Determine the values of n and w that satisfy the following system of linear equations simultaneously: 31n + 64w = 120 57w + 71n = 173
Claude-3 Sonnet:
To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by -71 and the second equation by 31 to eliminate the variable n. -71(31n + 64w = 120) ⇒ -2191n - 4464w = -8520 31(57w + 71n = 173) ⇒ 1767w + 2201n = 5353 Step 2: Add the two equations to eliminate n. (-2191n - 4464w = -8520) (+1767w + 2201n = 5353) -------------- -697w = -3167 Step 3: Solve for w. -697w = -3167 w = -3167/-697 w = 4.54 Step 4: Substitute the value of w into either of the original equations to find n. Let's use the first equation: 31n + 64(4.54) = 120 31n + 290.56 = 120 31n = -170.56 n = -5.5 Therefore, the values of n and w that satisfy the system of linear equations simultaneously are: n = -5.5 w = 4.54 ### Answer: n = -5.5, w = 4.54
Expected answer: {n: 4232/2777, w: 3157/2777}
Correct? No
Problem: Determine the values of u and a from the following system of linear equations: 1) 72 * a + 26 * u = 112 2) 76 * u - 77 * a = 31.
Claude-3 Sonnet:
To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 76 and the second equation by 26 to eliminate the variable u. 76 × (72a + 26u = 112) 5472a + 1976u = 8512 26 × (76u - 77a = 31) 1976u - 2002a = 806 Step 2: Subtract the second equation from the first equation to eliminate u. (5472a + 1976u) - (1976u - 2002a) = 8512 - 806 7474a = 7706 a = 7706 / 7474 a = 1.03 Step 3: Substitute the value of a into either of the original equations to find u. Let's use the first equation: 72(1.03) + 26u = 112 74.16 + 26u = 112 26u = 112 - 74.16 26u = 37.84 u = 37.84 / 26 u = 1.455 Therefore, the values of u and a are: u = 1.455 a = 1.03 ### Answer: u = 1.455, a = 1.03
Expected answer: {a: 3853/3737, u: 5428/3737}
Correct? No
Problem: Solve the following system of equations for the values of b and n: the first equation is 54n + 18b = 143 and the second equation is 87n - 93b = 48.
Claude-3 Sonnet:
To solve the system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 54n + 18b = 143 (1) 87n - 93b = 48 (2) Step 1: Multiply the first equation by (-87) and the second equation by 54 to eliminate the variable n. (-87)(54n + 18b = 143) -4698n - 1566b = -12441 54(87n - 93b = 48) 4698n - 5022b = 2592 Step 2: Add the two equations to eliminate n. -1566b = -12441 -5022b = 2592 ----------- -6588b = -9849 Step 3: Divide both sides by -6588 to find the value of b. b = -9849 / -6588 b = 1.5 Step 4: Substitute the value of b into either of the original equations to find the value of n. Let's use the first equation: 54n + 18(1.5) = 143 54n + 27 = 143 54n = 116 n = 116/54 n = 2.148148148... Therefore, the values of b and n are: b = 1.5 n = 2.148148148... ### Answer: b = 1.5, n = 2.148148148...
Expected answer: {b: 3283/2196, n: 4721/2196}
Correct? No
Problem: Given the following system of equations about a company's production economics, where e represents the cost of electricity and p is the cost of petroleum: 53 * e + 7 * p = 103 69 * e - 88 * p = 6 What are the costs of electricity (e) and petroleum (p)?
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 53e + 7p = 103 69e - 88p = 6 Step 1: Multiply the first equation by (-88) and the second equation by 7 to eliminate the variable p. (-88)(53e + 7p = 103) (-4664e - 616p = -9064) 7(69e - 88p = 6) 483e - 616p = 42 Step 2: Add the two equations to eliminate p. (-4664e - 616p = -9064) (483e - 616p = 42) ----------------- -4181e = -9022 Step 3: Solve for e. -4181e = -9022 e = 9022/4181 e = 2.16 Step 4: Substitute the value of e into either of the original equations to find p. Let's use the first equation: 53e + 7p = 103 53(2.16) + 7p = 103 114.48 + 7p = 103 7p = -11.48 p = -1.64 Therefore, the cost of electricity (e) is $2.16, and the cost of petroleum (p) is -$1.64. ### Answer: e = 2.16, p = -1.64
Expected answer: {e: 9106/5147, p: 6789/5147}
Correct? No
Problem: Consider the following pair of linear equations. First equation: ((18 * p) - (7 * s)) = 21 Second equation: ((51 * s) - (24 * p)) = 14 Find the values of p and s that satisfy both equations.
Claude-3 Sonnet:
To solve this system of linear equations, we can use the substitution method or the elimination method. Let's use the elimination method. Given equations: (18p - 7s) = 21 (51s - 24p) = 14 Step 1: Multiply the first equation by 24 and the second equation by 18 to eliminate the variable p. 24(18p - 7s) = 24(21) 432p - 168s = 504 18(51s - 24p) = 18(14) 918s - 432p = 252 Step 2: Add the two equations to eliminate p. 432p - 168s = 504 918s - 432p = 252 -------------- 750s = 756 Step 3: Solve for s. s = 756 / 750 s = 1 Step 4: Substitute the value of s into either of the original equations to find p. Let's use the first equation: (18p - 7(1)) = 21 18p - 7 = 21 18p = 28 p = 28/18 p = 7/4 Therefore, the values of p and s that satisfy both equations are: p = 7/4 s = 1 ### Answer: p = 7/4, s = 1
Expected answer: {p: 1169/750, s: 126/125}
Correct? No
Problem: Solve the following system of equations for c and w: The first equation is 59 times c minus 34 times w is equal to 84. The second equation is 11 times c plus 42 times w is equal to 98.
Claude-3 Sonnet:
To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 59c - 34w = 84 (Equation 1) 11c + 42w = 98 (Equation 2) Step 1: Multiply Equation 1 by 11 and Equation 2 by 59 to eliminate the variable c. 649c - 374w = 924 (11 × Equation 1) 649c + 2478w = 5782 (59 × Equation 2) Step 2: Subtract the second equation from the first equation to eliminate c. -2852w = -4858 w = 4858 / 2852 w = 1.7 Step 3: Substitute the value of w into either of the original equations to find c. Let's use Equation 1: 59c - 34(1.7) = 84 59c - 57.8 = 84 59c = 141.8 c = 141.8 / 59 c = 2.4 Therefore, the solution to the system of equations is: c = 2.4 w = 1.7 ### Answer: c = 2.4, w = 1.7
Expected answer: {c: 1715/713, w: 2429/1426}
Correct? No
Problem: Solve the following system of equations for the values of f and z: First equation is 45 * f + 42 * z = 101, Second equation is 60 * z - 42 * f = 13.
Claude-3 Sonnet:
To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 45f + 42z = 101 (1) 60z - 42f = 13 (2) Step 1: Multiply the first equation by (-42) and the second equation by 45 to eliminate the variable f. (-42)(45f + 42z = 101) -1890f - 1764z = -4242 45(60z - 42f = 13) 2700z - 1890f = 585 Step 2: Add the two equations to eliminate f. -1764z = -4242 +2700z = 585 ------- 936z = -3657 z = -3657/936 z = -3.91 (approximately) Step 3: Substitute the value of z into either of the original equations to find f. Let's use the first equation: 45f + 42(-3.91) = 101 45f - 164.22 = 101 45f = 265.22 f = 265.22/45 f = 5.89 (approximately) Therefore, the solution to the system of equations is: f ≈ 5.89 z ≈ -3.91 ### Answer: f ≈ 5.89, z ≈ -3.91
Expected answer: {f: 919/744, z: 1609/1488}
Correct? No
Problem: Calculate the values of p and u from the following system of equations: 26 * p - 21 * u = 6 15 * p + 58 * u = 83.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 26p - 21u = 6 15p + 58u = 83 Step 1: Multiply the first equation by 15 and the second equation by 26 to eliminate the variable p. 15(26p - 21u = 6) 390p - 315u = 90 26(15p + 58u = 83) 390p + 1508u = 2158 Step 2: Subtract the second equation from the first equation to eliminate p. 390p - 315u = 90 -(390p + 1508u = 2158) -------------- -1823u = -2068 Step 3: Divide both sides by -1823 to find the value of u. u = 2068/1823 u ≈ 1.134 Step 4: Substitute the value of u into either of the original equations to find the value of p. Let's use the first equation: 26p - 21(1.134) = 6 26p - 23.814 = 6 26p = 29.814 p = 29.814/26 p ≈ 1.147 Therefore, the values of p and u are: p ≈ 1.147 u ≈ 1.134 ### Answer: p ≈ 1.147, u ≈ 1.134
Expected answer: {p: 2091/1823, u: 2068/1823}
Correct? No
Problem: Solve for the values of n and s using these two simultaneous equations: 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75
Claude-3 Sonnet:
To solve for the values of n and s using the given simultaneous equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 1) 92n - 38s = 62 2) 59n + 8s = 75 Step 1: Multiply the first equation by 8 and the second equation by 38 to eliminate the variable s. 8 × (92n - 38s = 62) 736n - 304s = 496 38 × (59n + 8s = 75) 2242n + 304s = 2850 Step 2: Subtract the second equation from the first equation to eliminate s. (736n - 304s = 496) - (2242n + 304s = 2850) -1506n = -2354 n = 2354/1506 n = 1.56 (approximately) Step 3: Substitute the value of n into either of the original equations to find the value of s. Let's use the first equation: 92n - 38s = 62 92(1.56) - 38s = 62 143.52 - 38s = 62 -38s = -81.52 s = 2.14 (approximately) Therefore, the values of n and s are: n ≈ 1.56 s ≈ 2.14 ### Answer: n ≈ 1.56, s ≈ 2.14
Expected answer: {n: 1673/1489, s: 1621/1489}
Correct? No
Problem: Solving the following pairs of simultaneous equations, what would be the values for e and z? The equations are: 87 * z - 35 * e = 63 and 81 * z + 58 * e = 154.
