Claude-3 Opus 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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the second equation, and then substitute it into the first equation. From the second equation, let's solve for s in terms of q: (30 * s) + (12 * q) = 121 30s = 121 - 12q s = (121 - 12q) / 30 s = 121/30 - 12q/30 s = 121/30 - 2q/5 <<121/30=121/30=121/30=4.033333333333333>><<12/30=12/30=2/5=0.4>> s = 4.033333333333333 - 0.4q Now, substitute this expression for s into the first equation: (40 * s) - (91 * q) = 37 40 * (4.033333333333333 - 0.4q) - 91q = 37 161.33333333333333 - 16q - 91q = 37 161.33333333333333 - 107q = 37 -107q = -124.33333333333333 <<-124.33333333333333/-107=-124.33333333333333/-107=1.162790697674419>> q = 1.162790697674419 Now that we have the value of q, we can substitute it back into the equation for s: s = 4.033333333333333 - 0.4q s = 4.033333333333333 - 0.4 * 1.162790697674419 <<4.033333333333333-0.4*1.162790697674419=4.033333333333333-0.465116279069767=3.568217054263566>> s = 3.568217054263566 Therefore, the solution to the system of equations is: q ≈ 1.163 (rounded to 3 decimal places) s ≈ 3.568 (rounded to 3 decimal places) ### Answer: q ≈ 1.163, s ≈ 3.568
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 1. 4b + 44m = 180 2. 79m - 6b = 48 Step 1: Solve for b in terms of m using equation 1. 4b + 44m = 180 4b = 180 - 44m b = (180 - 44m) ÷ 4 b = 45 - 11m Step 2: Substitute the expression for b into equation 2. 79m - 6(45 - 11m) = 48 79m - 270 + 66m = 48 145m - 270 = 48 145m = 318 m = 318 ÷ 145 <<145m = 318 m = 318 ÷ 145 m = 2.19310344827586>>m ≈ 2.19 Step 3: Substitute the value of m into the expression for b from step 1. b = 45 - 11m b = 45 - 11(2.19) b = 45 - 24.09 <<b = 45 - 11(2.19) b = 45 - 24.09 b = 20.91>>b ≈ 20.91 Therefore, the solution to the system of equations is: m ≈ 2.19 b ≈ 20.91 ### Answer: m ≈ 2.19, b ≈ 20.91
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, let's solve for o: 42 * o - 15 * g = 28 42 * o = 28 + 15 * g o = (28 + 15 * g) / 42 o = (2 * g) / 3 + 2 / 3 Now, substitute the expression for o into the second equation: 47 * ((2 * g) / 3 + 2 / 3) + 29 * g = 141 (94 * g) / 3 + 94 / 3 + 29 * g = 141 (94 * g) / 3 + 29 * g = 141 - 94 / 3 (94 * g) / 3 + 29 * g = 423 / 3 - 94 / 3 (94 * g) / 3 + 29 * g = 329 / 3 (94 * g + 87 * g) / 3 = 329 / 3 181 * g / 3 = 329 / 3 181 * g = 329 g = 329 / 181 g = 1.81767955801105 Now that we have the value of g, we can substitute it back into the expression for o: o = (2 * 1.81767955801105) / 3 + 2 / 3 o = 3.63535911602209 / 3 + 2 / 3 o = 1.21178637200736 + 0.66666666666667 o = 1.87845303867403 Therefore, the solution to the system of equations is: g ≈ 1.81768 o ≈ 1.87845 ### Answer: g ≈ 1.81768, o ≈ 1.87845
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for v in the first equation and then substitute it into the second equation to solve for o. First equation: ((12 * v) - (6 * o)) = 21 12v - 6o = 21 12v = 21 + 6o v = (21 + 6o) / 12 v = (7/4) + (o/2) ... (1) Now, substitute (1) into the second equation: ((76 * v) - (45 * o)) = 50 76 * ((7/4) + (o/2)) - 45o = 50 133 + 38o - 45o = 50 133 - 7o = 50 -7o = -83 o = 83/7 o = 11.857... ≈ 11.86 Now that we have the value of o, we can substitute it back into equation (1) to find v: v = (7/4) + (o/2) v = 1.75 + (11.86/2) v = 1.75 + 5.93 v = 7.68 ### Answer: v ≈ 7.68, o ≈ 11.86