Claude-3 Sonnet:
To solve the given pair of simultaneous equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 87z - 35e = 63 81z + 58e = 154 Step 1: Multiply the first equation by 81 and the second equation by -87 to eliminate the variable z. 81(87z - 35e) = 7107z - 2835e = 5103 -87(81z + 58e) = -7047z - 5066e = -12113 Step 2: Add the two equations to eliminate z. 5103 + (-12113) = -7010 Step 3: Simplify the resulting equation to find the value of e. -7010 = 0 e = 200 Step 4: Substitute the value of e into either of the original equations to find the value of z. Let's use the first equation: 87z - 35(200) = 63 87z - 7000 = 63 87z = 7063 z = 81 Therefore, the values of e and z are: e = 200 z = 81 ### Answer: e = 200, z = 81
Expected answer: {e: 2765/2627, z: 9044/7881}
Correct? No
Problem: To find the values of h and t, solve the following system of simultaneous linear equations: (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18.
Claude-3 Sonnet:
To solve the given system of simultaneous linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18 Step 1: Multiply the first equation by -9 to eliminate the variable t. -9(11t + 44h = 110) -99t - 396h = -990 Step 2: Multiply the second equation by 1 to keep it unchanged. 98t - 36h = 18 Step 3: Add the two equations to eliminate the variable t. (-99t - 396h = -990) (98t - 36h = 18) ----------- -432h = -972 Step 4: Divide both sides by -432 to find the value of h. h = -972 / -432 h = 2.25 Step 5: Substitute the value of h into either of the original equations to find the value of t. Let's use the first equation: (11t) + (44 * 2.25) = 110 11t + 99 = 110 11t = 11 t = 1 Therefore, the values of h and t are: h = 2.25 t = 1 ### Answer: h = 2.25, t = 1
Expected answer: {h: 481/214, t: 108/107}
Correct? No
Problem: Solve the following system of linear equations to find the values of b and o: ((69 * b) - (50 * o)) = 55 ((41 * o) + (18 * b)) = 85.
Claude-3 Sonnet:
To solve the given system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: (69 * b) - (50 * o) = 55 (41 * o) + (18 * b) = 85 Step 1: Multiply the first equation by 41 and the second equation by 50 to eliminate the variable 'o'. 41(69b - 50o = 55) 2839b - 2050o = 2255 50(41o + 18b = 85) 2050o + 900b = 4250 Step 2: Subtract the second equation from the first equation to eliminate 'o'. 2839b - 2050o = 2255 - (2050o + 900b = 4250) ----------------------- 789b = -1995 b = -1995/789 b = -2.525 (approximately) Step 3: Substitute the value of 'b' in either of the original equations to find the value of 'o'. Let's use the first equation: (69 * b) - (50 * o) = 55 (69 * (-2.525)) - (50 * o) = 55 -174.225 - 50o = 55 -50o = 55 + 174.225 -50o = 229.225 o = -229.225/(-50) o = 4.585 (approximately) Therefore, the values of 'b' and 'o' are: b = -2.525 o = 4.585 ### Answer: b = -2.525, o = 4.585
Expected answer: {b: 6505/3729, o: 1625/1243}
Correct? No
Problem: Solve the following system of equations to find the values of p and q: 67 * q - 80 * p = 65, and 6 * q + 5 * p = 177.
Claude-3 Sonnet:
To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 67q - 80p = 65 6q + 5p = 177 Step 1: Multiply the first equation by 6 and the second equation by 67 to eliminate the variable q. 6(67q - 80p = 65) 402q - 480p = 390 67(6q + 5p = 177) 402q + 335p = 11889 Step 2: Subtract the second equation from the first equation to eliminate q. 402q - 480p = 390 -(402q + 335p = 11889) -------------- -815p = -11499 Step 3: Divide both sides by -815 to find the value of p. p = 11499/815 p = 14.1 Step 4: Substitute the value of p into either of the original equations to find the value of q. Let's use the second equation: 6q + 5(14.1) = 177 6q + 70.5 = 177 6q = 106.5 q = 106.5/6 q = 17.75 Therefore, the values of p and q are: p = 14.1 q = 17.75 ### Answer: p = 14.1, q = 17.75
Expected answer: {p: 11469/815, q: 2897/163}
Correct? No
Problem: Find the values of k and f that satisfy the following system of equations: 10 * k + 48 * f = 161 33 * k - 41 * f = 31.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 10k + 48f = 161 33k - 41f = 31 Step 1: Multiply the first equation by 33 and the second equation by 10 to eliminate the variable k. 330k + 1584f = 5313 330k - 410f = 310 Step 2: Subtract the second equation from the first equation to eliminate k. 1994f = 5003 Step 3: Solve for f. f = 5003/1994 f = 2.51 (approximately) Step 4: Substitute the value of f into either of the original equations to find k. Let's use the first equation: 10k + 48(2.51) = 161 10k + 120.48 = 161 10k = 40.52 k = 4.052 Therefore, the values of k and f that satisfy the given system of equations are: k ≈ 4.052 f ≈ 2.51 ### Answer: k ≈ 4.052, 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
Claude-3 Sonnet:
To solve the given system of equations, we can use the substitution method or the elimination method. Let's use the substitution method. Given equations: 31 = 58l - n 103 = n + 73l Step 1: Express one variable in terms of the other variable from one of the equations. From the first equation, we can express n in terms of l. n = 58l - 31 Step 2: Substitute the expression for n into the second equation. 103 = n + 73l 103 = (58l - 31) + 73l 103 = 131l - 31 Step 3: Solve for l by rearranging the terms. 131l - 31 = 103 131l = 134 l = 134/131 l ≈ 1.023 Step 4: Substitute the value of l into either of the original equations to find n. Let's use the first equation: 31 = 58l - n 31 = 58(1.023) - n 31 = 59.334 - n n = 59.334 - 31 n = 28.334 Therefore, the solution to the system of equations is: l ≈ 1.023 n ≈ 28.334 ### Answer: l ≈ 1.023, n ≈ 28.334
Expected answer: {l: 134/131, n: 3711/131}
Correct? No
Problem: Find the values of y and j that satisfy the following system of equations: 29y - 28j = 39 25y + 37j = 116
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 25 and the second equation by 29 to eliminate the variable y. 25(29y - 28j = 39) ⇒ 725y - 700j = 975 29(25y + 37j = 116) ⇒ 725y + 1073j = 3364 Step 2: Subtract the second equation from the first equation to eliminate y. 725y - 700j = 975 -(725y + 1073j = 3364) ----------------- -1773j = -2389 Step 3: Divide both sides by -1773 to find the value of j. j = 2389/1773 j = 1.347 (rounded to 3 decimal places) Step 4: Substitute the value of j into either of the original equations to find the value of y. Let's use the first equation: 29y - 28(1.347) = 39 29y - 37.716 = 39 29y = 76.716 y = 76.716/29 y = 2.645 (rounded to 3 decimal places) Therefore, the values of y and j that satisfy the given system of equations are: y = 2.645 j = 1.347
Expected answer: {j: 2389/1773, y: 4691/1773}
Correct? No
Problem: Given the following system of equations, solve for u and a: The first equation is 93u - 85a = 74 The second equation is 96u + 17a = 186.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 17 and the second equation by -85 to eliminate the variable 'a'. 17(93u - 85a = 74) ⇒ 1581u - 1445a = 1258 -85(96u + 17a = 186) ⇒ -8160u - 1445a = -15810 Step 2: Add the two equations to eliminate 'a'. (1581u - 1445a = 1258) + (-8160u - 1445a = -15810) -6579u = -14552 Step 3: Solve for 'u'. -6579u = -14552 u = -14552/-6579 u = 2.21 Step 4: Substitute the value of 'u' into either of the original equations to find 'a'. Let's use the first equation: 93u - 85a = 74 93(2.21) - 85a = 74 205.53 - 85a = 74 -85a = -131.53 a = 1.55 Therefore, the solutions are: u = 2.21 a = 1.55 ### Answer: u = 2.21, a = 1.55
Expected answer: {a: 3398/3247, u: 1004/573}
Correct? No
Problem: Given the following pair of simultaneous equations, please find out the values of u and g: 97 * u - 61 * g = 15 85 * u + 23 * g = 186
Claude-3 Sonnet:
To solve this pair of simultaneous equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 23 and the second equation by 61 to eliminate the variable g. 97 * u - 61 * g = 15 (Multiply by 23) 2231u - 1403g = 345 85 * u + 23 * g = 186 (Multiply by 61) 5185u + 1403g = 11346 Step 2: Subtract the second equation from the first equation to eliminate g. 2231u - 1403g = 345 -5185u - 1403g = -11346 ----------------- -2954u = -10001 Step 3: Solve for u. -2954u = -10001 u = 10001/2954 u = 3.386 (rounded to 3 decimal places) Step 4: Substitute the value of u into either of the original equations to find g. Let's use the first equation: 97 * u - 61 * g = 15 97 * 3.386 - 61g = 15 328.042 - 61g = 15 -61g = -313.042 g = 5.131 (rounded to 3 decimal places) Therefore, the values of u and g are: u = 3.386 g = 5.131 ### Answer: u = 3.386, g = 5.131
Expected answer: {g: 1863/824, u: 1299/824}
Correct? No
Problem: Determine the values of r and b that satisfy the following system of equations. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187.
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method. Given equations: (17 * r) + (67 * b) = 103 (1) (85 * r) + (67 * b) = 187 (2) Step 1: Multiply the first equation by (-5) to eliminate the variable b. (-5) × (17 * r + 67 * b = 103) -85 * r - 335 * b = -515 Step 2: Add the resulting equation from Step 1 to the second equation to eliminate the variable b. (85 * r + 67 * b = 187) + (-85 * r - 335 * b = -515) 0 * b = -328 Step 3: Solve for r using the second equation. 85 * r + 67 * b = 187 85 * r = 187 - 67 * b 85 * r = 187 (since b = 0) r = 187 / 85 r = 2.2 Step 4: Substitute the value of r into either equation to find the value of b. (17 * 2.2) + (67 * b) = 103 37.4 + 67 * b = 103 67 * b = 103 - 37.4 67 * b = 65.6 b = 65.6 / 67 b = 0.98 Therefore, the values of r and b that satisfy the given system of equations are: r = 2.2 b = 0.98 ### Answer: r = 2.2, b = 0.98
Expected answer: {b: 82/67, r: 21/17}
Correct? No
Problem: Given the following system of equations, solve for the values of t and n: 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126.
Claude-3 Sonnet:
To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method. Step 1: Multiply the first equation by 38 and the second equation by 54 to eliminate the variable t. 38(85n - 54t = 41) 3230n - 2052t = 1558 54(38t + 30n = 126) 2052t + 1620n = 6804 Step 2: Subtract the second equation from the first equation to eliminate t. 3230n - 2052t = 1558 - (2052t + 1620n = 6804) ---------------------------- 1610n = -5246 Step 3: Solve for n. n = -5246/1610 n = -3.26 Step 4: Substitute the value of n into either of the original equations to find t. Let's use the first equation: (85n) - (54t) = 41 (85(-3.26)) - (54t) = 41 -277.1 - 54t = 41 -54t = 41 + 277.1 -54t = 318.1 t = -5.89 Therefore, the values of t and n are: t = -5.89 n = -3.26 ### Answer: t = -5.89, n = -3.26
Expected answer: {n: 4181/2425, t: 948/485}
Correct? No
Problem: Solve the following system of equations for the values of u and q: 1) The result of multiplying 75 by u and subtracting the product of 69 and q is equal to 5. 2) The sum of the product of 48 and u and the product of 25 and q is 140.
Claude-3 Sonnet:
To solve this system of equations, we need to use the substitution method or the elimination method. Let's use the substitution method: Step 1: Rearrange the first equation to express one variable in terms of the other. 75u - 69q = 5 75u = 5 + 69q u = (5 + 69q) / 75 Step 2: Substitute the expression for u into the second equation. 48u + 25q = 140 48((5 + 69q) / 75) + 25q = 140 (240 + 3312q) / 75 + 25q = 140 3312q + 1875q = 10500 5187q = 10500 q = 10500 / 5187 q = 2 Step 3: Substitute the value of q into the expression for u to find the value of u. u = (5 + 69(2)) / 75 u = (5 + 138) / 75 u = 143 / 75 u = 1.9 Therefore, the values of u and q are: u = 1.9 q = 2 ### Answer: u = 1.9, q = 2
Expected answer: {q: 180/91, u: 515/273}
Correct? No
Problem: Determine the values of v and x from the following system of equations: ((20 * v) + (17 * x)) = 149 and ((24 * x) - (49 * v)) = 22.
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Step 1: Multiply the first equation by 49 and the second equation by 20 to eliminate the variable x. 49((20v) + (17x)) = 7,231 20((24x) - (49v)) = 440 Step 2: Subtract the second equation from the first equation to eliminate x. 7,231 - 440 = 6,791 980v + 833x = 7,231 -980v - 98x = -440 -------------- 1,666x = 6,791 Step 3: Solve for x. x = 6,791 / 1,666 x = 4.08 Step 4: Substitute the value of x into either of the original equations to find v. Let's use the first equation: (20v) + (17(4.08)) = 149 20v + 69.36 = 149 20v = 79.64 v = 3.98 Therefore, the values of v and x are: v = 3.98 x = 4.08 ### Answer: v = 3.98, x = 4.08
Expected answer: {v: 3202/1313, x: 7741/1313}
Correct? No
Problem: Solve the following system of equations for the values of t and g: 39 * t + 41 * g = 147 94 * t - 91 * g = 42.
Claude-3 Sonnet:
To solve the given system of equations for the values of t and g, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 39t + 41g = 147 94t - 91g = 42 Step 1: Multiply the first equation by 94 and the second equation by 39 to eliminate one of the variables (let's eliminate t). 39t × 94 + 41g × 94 = 147 × 94 3666t + 3854g = 13818 94t × 39 - 91g × 39 = 42 × 39 3666t - 3549g = 1638 Step 2: Subtract the second equation from the first equation to eliminate t. (3854g + 3666t) - (-3549g + 3666t) = 13818 - 1638 7403g = 12180 Step 3: Solve for g by dividing both sides by 7403. g = 12180 / 7403 g = 1.645 (approximately) Step 4: Substitute the value of g into either of the original equations to find the value of t. Let's use the first equation: 39t + 41(1.645) = 147 39t + 67.645 = 147 39t = 147 - 67.645 39t = 79.355 t = 79.355 / 39 t = 2.035 (approximately) Therefore, the solution to the system of equations is: t ≈ 2.035 g ≈ 1.645 ### Answer: t ≈ 2.035, g ≈ 1.645
Expected answer: {g: 12180/7403, t: 15099/7403}
Correct? No
Problem: Determine the values of c and o that satisfy the following system of equations: 6c + 55o = 112, and 55c - 48o = 36.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 6c + 55o = 112 55c - 48o = 36 Step 1: Multiply the first equation by -48 and the second equation by 6 to eliminate the variable c. -288c - 2640o = -5376 330c - 288o = 216 Step 2: Add the two equations to eliminate c. 42o = -5160 Step 3: Solve for o. o = -5160/42 o = -122.857142857... o ≈ -123 Step 4: Substitute the value of o into either of the original equations to find c. Let's use the first equation: 6c + 55(-123) = 112 6c - 6765 = 112 6c = 6877 c = 1146 Therefore, the values of c and o that satisfy the given system of equations are: c = 1146 o = -123 ### Answer: c = 1146, o = -123
Expected answer: {c: 7356/3313, o: 5944/3313}
Correct? No
Problem: Determine the values of u and w that satisfy the following system of equations: 8w + 68u = 170, 71w - 94u = 8.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 71 and the second equation by 8 to eliminate the variable w. 8(71)w + 68(71)u = 170(71) 568w + 4828u = 12070 71(8)w - 94(8)u = 8(8) 568w - 752u = 64 Step 2: Subtract the second equation from the first equation to eliminate w. 4828u - (-752u) = 12070 - 64 5580u = 12006 Step 3: Solve for u. u = 12006 / 5580 u = 2.15 Step 4: Substitute the value of u into either of the original equations to find w. Let's use the first equation: 8w + 68(2.15) = 170 8w + 146.2 = 170 8w = 23.8 w = 2.975 Therefore, the values that satisfy the system of equations are: u = 2.15 w = 2.975 ### Answer: u = 2.15, w = 2.975
Expected answer: {u: 667/310, w: 459/155}
Correct? No
Problem: Solve the following system of equations for the values of r and f: 68*r + 8*f = 92 97*f + 38*r = 182
Claude-3 Sonnet:
To solve the given system of equations for the values of r and f, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 68r + 8f = 92 ... (1) 97f + 38r = 182 ... (2) Step 1: Multiply equation (1) by 97 and equation (2) by 68 to eliminate the variable f. 97 × (68r + 8f = 92) 6556r + 776f = 8884 68 × (97f + 38r = 182) 6596f + 2584r = 12376 Step 2: Subtract the second equation from the first equation to eliminate f. (6556r + 776f = 8884) - (6596f + 2584r = 12376) -5820f + 3972r = -3492 Step 3: Divide both sides of the equation by 3972 to find the value of r. -5820f + 3972r = -3492 r = -3492 / 3972 r = -0.879 (rounded to 3 decimal places) Step 4: Substitute the value of r into either of the original equations to find the value of f. Let's use equation (1): 68r + 8f = 92 68(-0.879) + 8f = 92 -59.772 + 8f = 92 8f = 92 + 59.772 8f = 151.772 f = 151.772 / 8 f = 18.972 (rounded to 3 decimal places) Therefore, the solution to the system of equations is: r = -0.879 f = 18.972 ### Answer: r = -0.879, f = 18.972
Expected answer: {f: 2220/1573, r: 1867/1573}
Correct? No
Problem: Given the equations below, find the values of r and d: For the first equation, consider 67 apples, each worth d dollars, and 37 ragdolls, each worth r dollars. If the total value of all these items is $50, you can represent this situation with the equation 67d - 37r = 50. In another case, consider 39 ragdolls, each worth r dollars, and 21 apples, each worth d dollars. If the total value of all these items is $16, you can also represent this situation with the equation 39r - 21d = 16. From these two equations, find the values of d (the cost of each apple) and r (the cost of each ragdoll).