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 Opus:
To solve this system of equations, we can use the substitution method. 1. Solve the first equation for f in terms of g: 75 * f + 25 * g = 138 75 * f = 138 - 25 * g f = (138 - 25 * g) / 75 f = (138 - 25 * g) / 75 = <<(138 - 25 * g) / 75 = (138/75) - (25/75)*g>> (138/75) - (25/75)*g 2. Substitute the expression for f into the second equation: 2 * f + 90 * g = 143 2 * ((138/75) - (25/75)*g) + 90 * g = 143 (276/75) - (50/75)*g + 90 * g = 143 (276/75) + (6750/75)*g - (50/75)*g = 143 (276/75) + (6700/75)*g = 143 3. Solve the equation for g: (276/75) + (6700/75)*g = 143 (6700/75)*g = 143 - (276/75) (6700/75)*g = (10725/75) - (276/75) = <<(10725/75) - (276/75) = 10449/75>>10449/75 g = (10449/75) / (6700/75) g = <<(10449/75) / (6700/75) = 10449/6700>>10449/6700 4. Substitute the value of g into the expression for f: f = (138/75) - (25/75)*g f = (138/75) - (25/75)*(10449/6700) f = (138/75) - (261225/502500) = <<(138/75) - (261225/502500) = (69000/502500) - (261225/502500) = -192225/502500>>-192225/502500 Therefore, the values of f and g are: g = 10449/6700 f = -192225/502500 ### Answer: f = -192225/502500, g = 10449/6700
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. First equation: 194 = 4y + 14n 194 - 14n = 4y (194 - 14n) / 4 = y y = (194 - 14n) / 4 ... (1) Second equation: 7 = 65n - 79y Substitute (1) into the second equation: 7 = 65n - 79((194 - 14n) / 4) 7 = 65n - 79(194 - 14n) / 4 7 = 65n - (15286 - 1106n) / 4 28 = 260n - 15286 + 1106n 28 = 1366n - 15286 15314 = 1366n n = 15314 / 1366 n = 11.2116251831 ... (2) Substitute (2) into (1) to find y: y = (194 - 14 × 11.2116251831) / 4 y = (194 - 156.962752564) / 4 y = 37.037247436 / 4 y = 9.25931185898 ### Answer: y ≈ 9.25931185898, n ≈ 11.2116251831
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 Opus:
To solve for l and g, we can use substitution or elimination methods. Let's use the substitution method here. From the second equation, we can express l in terms of g: 6 = 41 * l - 53 * g 41 * l = 53 * g + 6 l = (53 * g + 6) / 41 Now, substitute this expression for l into the first equation: 146 = 28 * g + 26 * ((53 * g + 6) / 41) Simplify: 146 = 28 * g + (1378 * g + 156) / 41 146 = 28 * g + (1378 * g) / 41 + 156 / 41 146 = 28 * g + 33.6097560976 * g + 3.8048780488 146 = 61.6097560976 * g + 3.8048780488 Subtract 3.8048780488 from both sides: 142.1951219512 = 61.6097560976 * g Divide both sides by 61.6097560976: g = 142.1951219512 / 61.6097560976 g ≈ 2.3081395348 Now that we have the value of g, we can substitute it back into the expression for l: l = (53 * 2.3081395348 + 6) / 41 l ≈ 3.0000000001 Therefore, the values of l and g that satisfy the given equations are approximately: g ≈ 2.3081395348 l ≈ 3.0000000001 ### Answer: g ≈ 2.3081395348, l ≈ 3.0000000001
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one of the variables in the first equation and substitute it into the second equation. From equation 1, solve for y: 20 * p + 56 * y = 90 56 * y = 90 - 20 * p y = (90 - 20 * p) / 56 y = 45/28 - (5/14) * p Substitute the expression for y into equation 2: 89 * p + 7 * (45/28 - (5/14) * p) = 98 89 * p + 315/28 - 35/14 * p = 98 89 * p - 35/14 * p = 98 - 315/28 (89 - 35/14) * p = 98 - 315/28 (1246/14 - 35/14) * p = 98 - 315/28 (1211/14) * p = 98 - 315/28 1211 * p = 1372 - 315 1211 * p = 1057 p = 1057/1211 Substitute p = 1057/1211 into the expression for y: y = 45/28 - (5/14) * (1057/1211) y = 45/28 - 5285/16954 y = 12135/16954 - 5285/16954 y = 6850/16954 Therefore, the solution to the system of equations is: p = 1057/1211 y = 6850/16954 ### Answer: p = 1057/1211, y = 6850/16954
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 Opus:
To solve the given system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 1. 72*k - 22*d = 17 2. 6*d + 69*k = 96 Step 1: Solve for d in terms of k using equation 1. 72*k - 22*d = 17 -22*d = -72*k + 17 d = (72*k - 17) / 22 d = (72*k - 17) / 22 Step 2: Substitute the expression for d into equation 2. 6*d + 69*k = 96 6*((72*k - 17) / 22) + 69*k = 96 (432*k - 102) / 22 + 69*k = 96 432*k - 102 + 1518*k = 2112 1950*k - 102 = 2112 1950*k = 2214 k = 2214 / 1950 k = <<2214/1950=1.135384615384615>>1.135384615384615 Step 3: Substitute the value of k into the expression for d. d = (72*k - 17) / 22 d = (72 * 1.135384615384615 - 17) / 22 d = (81.74769230769231 - 17) / 22 d = 64.74769230769231 / 22 d = <<64.74769230769231/22=2.943076923076923>>2.943076923076923 Therefore, the values of the variables are: k ≈ 1.135384615384615 d ≈ 2.943076923076923 ### Answer: k ≈ 1.135384615384615, d ≈ 2.943076923076923
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 73 * h - 33 * d = 48 10 * d + 14 * h = 91 Step 1: Solve the first equation for h in terms of d. 73 * h - 33 * d = 48 73 * h = 33 * d + 48 h = (33 * d + 48) / 73 Step 2: Substitute the expression for h into the second equation. 10 * d + 14 * ((33 * d + 48) / 73) = 91 Step 3: Simplify the equation and solve for d. 10 * d + (462 * d + 672) / 73 = 91 730 * d + 462 * d + 672 = 6643 1192 * d + 672 = 6643 1192 * d = 5971 d = 5971 / 1192 d = 5 Step 4: Substitute the value of d into the expression for h from Step 1. h = (33 * 5 + 48) / 73 h = 213 / 73 h = 2.9178082191780823 (rounded to 16 decimal places) Therefore, the solution to the system of equations is: d = 5 h ≈ 2.9178082191780823 ### Answer: d = 5, h ≈ 2.9178082191780823
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 Opus:
To solve the system of equations, we can use the substitution method. From equation 1, we can express q in terms of e: 43 * q - 44 * e = 29 43 * q = 29 + 44 * e q = (29 + 44 * e) / 43 Now, substitute this expression for q into equation 2: 30 * ((29 + 44 * e) / 43) + 43 * e = 130 Simplify: (870 + 1320 * e) / 43 + 43 * e = 130 870 / 43 + 1320 * e / 43 + 43 * e = 130 20.23 + 30.70 * e + 43 * e = 130 73.70 * e = 109.77 e = 109.77 / 73.70 e ≈ 1.49 Now that we have the value of e, we can substitute it back into the expression for q: q = (29 + 44 * 1.49) / 43 q = (29 + 65.56) / 43 q = 94.56 / 43 q ≈ 2.20 ### Answer: q ≈ 2.20, e ≈ 1.49
Expected answer: {e: 4720/3169, q: 6967/3169}
Correct? No
Problem: Find the values of r and a in the following system of equations: 53 = ((89 * r) - (74 * a)) 109 = ((37 * a) + (6 * r))
Claude-3 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the second equation, then substitute it into the first equation to solve for the remaining variable. From the second equation, let's solve for r in terms of a: 109 = 37a + 6r 6r = 109 - 37a r = (109 - 37a) / 6 r = (109/6) - (37/6)a r = 18.1666... - 6.1666...a Now, substitute this expression for r into the first equation: 53 = 89r - 74a 53 = 89(18.1666... - 6.1666...a) - 74a 53 = 1616.8333... - 549.8333...a - 74a 53 = 1616.8333... - 623.8333...a 623.8333...a = 1616.8333... - 53 623.8333...a = 1563.8333... a = 1563.8333... / 623.8333... a = 2.5 Now that we have the value of a, we can substitute it back into the expression for r: r = 18.1666... - 6.1666...(2.5) r = 18.1666... - 15.4166... r = 2.75 Therefore, the solution to the system of equations is: a = 2.5 r = 2.75 ### Answer: a = 2.5, r = 2.75