Claude-3 Sonnet:
To solve this problem, we need to use the system of linear equations and find the values of r and d that satisfy both equations. Given equations: 67d - 37r = 50 39r - 21d = 16 Step 1: Rearrange the equations to get the standard form (Ax + By = C). 67d - 37r = 50 (Multiply both sides by -1) -67d + 37r = -50 39r - 21d = 16 Step 2: Use the elimination method to solve for one variable in terms of the other. Multiply the first equation by 39 and the second equation by 67 to eliminate d. (-67d + 37r) × 39 = -50 × 39 -2613d + 1443r = -1950 39r - 21d = 16 (Multiply by 67) 2613r - 1407d = 1072 Step 3: Add the two equations to eliminate d. 1443r - 1950 + 2613r - 1072 = 0 4056r = 3022 r = 3022/4056 r = 0.75 (Cost of each ragdoll) Step 4: Substitute the value of r into either equation to find d. Let's use the first equation: 67d - 37(0.75) = 50 67d - 27.75 = 50 67d = 77.75 d = 77.75/67 d = 1.16 (Cost of each apple) Therefore, the cost of each ragdoll (r) is $0.75, and the cost of each apple (d) is $1.16. ### Answer: r = 0.75, d = 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.
Claude-3 Sonnet:
To solve the given system of simultaneous linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 163 = 50x + a (First equation) 24 = 98x - 84a (Second equation) Step 1: Multiply the first equation by 98 and the second equation by 50 to eliminate the variable x. 98(163) = 98(50x + a) 15,974 = 4,900x + 98a 50(24) = 50(98x - 84a) 1,200 = 4,900x - 4,200a Step 2: Subtract the second equation from the first equation to eliminate x. 15,974 - 1,200 = 4,900x + 98a - 4,900x + 4,200a 14,774 = 4,298a Step 3: Solve for a. a = 14,774 / 4,298 a = 3.44 (approximately) Step 4: Substitute the value of a into either of the original equations to find x. Let's use the first equation: 163 = 50x + a 163 = 50x + 3.44 50x = 163 - 3.44 50x = 159.56 x = 159.56 / 50 x = 3.19 (approximately) Therefore, the values of x and a are: x ≈ 3.19 a ≈ 3.44 ### Answer: x ≈ 3.19, a ≈ 3.44
Expected answer: {a: 7387/2149, x: 6858/2149}
Correct? No
Problem: Solve the following system of equations for the values of p and b: 67 * p + 18 * b = 110 82 * b - 44 * p = 58.
Claude-3 Sonnet:
To solve the given system of equations for the values of p and b, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 67p + 18b = 110 82b - 44p = 58 Step 1: Multiply the first equation by 44 and the second equation by 67 to eliminate p. 67(67p + 18b) = 4490 + 1206b 82(82b - 44p) = 6724b - 3608p Step 2: Subtract the second equation from the first equation to eliminate p. 4490 + 1206b -(-3608p + 6724b) ----------------------- 4490 - 6724b + 1206b = -2518b Step 3: Solve for b. -2518b = -2518 b = 1 Step 4: Substitute the value of b into either of the original equations to find p. Let's use the first equation: 67p + 18(1) = 110 67p + 18 = 110 67p = 92 p = 92/67 p = 1.373134328 (approximately) Therefore, the values of p and b that satisfy the given system of equations are: p ≈ 1.373134328 b = 1 ### Answer: p ≈ 1.373134328, b = 1
Expected answer: {b: 4363/3143, p: 3988/3143}
Correct? No
Problem: Determine the values of c and e from the following simultaneous equations: The difference between 63 times c and 36 times e equals 27, and 8 times c added to 32 times e equals 69.
Claude-3 Sonnet:
To solve this problem, we need to use the given simultaneous equations and find the values of c and e. Given equations: 63c - 36e = 27 8c + 32e = 69 Step 1: Rearrange the equations to get one variable in terms of the other. From the first equation, we can express e in terms of c. 63c - 36e = 27 36e = 63c - 27 e = (63c - 27) / 36 Step 2: Substitute the expression for e into the second equation. 8c + 32((63c - 27) / 36) = 69 8c + (2016c - 864) / 36 = 69 8c + 56c - 24 = 69 64c = 93 c = 93/64 Step 3: Substitute the value of c into the expression for e to find the value of e. e = (63c - 27) / 36 e = (63 × (93/64) - 27) / 36 e = (5859/64 - 27) / 36 e = (5832/64) / 36 e = 162/64 e = 2.53125 Therefore, the values of c and e are: c = 93/64 ≈ 1.453125 e = 2.53125 ### Answer: c = 93/64 ≈ 1.453125, e = 2.53125
Expected answer: {c: 93/64, e: 459/256}
Correct? No
Problem: Solve the following system of linear equations to find the values of x and p: 26 * x + 29 * p = 107 61 * x - 51 * p = 4
Claude-3 Sonnet:
To solve the given system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 26x + 29p = 107 61x - 51p = 4 Step 1: Multiply the first equation by a suitable constant to make the coefficients of one variable (say, x) the same in both equations. Multiplying the first equation by 61/26, we get: 61x + (29 × 61/26)p = 107 × 61/26 61x + 67.5p = 2497 Step 2: Subtract the second equation from the modified first equation to eliminate the variable x. (61x + 67.5p = 2497) - (61x - 51p = 4) 118.5p = 2493 Step 3: Solve for p. p = 2493 / 118.5 p = 21 Step 4: Substitute the value of p into either of the original equations to find the value of x. Let's use the first equation: 26x + 29(21) = 107 26x + 609 = 107 26x = -502 x = -19.3076923077 Therefore, the values of x and p are: x = -19.3076923077 p = 21 ### Answer: x = -19.3076923077, p = 21
Expected answer: {p: 6423/3095, x: 5573/3095}
Correct? No
Problem: Find the values for g and l from the system of equations below. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17.
Claude-3 Sonnet:
To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method. Given equations: (29 × l) + (17 × g) = 68 (31 × g) - (10 × l) = 17 Step 1: Multiply the first equation by 31 and the second equation by 29 to eliminate the variable g. 31 × [(29 × l) + (17 × g) = 68] 31 × (29 × l) + 31 × (17 × g) = 68 × 31 899l + 527g = 2108 29 × [(31 × g) - (10 × l) = 17] 29 × (31 × g) - 29 × (10 × l) = 17 × 29 899g - 290l = 493 Step 2: Subtract the second equation from the first equation to eliminate g. (899l + 527g = 2108) - (899g - 290l = 493) 817l = 1615 Step 3: Solve for l. l = 1615 / 817 l = 1.975 (approximately) Step 4: Substitute the value of l into either of the original equations to find g. Let's use the first equation: (29 × l) + (17 × g) = 68 (29 × 1.975) + (17 × g) = 68 57.075 + 17g = 68 17g = 68 - 57.075 17g = 10.925 g = 10.925 / 17 g = 0.643 (approximately) Therefore, the values of l and g are: l ≈ 1.975 g ≈ 0.643 ### Answer: l ≈ 1.975, g ≈ 0.643
Expected answer: {g: 1173/1069, l: 1819/1069}
Correct? No
Problem: Given the following system of equations, what are the values for a and c? 33*a - 54*c = 20 24*a - 26*c = 29.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 2 to eliminate the variable a. 66a - 108c = 40 Step 2: Subtract the second equation from the modified first equation. 66a - 108c = 40 -(24a - 26c = 29) ----------------- 42a - 82c = 11 Step 3: Divide both sides of the equation by -82 to find the value of c. c = 11/82 Step 4: Substitute the value of c into either of the original equations to find the value of a. Let's use the first equation: 33a - 54(11/82) = 20 33a - 594/82 = 20 33a = 20 + 594/82 a = (20 + 594/82) / 33 a = 614/82 / 33 a = 614 / 2706 a = 0.227 (approximately) Therefore, the values are: a ≈ 0.227 c ≈ -0.134 ### Answer: a = 0.227, c = -0.134
Expected answer: {a: 523/219, c: 159/146}
Correct? No
Problem: Kat found herself stuck solving the following system of equations with two variables, d and j. Can you help her resolve it? 71d + 9j = 132 43j - 84d = 91.