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 Opus:
Let's solve this step by step. We need to find the weight of one unit of 'g' and one unit of 'q'. From equation 1: 42g + 12q = 169 ... (1) From equation 2: 12q - 50g = 19 ... (2) Step 1: Multiply equation (2) by -1 to eliminate the negative sign. 50g - 12q = -19 ... (3) Step 2: Add equations (1) and (3) to eliminate 'q'. 42g + 12q = 169 50g - 12q = -19 _______________ 92g = 150 Step 3: Solve for 'g'. g = 150 ÷ 92 g = 1.63043478 ... (4) Step 4: Substitute the value of 'g' in equation (1) to solve for 'q'. 42(1.63043478) + 12q = 169 68.47826076 + 12q = 169 12q = 169 - 68.47826076 12q = 100.5217392 q = 100.5217392 ÷ 12 q = 8.37681160 ... (5) Therefore, the weight of one unit of grain 'g' is approximately 1.63 kg, and the weight of one unit of grain 'q' is approximately 8.38 kg. ### Answer: g ≈ 1.63 kg, q ≈ 8.38 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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for s in the first equation and substitute it into the second equation to solve for v. 1. Solve for s in the first equation: 37 * s - 11 * v = 54 37 * s = 54 + 11 * v s = (54 + 11 * v) / 37 2. Substitute the expression for s into the second equation: 82 * s - 62 * v = 41 82 * ((54 + 11 * v) / 37) - 62 * v = 41 3. Simplify the equation: (82 * (54 + 11 * v)) / 37 - 62 * v = 41 (4428 + 902 * v) / 37 - 62 * v = 41 4. Multiply both sides by 37 to eliminate fractions: 4428 + 902 * v - 2294 * v = 1517 4428 - 1392 * v = 1517 5. Subtract 4428 from both sides: -1392 * v = -2911 6. Divide both sides by -1392 to solve for v: v = 2911 / 1392 v ≈ 2.09 7. Substitute v ≈ 2.09 into the equation from step 1 to solve for s: s = (54 + 11 * 2.09) / 37 s = (54 + 22.99) / 37 s = 76.99 / 37 s ≈ 2.08 Therefore, the approximate values for the system of equations are: s ≈ 2.08 v ≈ 2.09 ### Answer: s ≈ 2.08, v ≈ 2.09
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 Opus:
To solve for c and z, we can use the substitution method. Let's solve for c in the first equation and substitute it into the second equation to solve for z. From the first equation, 41 * c + 65 * z = 151, we can express c in terms of z: 41 * c = 151 - 65 * z c = (151 - 65 * z) / 41 Now, let's substitute this expression for c into the second equation, 78 * z - 48 * c = 48: 78 * z - 48 * ((151 - 65 * z) / 41) = 48 78 * z - (48 * 151 - 48 * 65 * z) / 41 = 48 78 * z - (7248 - 3120 * z) / 41 = 48 3198 * z - 7248 / 41 + 3120 * z / 41 = 48 (3198 * 41 * z - 7248 + 3120 * z) / 41 = 48 (131118 * z - 7248 + 3120 * z) / 41 = 48 (134238 * z - 7248) / 41 = 48 134238 * z - 7248 = 1968 134238 * z = 9216 z = 9216 / 134238 z = <<9216/134238=0.0686275>>0.0686275 Now that we have the value of z, we can substitute it back into the equation for c: c = (151 - 65 * 0.0686275) / 41 c = (151 - 4.460788) / 41 c = 146.539212 / 41 c = <<146.539212/41=3.574371>>3.574371 Therefore, the values for c and z are: z ≈ 0.0686275 c ≈ 3.574371 ### Answer: c ≈ 3.574371, z ≈ 0.0686275
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 60c - 91i = 7 ... (1) 7i + 76c = 148 ... (2) Step 1: Solve for c in terms of i using equation (1). 60c - 91i = 7 60c = 91i + 7 c = (91i + 7) / 60 ... (3) Step 2: Substitute the expression for c from equation (3) into equation (2). 7i + 76c = 148 7i + 76 * ((91i + 7) / 60) = 148 7i + (76 * (91i + 7)) / 60 = 148 7i + (6916i + 532) / 60 = 148 7i + 115.2667i + 8.8667 = 148 122.2667i + 8.8667 = 148 122.2667i = 139.1333 i = 139.1333 / 122.2667 i ≈ 1.1385 Step 3: Substitute the value of i into equation (3) to solve for c. c = (91i + 7) / 60 c = (91 * 1.1385 + 7) / 60 c = (103.6035 + 7) / 60 c = 110.6035 / 60 c ≈ 1.8434 Therefore, the values of c and i are: i ≈ 1.1385 c ≈ 1.8434 ### Answer: i ≈ 1.1385, c ≈ 1.8434
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 Opus:
To solve the given simultaneous linear equations, we can use the substitution method. From equation 1, we can express v in terms of e: 71*v - 37*e = 69 71*v = 37*e + 69 v = (37*e + 69) / 71 Now, substitute this expression for v into equation 2: 41*e + 76*v = 184 41*e + 76*((37*e + 69) / 71) = 184 Simplify the equation: 41*e + (76*37*e + 76*69) / 71 = 184 41*e + (2812*e + 5244) / 71 = 184 Multiply both sides by 71: 2911*e + 2812*e + 5244 = 13064 5723*e + 5244 = 13064 5723*e = 7820 e = 7820 / 5723 Solve for e: e = <<7820/5723=1.3664>>1.3664 (rounded to 4 decimal places) Substitute e = 1.3664 into the expression for v: v = (37*1.3664 + 69) / 71 v = 119.5568 / 71 Solve for v: v = <<119.5568/71=1.6839>>1.6839 (rounded to 4 decimal places) ### Answer: e = 1.3664, v = 1.6839
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 Opus:
Let's solve this system of equations step by step using the substitution method. Given equations: 1. 25v + 69d = 113 2. 42v + 26d = 85 Step 1: Solve equation 1 for v in terms of d. 25v + 69d = 113 25v = 113 - 69d v = (113 - 69d) / 25 Step 2: Substitute the expression for v into equation 2. 42((113 - 69d) / 25) + 26d = 85 Step 3: Simplify the equation and solve for d. (4746 - 2898d) / 25 + 26d = 85 4746 - 2898d + 650d = 2125 4746 - 2248d = 2125 -2248d = -2621 d = -2621 / -2248 d = <<-2621/-2248=1.165925266903915>>1.165925266903915 Step 4: Substitute the value of d into the expression for v from step 1. v = (113 - 69(1.165925266903915)) / 25 v = (113 - 80.44834334623217) / 25 v = 32.55165665376783 / 25 v = <<32.55165665376783/25=1.302066266150713>>1.302066266150713 Therefore, the values of d and v are: d ≈ 1.165925266903915 v ≈ 1.302066266150713 ### Answer: d ≈ 1.165925266903915, v ≈ 1.302066266150713
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 Opus:
To solve the system of equations for g and z, we can use the substitution method or the elimination method. Let's use the substitution method here. From the first equation, we can express z in terms of g: 43 * z - 91 * g = 33 43 * z = 91 * g + 33 z = (91 * g + 33) / 43 Now, substitute this expression for z into the second equation: 72 * g - 3 * ((91 * g + 33) / 43) = 68 Simplify: 72 * g - (273 * g + 99) / 43 = 68 3096 * g - 273 * g - 99 = 2924 2823 * g - 99 = 2924 2823 * g = 3023 g = 3023 / 2823 Now that we have the value of g, we can substitute it back into the expression for z: z = (91 * (3023 / 2823) + 33) / 43 Simplify: z = (273993 / 2823 + 33) / 43 z = (273993 + 93159) / 121969 z = 367152 / 121969 Therefore, the solutions for g and z are: g = 3023 / 2823 z = 367152 / 121969 ### Answer: g = 3023 / 2823, z = 367152 / 121969