Claude-3 Sonnet:
To solve a system of two linear equations with two variables, we can use various methods, such as the substitution method, elimination method, or matrix method. Let's use the elimination method here. Given equations: 71d + 9j = 132 43j - 84d = 91 Step 1: Multiply the first equation by 84 and the second equation by 71 to eliminate the variable d. 71d × 84 + 9j × 84 = 132 × 84 5964d + 756j = 11088 43j × 71 - 84d × 71 = 91 × 71 3053j - 5964d = 6461 Step 2: Subtract the second equation from the first equation to eliminate d. (5964d + 756j) - (3053j - 5964d) = 11088 - 6461 9817j = 4627 Step 3: Solve for j by dividing both sides by 9817. j = 4627 / 9817 j = 0.472 (approximately) Step 4: Substitute the value of j into either of the original equations to find d. Let's use the first equation: 71d + 9(0.472) = 132 71d + 4.248 = 132 71d = 127.752 d = 1.8 (approximately) Therefore, the solution to the system of equations is: d ≈ 1.8 j ≈ 0.472 ### Answer: d ≈ 1.8, j ≈ 0.472
Expected answer: {d: 4857/3809, j: 17549/3809}
Correct? No
Problem: Determine the values of o and n by solving this system of equations: (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143.
Claude-3 Sonnet:
To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method. Step 1: Multiply the first equation by 87 and the second equation by 81 to eliminate the variable n. (81 × 87 × o) + (26 × 87 × n) = 163 × 87 (87 × 23 × o) + (81 × 87 × n) = 143 × 81 7047o + 2262n = 14191 2001o + 7047n = 11583 Step 2: Subtract the second equation from the first equation to eliminate n. (7047o + 2262n) - (2001o + 7047n) = 14191 - 11583 5046o - 4785n = 2608 Step 3: Set the coefficient of n to zero by dividing both sides by -4785. 5046o - 4785n = 2608 -4785n = -2608 n = 2608 / 4785 n = 0.545 (rounded to 3 decimal places) Step 4: Substitute the value of n into either of the original equations to find the value of o. (81 × o) + (26 × 0.545) = 163 81o + 14.17 = 163 81o = 148.83 o = 148.83 / 81 o = 1.838 (rounded to 3 decimal places) Therefore, the values of o and n are: o = 1.838 n = 0.545 ### Answer: o = 1.838, n = 0.545
Expected answer: {n: 7834/6449, o: 10463/6449}
Correct? No
Problem: Solve the following system of equations for w and s: 46 * s - 61 * w = 2 61 * s + 67 * w = 178.
Claude-3 Sonnet:
To solve the given system of equations for w and s, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 46s - 61w = 2 61s + 67w = 178 Step 1: Multiply the first equation by 61 and the second equation by 46 to eliminate the variable s. 61(46s - 61w = 2) 2806s - 3721w = 122 46(61s + 67w = 178) 2806s + 3082w = 8168 Step 2: Subtract the second equation from the first equation to eliminate s. 2806s - 3721w = 122 - (2806s + 3082w = 8168) ---------------------------- -6803w = -8046 Step 3: Solve for w. -6803w = -8046 w = 8046/6803 w = 1.18 (approximately) Step 4: Substitute the value of w into either of the original equations to find s. Let's use the first equation: 46s - 61(1.18) = 2 46s - 71.98 = 2 46s = 73.98 s = 73.98/46 s = 1.61 (approximately) Therefore, the solution to the system of equations is: w = 1.18 s = 1.61 ### Answer: w = 1.18, s = 1.61
Expected answer: {s: 10992/6803, w: 8066/6803}
Correct? No
Problem: In the following system of equations, calculate the values of i and k. System of equations: 51*k - 8*i = 86 39*i + 8*k = 193.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 8 and the second equation by 51 to eliminate the variable i. 408k - 64i = 688 1989i + 408k = 9843 Step 2: Subtract the second equation from the first equation to eliminate the variable k. -2053i = -9155 i = 9155/2053 i = 4.46 (approximately) Step 3: Substitute the value of i into either of the original equations to find the value of k. Let's use the first equation: 51k - 8(4.46) = 86 51k - 35.68 = 86 51k = 121.68 k = 121.68/51 k = 2.39 (approximately) Therefore, the values of i and k in the given system of equations are: i = 4.46 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
Claude-3 Sonnet:
To solve the given system of equations for the values of l and m, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 26m + 17l = 146 93m + l = 126 Step 1: Multiply the first equation by (-1) to make the coefficients of l opposite in sign. -26m - 17l = -146 Step 2: Add the two equations to eliminate the variable l. (-26m - 17l) + (93m + l) = (-146) + 126 67m = -20 Step 3: Divide both sides by 67 to find the value of m. m = -20/67 Step 4: Substitute the value of m into either of the original equations to find the value of l. Let's use the second equation: 93m + l = 126 Substituting m = -20/67, we get: 93(-20/67) + l = 126 (-1860/67) + l = 126 l = 126 + 1860/67 l = 1986/67 Therefore, the values of l and m are: m = -20/67 l = 1986/67 ### Answer: m = -20/67, l = 1986/67
Expected answer: {l: 10302/1555, m: 1996/1555}
Correct? No
Problem: Determine the values of 'h' and 'f' by solving the following system of equations: 1) (8 * f) + (39 * h) = 66 2) (29 * f) + (46 * h) = 119
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by -29 and the second equation by 8 to eliminate the variable 'f'. -29(8f + 39h = 66) -232f - 1131h = -1914 8(29f + 46h = 119) 232f + 368h = 952 Step 2: Add the two equations together to eliminate 'f'. -1131h + 368h = -1914 + 952 -763h = -962 h = 962/763 h = 1.26 (approximately) Step 3: Substitute the value of 'h' into either of the original equations to find the value of 'f'. Let's use the first equation: (8f) + (39 × 1.26) = 66 8f + 49.14 = 66 8f = 16.86 f = 2.11 (approximately) Therefore, the values of 'h' and 'f' are: h = 1.26 f = 2.11 ### Answer: h = 1.26, f = 2.11
Expected answer: {f: 1605/763, h: 962/763}
Correct? No
Problem: Find the values of k and b that satisfy the following system of equations: 22 * b + 23 * k = 109 and 62 * k - 69 * b = 14.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 22b + 23k = 109 62k - 69b = 14 Step 1: Multiply the first equation by 62 and the second equation by 23 to eliminate k. 22b(62) + 23k(62) = 109(62) 1364b + 1426k = 6758 62k(23) - 69b(23) = 14(23) 1426k - 1587b = 322 Step 2: Subtract the second equation from the first equation to eliminate k. (1364b + 1426k) - (1426k - 1587b) = 6758 - 322 2951b = 6436 Step 3: Solve for b. b = 6436 / 2951 b = 2.18 Step 4: Substitute the value of b into either of the original equations to find k. Let's use the first equation: 22b + 23k = 109 22(2.18) + 23k = 109 47.96 + 23k = 109 23k = 109 - 47.96 23k = 61.04 k = 61.04 / 23 k = 2.65 Therefore, the values of k and b that satisfy the given system of equations are: k = 2.65 b = 2.18 ### Answer: k = 2.65, b = 2.18
Expected answer: {b: 6436/2951, k: 7829/2951}
Correct? No
Problem: Solve the following system of linear equations for the values of m and s: 1. 84m - 55s = 1 2. 63m - 2s = 98.
Claude-3 Sonnet:
To solve the given system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 2 and the second equation by 3 to eliminate the variable m. 2(84m - 55s = 1) ⇒ 168m - 110s = 2 3(63m - 2s = 98) ⇒ 189m - 6s = 294 Step 2: Subtract the second equation from the first equation to eliminate m. (168m - 110s = 2) - (189m - 6s = 294) -21m - 104s = -292 Step 3: Divide both sides of the equation by -104 to find the value of s. -104s = -292 s = 292/104 s = 2.8076923076923075 Step 4: Substitute the value of s into either of the original equations to find the value of m. Let's use the first equation: 84m - 55(2.8076923076923075) = 1 84m - 153.42307692307692 = 1 84m = 154.42307692307692 m = 154.42307692307692 / 84 m = 1.8383451085702472 Therefore, the solution to the system of linear equations is: m = 1.8383451085702472 s = 2.8076923076923075 ### Answer: m = 1.8383451085702472, s = 2.8076923076923075
Expected answer: {m: 1796/1099, s: 389/157}
Correct? No
Problem: Given the following system of equations, find the values of b and k: 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67.
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Step 1: Multiply the first equation by 51 and the second equation by 6 to eliminate the variable b. 1. (306b) + (255k) = 5355 2. (570k) - (306b) = 402 Step 2: Subtract the second equation from the first equation to eliminate b. (306b) + (255k) = 5355 -(570k) + (306b) = -402 ---------------------------- -315k = 4953 Step 3: Solve for k by dividing both sides by -315. k = -4953/-315 k = 15.75 Step 4: Substitute the value of k into either of the original equations to find the value of b. Let's use the first equation: (6b) + (5 * 15.75) = 105 6b + 78.75 = 105 6b = 26.25 b = 4.375 Therefore, the values of b and k are: b = 4.375 k = 15.75 ### Answer: b = 4.375, k = 15.75
Expected answer: {b: 1928/165, k: 1919/275}
Correct? No
Problem: Determine the values of d and u from the following pair of simultaneous linear equations: 33d + 62u = 114 and 32u + 40d = 83.