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for c in the second equation and substitute it into the first equation to solve for f. From the second equation: 36c - 21f = 34 36c = 34 + 21f c = (34 + 21f) / 36 c = (17 + 21f) / 18 (simplifying the fraction) Substituting this expression for c into the first equation: 50c + 12f = 92 50((17 + 21f) / 18) + 12f = 92 (850 + 1050f) / 18 + 12f = 92 850 + 1050f + 216f = 1656 1266f = 806 f = 806 / 1266 f = 62 / 79 Now that we have the value of f, we can substitute it back into the equation for c: c = (17 + 21f) / 18 c = (17 + 21(62/79)) / 18 c = (1343 + 1302) / 1422 c = 2645 / 1422 c = 1 76/79 Therefore, the quantity of cherries (c) is 1 76/79, and the quantity of figs (f) is 62/79. ### Answer: c = 1 76/79, f = 62/79
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 Opus:
Great! Let's solve this system of equations step by step to find the values of u and d. Given equations: 1) 87u - 75d = 7 2) 85d - 82u = 54 Step 1: Multiply equation (1) by 82 and equation (2) by 87 to eliminate the variable u. Equation (1): 82(87u - 75d = 7) → 7134u - 6150d = 574 Equation (2): 87(85d - 82u = 54) → 7395d - 7134u = 4698 Step 2: Add the resulting equations from step 1 to eliminate the variable u. 7134u - 6150d = 574 7395d - 7134u = 4698 1245d = 5272 Step 3: Solve for d by dividing both sides by 1245. d = 5272 ÷ 1245 d = 4.23 Step 4: Substitute the value of d into equation (1) to solve for u. 87u - 75(4.23) = 7 87u - 317.25 = 7 87u = 324.25 u = 324.25 ÷ 87 u = 3.73 Therefore, the values of u and d are: u = 3.73 d = 4.23 ### Answer: u = 3.73, d = 4.23
Expected answer: {d: 5272/1245, u: 929/249}
Correct? No
Problem: Bill is trying to decipher a secret code. He manages to translate two sentences into the following mathematical expressions: (75 * his unknown number a) - (4 * his unknown number i) equals 42, and (10 * i) + (13 * a) equals 160. Using these equations, can you figure out the values of i and a for him?
Claude-3 Opus:
Let's solve this problem step by step using a system of equations. We have two equations with two unknown variables, a and i. Equation 1: (75 * a) - (4 * i) = 42 Equation 2: (10 * i) + (13 * a) = 160 Step 1: Simplify Equation 1 by dividing both sides by 3. (25 * a) - (4/3 * i) = 14 Step 2: Substitute the value of a from Equation 2 into the simplified Equation 1. (25 * ((160 - 10 * i) / 13)) - (4/3 * i) = 14 Step 3: Solve for i. (25 * (160 - 10 * i)) / 13 - (4/3 * i) = 14 (4000 - 250 * i) / 13 - (4/3 * i) = 14 (4000 - 250 * i) / 13 - (52/3 * i) = 14 (12000 - 750 * i) / 39 - (52/3 * i) = 14 (12000 - 750 * i) / 39 - (156/9 * i) = 14 (108000 - 6750 * i) / 351 - (156/9 * i) = 14 (108000 - 6750 * i) / 351 - (6084/351 * i) = 14 (108000 - 6750 * i - 6084 * i) / 351 = 14 (108000 - 12834 * i) / 351 = 14 108000 - 12834 * i = 4914 -12834 * i = -103086 i = 8 Step 4: Substitute the value of i into Equation 2 to solve for a. (10 * 8) + (13 * a) = 160 80 + 13 * a = 160 13 * a = 80 a = 80 / 13 Therefore, the values are: i = 8 a = 80 / 13 ### Answer: i = 8, a = 80 / 13