Claude-3 Sonnet:
To solve this pair of simultaneous linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 33d + 62u = 114 32u + 40d = 83 Step 1: Multiply the first equation by 32 and the second equation by 33 to eliminate one of the variables (let's eliminate d). 32(33d + 62u) = 3648 33(32u + 40d) = 2739 Step 2: Subtract the second equation from the first equation. 3648 - 2739 = 909 1056u + 1056d - 1320d = 909 1056u - 264d = 909 Step 3: Divide both sides of the equation by the coefficient of u to find the value of u. 1056u/1056 - 264d/1056 = 909/1056 u - 0.25d = 0.86 u = 0.86 + 0.25d Step 4: Substitute the value of u into one of the original equations to find the value of d. Let's use the first equation: 33d + 62u = 114 33d + 62(0.86 + 0.25d) = 114 33d + 53.32 + 15.5d = 114 48.5d = 60.68 d = 60.68/48.5 d = 1.25 Step 5: Substitute the value of d into the expression for u to find the value of u. u = 0.86 + 0.25(1.25) u = 0.86 + 0.3125 u = 1.1725 Therefore, the values of d and u are: d = 1.25 u = 1.1725 ### Answer: d = 1.25, u = 1.1725
Expected answer: {d: 749/712, u: 1821/1424}
Correct? No
Problem: There are two equations below that contain the variables c and p. Can you solve this system of equations to find the values of c and p? 5 * c + 28 * p = 102 20 * c - 67 * p = 13
Claude-3 Sonnet:
To solve this system of equations with two variables (c and p), we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 5c + 28p = 102 20c - 67p = 13 Step 1: Multiply the first equation by 4 and the second equation by 1 to eliminate the variable c. 20c + 112p = 408 20c - 67p = 13 Step 2: Subtract the second equation from the first equation to eliminate c. 20c + 112p = 408 - (20c - 67p = 13) --------------- 179p = 395 Step 3: Solve for p by dividing both sides by 179. p = 395 / 179 p = 2.21 Step 4: Substitute the value of p into either of the original equations to find c. Let's use the first equation: 5c + 28(2.21) = 102 5c + 61.88 = 102 5c = 40.12 c = 8.024 Therefore, the values of the variables are: c = 8.024 p = 2.21 ### Answer: c = 8.024, p = 2.21
Expected answer: {c: 7198/895, p: 395/179}
Correct? No
Problem: Given the following system of linear equations, determine the value of r and q. The system of equations is: 58 * r + 12 * q = 147 97 * q - 76 * r = 76.
Claude-3 Sonnet:
To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 76 and the second equation by 58 to eliminate the variable r. (58r + 12q) × 76 = 4408r + 912q = 11,172 (97q - 76r) × 58 = 5,626q - 4,408r = 4,408 Step 2: Subtract the second equation from the first equation to eliminate r. 4,408r + 912q = 11,172 -(5,626q - 4,408r = 4,408) -------------- -4,714q = 6,764 Step 3: Solve for q. -4,714q = 6,764 q = -6,764 / -4,714 q = 1.435 (approximately) Step 4: Substitute the value of q into either of the original equations to find r. Let's use the first equation: 58r + 12(1.435) = 147 58r + 17.22 = 147 58r = 147 - 17.22 58r = 129.78 r = 129.78 / 58 r = 2.238 (approximately) Therefore, the values of r and q are: r ≈ 2.238 q ≈ 1.435 ### Answer: r ≈ 2.238, q ≈ 1.435
Expected answer: {q: 7790/3269, r: 13347/6538}
Correct? No
Problem: Resolve the provided system of simultaneous linear equations for v and k: 47 * v - 3 * k = 10 73 * v + 2 * k = 128.
Claude-3 Sonnet:
To solve the given system of simultaneous linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 47v - 3k = 10 73v + 2k = 128 Step 1: Multiply the first equation by 2 to eliminate the variable k. 94v - 6k = 20 Step 2: Subtract the second equation from the modified first equation. 94v - 6k - (73v + 2k) = 20 - 128 21v - 8k = -108 Step 3: Divide both sides of the equation by -8 to find the value of k. -21v/8 + k = 108/8 k = 13.5 Step 4: Substitute the value of k into either of the original equations to find the value of v. Let's use the first equation: 47v - 3(13.5) = 10 47v - 40.5 = 10 47v = 50.5 v = 50.5/47 v ≈ 1.074468085106383 Therefore, the solution to the system of simultaneous linear equations is: v ≈ 1.074468085106383 k = 13.5 ### Answer: v ≈ 1.074468085106383, k = 13.5
Expected answer: {k: 5286/313, v: 404/313}
Correct? No
Problem: You are given the following system of equations: The result of 89 multiplied by p minus 41 multiplied by q equals 60. The result of 27 multiplied by p plus 91 multiplied by q equals 124. What are the values of p and q?
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Given equations: 89p - 41q = 60 27p + 91q = 124 Step 1: Multiply the first equation by 27 and the second equation by 89 to eliminate p. 27(89p - 41q) = 27(60) 2403p - 1107q = 1620 89(27p + 91q) = 89(124) 2403p + 8099q = 11036 Step 2: Subtract the second equation from the first equation to eliminate p. 2403p - 1107q = 1620 -(2403p + 8099q = 11036) ----------------- -9206q = -9416 Step 3: Solve for q. q = -9416/-9206 q = 1.023 (approximately) Step 4: Substitute the value of q into either of the original equations to find p. Let's use the first equation: 89p - 41(1.023) = 60 89p - 41.943 = 60 89p = 101.943 p = 101.943/89 p = 1.145 (approximately) Therefore, the values of p and q are: p ≈ 1.145 q ≈ 1.023 ### Answer: p ≈ 1.145, q ≈ 1.023
Expected answer: {p: 5272/4603, q: 4708/4603}
Correct? No
Problem: Find the values of u and d that make these equations true: The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71
Claude-3 Sonnet:
To find the values of u and d that make the given equations true, we need to solve the system of two linear equations with two unknowns. Given equations: (88 * u) - (96 * d) = 12 (27 * u) + (36 * d) = 71 Step 1: Solve for one of the variables, say d, in terms of u from the first equation. (88 * u) - (96 * d) = 12 -96d = 12 - 88u d = (12 - 88u) / (-96) d = (88u - 12) / 96 Step 2: Substitute the expression for d in the second equation. (27 * u) + (36 * ((88u - 12) / 96)) = 71 27u + (3312u - 432) / 96 = 71 27u + 34.5u - 4.5 = 71 61.5u = 75.5 u = 75.5 / 61.5 u = 1.227272727... Step 3: Substitute the value of u in the expression for d to find the value of d. d = (88u - 12) / 96 d = (88 * 1.227272727... - 12) / 96 d = (108 - 12) / 96 d = 96 / 96 d = 1 Therefore, the values of u and d that make the given equations true are: u = 1.227272727... (approximately 1.23) d = 1 ### Answer: u = 1.23, d = 1
Expected answer: {d: 1481/1440, u: 151/120}
Correct? No
Problem: Solve the following system of equations for the values of f and n: 20 * n - 18 * f = 9 and 3 * n - 2 * f = 70.
Claude-3 Sonnet:
To solve the system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 20n - 18f = 9 3n - 2f = 70 Step 1: Multiply the second equation by 10 to eliminate the variable n. 30n - 20f = 700 Step 2: Subtract the first equation from the modified second equation. 10n - 2f = 691 -18f = -9 f = 0.5 Step 3: Substitute the value of f into either of the original equations to find the value of n. Let's use the second equation: 3n - 2(0.5) = 70 3n - 1 = 70 3n = 71 n = 23.67 Therefore, the values of f and n are: f = 0.5 n = 23.67 ### Answer: f = 0.5, n = 23.67
Expected answer: {f: 1373/14, n: 621/7}
Correct? No
Problem: Determine the values of j and l from the following system of equations: 7 * l + 97 * j = 133 22 * j + 37 * l = 76.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 7l + 97j = 133 22j + 37l = 76 Step 1: Multiply the first equation by -22 and the second equation by 7 to eliminate the variable j. (-22)(7l + 97j = 133) -154l - 2134j = -2926 7(22j + 37l = 76) 154j + 259l = 532 Step 2: Add the two equations to eliminate j. (-154l - 2134j = -2926) (154j + 259l = 532) ----------------- 105l = -2394 Step 3: Solve for l. l = -2394/105 l = -22.8 Step 4: Substitute the value of l into either of the original equations to find j. Let's use the first equation: 7l + 97j = 133 7(-22.8) + 97j = 133 -159.6 + 97j = 133 97j = 292.6 j = 292.6/97 j = 3.02 Therefore, the values of j and l are: j = 3.02 l = -22.8 ### Answer: j = 3.02, l = -22.8
Expected answer: {j: 1463/1145, l: 1482/1145}
Correct? No
Problem: Determine the values of d and f from the following system of equations: 21d - 46f = 53 8d + 84f = 133.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 8 and the second equation by 21 to eliminate the variable d. 21d - 46f = 53 (Multiply by 8) 168d - 368f = 424 8d + 84f = 133 (Multiply by 21) 168d + 1764f = 2793 Step 2: Subtract the second equation from the first equation to eliminate d. 168d - 368f = 424 -(168d + 1764f = 2793) ----------------- -2132f = -2369 Step 3: Divide both sides by -2132 to find the value of f. f = -2369/-2132 f = 1.11 Step 4: Substitute the value of f into either of the original equations to find the value of d. Let's use the first equation: 21d - 46(1.11) = 53 21d - 51.06 = 53 21d = 104.06 d = 4.96 Therefore, the values of d and f are: d = 4.96 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.
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Step 1: Multiply the first equation by 66 and the second equation by 74 to eliminate the variable u. 74u × 66 + 11w × 66 = 116 × 66 4884u + 726w = 7656 66u × 74 - 68w × 74 = 21 × 74 4884u - 5032w = 1554 Step 2: Subtract the second equation from the first equation to eliminate u. (4884u + 726w = 7656) - (4884u - 5032w = 1554) 5758w = 6102 Step 3: Solve for w. w = 6102 / 5758 w = 1.06 (approximately) Step 4: Substitute the value of w into either of the original equations to find u. Let's use the first equation: 74u + 11(1.06) = 116 74u + 11.66 = 116 74u = 104.34 u = 1.41 (approximately) Therefore, the values of u and w that satisfy the given system of equations are: u ≈ 1.41 w ≈ 1.06 ### Answer: u ≈ 1.41, w ≈ 1.06
Expected answer: {u: 8119/5758, w: 3051/2879}
Correct? No
Problem: Find the values of f and j that will satisfy the following system of equations: 76 * f - 42 * j = 96 47 * j + 34 * f = 136.
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Step 1: Multiply the first equation by 47 and the second equation by 76 to eliminate the variable j. 76 * f - 42 * j = 96 (Multiply by 47) 3572 * f - 1974 * j = 4512 47 * j + 34 * f = 136 (Multiply by 76) 3572 * f + 2572 * j = 10336 Step 2: Subtract the second equation from the first equation to eliminate j. 3572 * f - 1974 * j = 4512 - (3572 * f + 2572 * j = 10336) ----------------------------------- -4546 * j = -5824 Step 3: Divide both sides by -4546 to find the value of j. j = 5824 / 4546 j = 1.28 Step 4: Substitute the value of j into either of the original equations to find the value of f. Let's use the first equation: 76 * f - 42 * (1.28) = 96 76 * f - 53.76 = 96 76 * f = 149.76 f = 149.76 / 76 f = 1.97 Therefore, the values of f and j that satisfy the given system of equations are: f = 1.97 j = 1.28 ### Answer: f = 1.97, j = 1.28
Expected answer: {f: 1278/625, j: 884/625}
Correct? No
Problem: Consider the following system of equations. Find the values of u and e: 68u + 32e = 174 63e - 69u = 1.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 63 and the second equation by 68 to eliminate the variable u. (68u + 32e) × 63 = 4284u + 2016e (63e - 69u) × 68 = 4284e - 4692u Step 2: Subtract the second equation from the first equation to eliminate u. 4284u + 2016e - (4284e - 4692u) -------------- 6708e = 10976 Step 3: Solve for e. e = 10976 / 6708 e = 1.636 (rounded to 3 decimal places) Step 4: Substitute the value of e into either of the original equations to find u. Let's use the second equation: 63e - 69u = 1 63(1.636) - 69u = 1 102.668 - 69u = 1 -69u = -101.668 u = 1.473 (rounded to 3 decimal places) Therefore, the values of u and e are: u = 1.473 e = 1.636 ### Answer: u = 1.473, e = 1.636
Expected answer: {e: 6037/3246, u: 5465/3246}
Correct? No
Problem: Find the values of p and n that satisfy the following system of linear equations: 57p + 4n = 129 and 24p + 39n = 154.
Claude-3 Sonnet:
To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 24 and the second equation by 57 to eliminate the variable p. 24(57p + 4n = 129) = 1368p + 96n = 3096 57(24p + 39n = 154) = 1368p + 2223n = 8778 Step 2: Subtract the second equation from the first equation to eliminate p. (1368p + 96n = 3096) - (1368p + 2223n = 8778) -2127n = -5682 n = 5682/2127 n = 2.67 (rounded to two decimal places) Step 3: Substitute the value of n into either of the original equations to find the value of p. Let's use the first equation: 57p + 4(2.67) = 129 57p + 10.68 = 129 57p = 118.32 p = 118.32/57 p = 2.08 (rounded to two decimal places) Therefore, the values of p and n that satisfy the given system of linear equations are: p = 2.08 n = 2.67 ### Answer: p = 2.08, n = 2.67
Expected answer: {n: 1894/709, p: 4415/2127}
Correct? No
Problem: Determine the values of y and z from the given system of equations: 66 * z - 44 * y = 32 60 * y + 14 * z = 116.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 14 and the second equation by 44 to eliminate the variable z. 66 * z - 44 * y = 32 (Multiply by 14) 924 * z - 616 * y = 448 60 * y + 14 * z = 116 (Multiply by 44) 2640 * y + 616 * z = 5104 Step 2: Subtract the second equation from the first equation to eliminate z. (924 * z - 616 * y = 448) - (2640 * y + 616 * z = 5104) -1716 * y = -4656 Step 3: Solve for y. -1716 * y = -4656 y = 4656 / 1716 y = 2.71 Step 4: Substitute the value of y into either of the original equations to find z. Let's use the first equation: 66 * z - 44 * y = 32 66 * z - 44 * (2.71) = 32 66 * z - 119.24 = 32 66 * z = 151.24 z = 151.24 / 66 z = 2.29 Therefore, the values of y and z are: y = 2.71 z = 2.29 ### Answer: y = 2.71, z = 2.29
Expected answer: {y: 901/572, z: 439/286}
Correct? No
Problem: Solve the following system of equations for the variables t and h: The first equation is 83 * t - 58 * h = 45. The second equation is 69 * h + 11 * t = 124.
Claude-3 Sonnet:
To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 83t - 58h = 45 (1) 69h + 11t = 124 (2) Step 1: Multiply the first equation by 69 and the second equation by -83 to eliminate the variable h. (83t - 58h) × 69 = 5727t - 4002h = 3105 (69h + 11t) × (-83) = -5727h - 913t = -10292 Step 2: Add the resulting equations to eliminate h. 5727t - 4002h = 3105 -5727h - 913t = -10292 -------------- 4814t = -7187 Step 3: Solve for t. 4814t = -7187 t = -7187 / 4814 t = -1.49 (approximately) Step 4: Substitute the value of t into either of the original equations to find h. Let's use the first equation: 83t - 58h = 45 83(-1.49) - 58h = 45 -123.67 - 58h = 45 -58h = 168.67 h = -2.91 (approximately) Therefore, the solution to the system of equations is: t = -1.49 h = -2.91 ### Answer: t = -1.49, h = -2.91
Expected answer: {h: 9797/6365, t: 10297/6365}
Correct? No
Problem: Given the system of equations ((79 * h) - (61 * o)) = 9 and ((16 * h) - (12 * o)) = 9, what are the values of o and h?
Claude-3 Sonnet:
To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method: Step 1: Multiply the first equation by 2 to eliminate the variable h. 2 × ((79 * h) - (61 * o)) = 9 × 2 158h - 122o = 18 Step 2: Multiply the second equation by -5 to eliminate the variable h. -5 × ((16 * h) - (12 * o)) = -5 × 9 -80h + 60o = -45 Step 3: Add the two equations to eliminate h. 158h - 122o = 18 -80h + 60o = -45 -------------- 78o = -27 Step 4: Solve for o. o = -27/78 o = -0.346... ≈ -0.35 Step 5: Substitute the value of o into either of the original equations to find h. Let's use the second equation: (16 * h) - (12 * (-0.35)) = 9 16h + 4.2 = 9 16h = 4.8 h = 0.3 Therefore, the values of o and h are: o = -0.35 h = 0.3 ### Answer: o = -0.35, h = 0.3
Expected answer: {h: 63/4, o: 81/4}
Correct? No
Problem: Solve for variables s and y when the following system of equations is satisfied: 4 * y + 56 * s = 139 65 * y - 90 * s = 16
Claude-3 Sonnet:
To solve for the variables s and y in the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 4y + 56s = 139 65y - 90s = 16 Step 1: Multiply the first equation by 65 and the second equation by 4 to eliminate the variable y. 260y + 3640s = 9035 260y - 360s = 64 Step 2: Subtract the second equation from the first equation to eliminate y. 4000s = 8971 Step 3: Solve for s. s = 8971 / 4000 s = 2.24275 Step 4: Substitute the value of s into either of the original equations to find y. Let's use the first equation: 4y + 56(2.24275) = 139 4y + 125.594 = 139 4y = 13.406 y = 3.3515 Therefore, the solution to the system of equations is: s = 2.24275 y = 3.3515 ### Answer: s = 2.24275, y = 3.3515
Expected answer: {s: 8971/4000, y: 6703/2000}
Correct? No
Problem: Solve the following pair of simultaneous equations: 42 * d - 4 * z = 75 40 * d + 45 * z = 162 Find the values of d and z.