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 37 * p + 74 * c = 183 ... (1) 27 * c - 18 * p = 13 ... (2) Step 1: Solve equation (1) for p in terms of c. 37 * p + 74 * c = 183 37 * p = 183 - 74 * c p = (183 - 74 * c) / 37 Step 2: Substitute the expression for p into equation (2). 27 * c - 18 * ((183 - 74 * c) / 37) = 13 Step 3: Solve the equation for c. 27 * c - (3294 - 1332 * c) / 37 = 13 999 * c - (3294 - 1332 * c) / 37 = 481 36963 * c - 3294 + 1332 * c = 17797 38295 * c = 21091 c = 21091 / 38295 c = 0.5508 (rounded to 4 decimal places) Step 4: Substitute the value of c into the expression for p from Step 1. p = (183 - 74 * 0.5508) / 37 p = (183 - 40.7592) / 37 p = 142.2408 / 37 p = 3.8444 (rounded to 4 decimal places) Therefore, the solution to the system of equations is: p ≈ 3.8444 c ≈ 0.5508 ### Answer: p ≈ 3.8444, c ≈ 0.5508
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 Opus:
To solve for k and p, we can use the substitution method. Let's solve for k in the first equation and substitute it into the second equation to solve for p. From the first equation: (48 * k) - (30 * p) = 37 48k - 30p = 37 48k = 37 + 30p k = (37 + 30p) / 48 Now, substitute the expression for k into the second equation: (16 * p) + (33 * k) = 105 16p + 33 * ((37 + 30p) / 48) = 105 16p + (1221 + 990p) / 48 = 105 16p + 1221/48 + 990p/48 = 105 16p + 25.4375 + 20.625p = 105 36.625p + 25.4375 = 105 36.625p = 79.5625 p = 79.5625 / 36.625 p = 2.1725 Now that we have the value of p, we can substitute it back into the expression for k: k = (37 + 30p) / 48 k = (37 + 30 * 2.1725) / 48 k = (37 + 65.175) / 48 k = 102.175 / 48 k = 2.1286 Therefore, the values of k and p are approximately: p ≈ 2.1725 k ≈ 2.1286 ### Answer: k ≈ 2.1286, p ≈ 2.1725
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 Opus:
To solve for v and f in the given pair of simultaneous equations, we can use the substitution method. Given equations: 1. 75v - 69f = 14 2. 46f + 53v = 110 Step 1: Solve equation 2 for f in terms of v. 46f + 53v = 110 46f = 110 - 53v f = (110 - 53v) ÷ 46 f = 110/46 - 53v/46 f = 110/46 - 53/46 × v <<110/46=2.391304348, 53/46=1.152173913>> f = 2.391304348 - 1.152173913v Step 2: Substitute the expression for f into equation 1. 75v - 69f = 14 75v - 69(2.391304348 - 1.152173913v) = 14 75v - 165 + 79.5v = 14 154.5v - 165 = 14 154.5v = 179 v = 179 ÷ 154.5 <<179/154.5=1.158576052>> v = 1.158576052 Step 3: Substitute the value of v into the expression for f from step 1. f = 2.391304348 - 1.152173913v f = 2.391304348 - 1.152173913 × 1.158576052 <<2.391304348-1.152173913*1.158576052=1.057471264>> f = 1.057471264 Therefore, the values that satisfy the given simultaneous equations are: v ≈ 1.158576052 f ≈ 1.057471264 ### Answer: v ≈ 1.158576052, f ≈ 1.057471264
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, we can solve for o: 7 * o + 13 * g = 131 7 * o = 131 - 13 * g o = (131 - 13 * g) / 7 Now, let's substitute this expression for o into the second equation: 89 * g - 70 * ((131 - 13 * g) / 7) = 37 Simplify the equation: 89 * g - 10 * (131 - 13 * g) = 37 89 * g - 1310 + 130 * g = 37 219 * g - 1310 = 37 219 * g = 1347 g = 1347 / 219 g = <<1347/219=6.15068493150685>>6.15068493150685 Now that we have the value of g, we can substitute it back into the expression for o: o = (131 - 13 * 6.15068493150685) / 7 o = (131 - 80) / 7 o = 51 / 7 o = <<51/7=7.28571428571429>>7.28571428571429 ### Answer: o = 7.28571428571429, g = 6.15068493150685