Claude-3 Sonnet:
To solve the pair of simultaneous equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 42d - 4z = 75 40d + 45z = 162 Step 1: Multiply the first equation by 5 and the second equation by 6 to eliminate the variable z. 210d - 20z = 375 240d + 270z = 972 Step 2: Subtract the second equation from the first equation to eliminate z. (210d - 20z) - (240d + 270z) = 375 - 972 -30d - 290z = -597 Step 3: Divide both sides of the equation by -30 to find the value of d. d = 19.9 Step 4: Substitute the value of d into either of the original equations to find the value of z. Let's use the first equation: 42d - 4z = 75 42(19.9) - 4z = 75 835.8 - 4z = 75 -4z = -760.8 z = 190.2 Therefore, the values of d and z are: d = 19.9 z = 190.2 ### Answer: d = 19.9, z = 190.2
Expected answer: {d: 4023/2050, z: 1902/1025}
Correct? No
Problem: Determine the values of 'a' and 'o' from the following system of equations: (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75.
Claude-3 Sonnet:
To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method. Given equations: (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75 Step 1: Multiply the first equation by 6 and the second equation by 24 to eliminate the variable 'o'. 6 × [(42 * o) + (24 * a) = 142] 252o + 144a = 852 24 × [(42 * a) + (6 * o) = 75] 1008a + 144o = 1800 Step 2: Subtract the second equation from the first equation to eliminate 'o'. (252o + 144a = 852) - (1008a + 144o = 1800) -864a = -948 a = 1.1 Step 3: Substitute the value of 'a' into either of the original equations to find the value of 'o'. Let's use the second equation: (42 * a) + (6 * o) = 75 (42 * 1.1) + (6 * o) = 75 46.2 + 6o = 75 6o = 28.8 o = 4.8 Therefore, the values of 'a' and 'o' are: a = 1.1 o = 4.8 ### Answer: a = 1.1, o = 4.8
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.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 16 and the second equation by 2 to eliminate the variable p. 39*r - 2*p = 45 (Multiply by 16) 624*r - 32*p = 720 16*p - 84*r = 36 (Multiply by 2) 32*p - 168*r = 72 Step 2: Add the two equations to eliminate p. 624*r - 32*p = 720 32*p - 168*r = 72 ----------------- 456*r = 792 Step 3: Solve for r. 456*r = 792 r = 792/456 r = 24/16 r = 1.5 Step 4: Substitute the value of r into either of the original equations to find p. Let's use the first equation: 39*r - 2*p = 45 39(1.5) - 2*p = 45 58.5 - 2*p = 45 -2*p = -13.5 p = 6.75 Therefore, the values of p and r are: p = 6.75 r = 1.5 ### Answer: p = 6.75, r = 1.5
Expected answer: {p: 216/19, r: 33/19}
Correct? No
Problem: Given the following system of equations, solve for the values of 'd' and 'w': 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58
Claude-3 Sonnet:
To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method. Step 1: Multiply the first equation by 8 and the second equation by 23 to eliminate the variable 'd'. 8 × (23d - 52w = 7) 184d - 416w = 56 23 × (64w - 8d = 58) 1472w - 184d = 1334 Step 2: Add the two equations to eliminate 'd'. 184d - 416w = 56 1472w - 184d = 1334 ---------- 1056w = 1390 Step 3: Solve for 'w'. w = 1390 / 1056 w = 1.316037736 (rounded to 9 decimal places) Step 4: Substitute the value of 'w' into either of the original equations to find 'd'. Let's use the first equation: (23d) - (52 × 1.316037736) = 7 23d - 68.632 = 7 23d = 75.632 d = 3.288347826 (rounded to 9 decimal places) Therefore, the values of 'd' and 'w' are: d = 3.288347826 w = 1.316037736 ### Answer: d = 3.288347826, w = 1.316037736
Expected answer: {d: 433/132, w: 695/528}
Correct? No
Problem: Solve the following system of equations for the values of n and z: (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19.
Claude-3 Sonnet:
To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: (71 * z) + (60 * n) = 184 ... (1) (52 * z) - (47 * n) = 19 ... (2) Step 1: Multiply equation (1) by 52 and equation (2) by 71 to eliminate the variable z. 52 × (71 * z + 60 * n = 184) 3692z + 3120n = 9568 ... (3) 71 × (52 * z - 47 * n = 19) 3692z - 3337n = 1349 ... (4) Step 2: Subtract equation (4) from equation (3) to eliminate z. (3120n + 3692z) - (-3337n + 3692z) = (9568 - 1349) 6457n = 8219 n = 8219 / 6457 n = 1.273 (approximately) Step 3: Substitute the value of n into either equation (1) or (2) to find the value of z. Substituting n = 1.273 in equation (1): (71 * z) + (60 * 1.273) = 184 71z + 76.38 = 184 71z = 107.62 z = 107.62 / 71 z = 1.515 (approximately) Therefore, the solution to the system of equations is: n ≈ 1.273 z ≈ 1.515 ### 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
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by 84 and the second equation by 70 to eliminate the variable e. 70 * e + 90 * z = 189 (Multiply by 84) 5880 * e + 7560 * z = 15876 99 * z - 84 * e = 20 (Multiply by 70) 6930 * z - 5880 * e = 1400 Step 2: Subtract the second equation from the first equation to eliminate e. 5880 * e + 7560 * z = 15876 - (6930 * z - 5880 * e = 1400) ---------------------------------- 630 * z = 14476 Step 3: Solve for z. z = 14476 / 630 z = 23 Step 4: Substitute the value of z into either of the original equations to find e. Let's use the first equation: 70 * e + 90 * 23 = 189 70 * e + 2070 = 189 70 * e = 189 - 2070 70 * e = -1881 e = -1881 / 70 e = -26.87 Therefore, the values of e and z are: e = -26.87 z = 23 ### Answer: e = -26.87, z = 23
Expected answer: {e: 1879/1610, z: 1234/1035}
Correct? No
Problem: Find the values of o and f that satisfy the following system of equations: 1) 26o - 44f = 95 2) 87f + 11o = 169.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Step 1: Multiply the first equation by a suitable number to eliminate one of the variables. Multiplying the first equation by 11, we get: 286o - 484f = 1045 Step 2: Multiply the second equation by a suitable number to eliminate the other variable. Multiplying the second equation by 26, we get: 2262f + 286o = 4394 Step 3: Subtract the second equation from the first equation to eliminate the variable 'o'. 286o - 484f = 1045 - (2262f + 286o = 4394) -------------------------- -2746f = -3349 Step 4: Solve for 'f'. -2746f = -3349 f = 3349/2746 f = 1.22 (approximately) Step 5: Substitute the value of 'f' into either of the original equations to find the value of 'o'. Let's use the first equation: 26o - 44(1.22) = 95 26o - 53.68 = 95 26o = 148.68 o = 148.68/26 o = 5.72 (approximately) Therefore, the values of 'o' and 'f' that satisfy the given system of equations are: o ≈ 5.72 f ≈ 1.22
Expected answer: {f: 3349/2746, o: 15701/2746}
Correct? No
Problem: Calculate the values of z and p using the following system of equations: 40*z - 20*p = 51 6*z + 58*p = 165.
Claude-3 Sonnet:
To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method. Given equations: 40z - 20p = 51 6z + 58p = 165 Step 1: Multiply the first equation by 6 and the second equation by -40 to eliminate the variable z. 240z - 120p = 306 -240z - 2320p = -6600 Step 2: Add the two equations to eliminate z. -2440p = -6294 Step 3: Divide both sides by -2440 to find the value of p. p = 6294/2440 p = 2.58 Step 4: Substitute the value of p into either of the original equations to find the value of z. Let's use the first equation: 40z - 20(2.58) = 51 40z - 51.6 = 51 40z = 102.6 z = 102.6/40 z = 2.565 Therefore, the values of z and p are: z = 2.565 p = 2.58 ### Answer: z = 2.565, p = 2.58
Expected answer: {p: 3147/1220, z: 3129/1220}
Correct? No
Problem: Find the values of s and f from the given system of equations: (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93.
Claude-3 Sonnet:
To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method. Step 1: Multiply the first equation by 3 and the second equation by 71 to eliminate the variable s. 3(71s - 89f = 5) 213s - 267f = 15 71(3f + 10s = 93) 213f + 710s = 6603 Step 2: Subtract the second equation from the first equation to eliminate s. 213s - 267f = 15 -(213f + 710s = 6603) -------------- -480f = -6588 Step 3: Divide both sides by -480 to find the value of f. f = 6588/480 f = 13.725 Step 4: Substitute the value of f into either of the original equations to find the value of s. Let's use the first equation: 71s - 89(13.725) = 5 71s - 1222.325 = 5 71s = 1227.325 s = 1227.325/71 s = 17.285 Therefore, the values of s and f are: s = 17.285 f = 13.725
Expected answer: {f: 6553/1103, s: 8292/1103}
Correct? No