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for d in terms of k in the first equation, then substitute it into the second equation to solve for k. Finally, we'll substitute the value of k back into the equation for d to find its value. 1. Solve the first equation for d in terms of k: 88 * d - 60 * k = 38 88 * d = 60 * k + 38 d = (60 * k + 38) / 88 2. Substitute the expression for d into the second equation: 33 * k - 11 * ((60 * k + 38) / 88) = 46 33 * k - (660 * k + 418) / 88 = 46 2904 * k - 660 * k - 418 = 4048 2244 * k - 418 = 4048 2244 * k = 4466 k = 4466 / 2244 <<4466/2244=1.99>>k = 1.99 3. Substitute k = 1.99 into the equation for d: d = (60 * 1.99 + 38) / 88 d = 157.4 / 88 <<157.4/88=1.79>>d = 1.79 Therefore, the solution to the system of equations is: ### Answer: d = 1.79, k = 1.99
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for z in terms of y using the second equation and then substitute it into the first equation to solve for y. Given equations: 1. 24 * z + 72 * y = 139 2. 2 * z + 82 * y = 96 Step 1: Solve for z in terms of y using the second equation. 2 * z + 82 * y = 96 2 * z = 96 - 82 * y z = (96 - 82 * y) / 2 z = 48 - 41 * y Step 2: Substitute the expression for z into the first equation and solve for y. 24 * z + 72 * y = 139 24 * (48 - 41 * y) + 72 * y = 139 1152 - 984 * y + 72 * y = 139 1152 - 912 * y = 139 -912 * y = -1013 y = -1013 / -912 y = <<-1013/-912=1.11>>1.11 (rounded to two decimal places) Step 3: Substitute the value of y into the expression for z from Step 1. z = 48 - 41 * y z = 48 - 41 * 1.11 z = 48 - 45.51 z = <<48-45.51=2.49>>2.49 (rounded to two decimal places) ### 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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 1. (87 * o) - (29 * v) = 1 2. (8 * v) + (30 * o) = 91 Step 1: Solve for v in terms of o using equation 1. (87 * o) - (29 * v) = 1 -29 * v = 1 - (87 * o) v = (1 - (87 * o)) / (-29) v = (87 * o - 1) / 29 Step 2: Substitute the expression for v into equation 2. (8 * ((87 * o - 1) / 29)) + (30 * o) = 91 Step 3: Solve for o. (8 * (87 * o - 1) / 29) + (30 * o) = 91 (696 * o - 8) / 29 + (30 * o) = 91 (696 * o - 8 + 870 * o) / 29 = 91 (1566 * o - 8) / 29 = 91 1566 * o - 8 = 91 * 29 1566 * o - 8 = 2639 1566 * o = 2647 o = 2647 / 1566 o = 1.69 (rounded to two decimal places) Step 4: Substitute the value of o into the expression for v from step 1. v = (87 * 1.69 - 1) / 29 v = (147.03 - 1) / 29 v = 146.03 / 29 v = 5.04 (rounded to two decimal places) Therefore, the solution to the system of equations is: o ≈ 1.69 v ≈ 5.04 ### Answer: o ≈ 1.69, v ≈ 5.04
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. First equation: (23 * h) - (46 * p) = 58 23h - 46p = 58 23h = 58 + 46p h = (58 + 46p) / 23 h = 58/23 + 2p Now, substitute the expression for h into the second equation: Second equation: (11 * p) + (23 * h) = 124 11p + 23(58/23 + 2p) = 124 11p + 58 + 46p = 124 57p + 58 = 124 57p = 66 p = 66/57 Now that we have the value of p, we can substitute it back into the expression for h: h = 58/23 + 2p h = 58/23 + 2(66/57) h = 58/23 + 132/57 h = (58*57 + 132*23) / (23*57) h = 3306 + 3036 / 1311 h = 6342 / 1311 Therefore, the solution to the system of equations is: p = 66/57 h = 6342/1311 ### Answer: p = 66/57, h = 6342/1311
Expected answer: {h: 2114/437, p: 22/19}
Correct? Yes
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve equation 1) for d in terms of f, then substitute the expression for d into equation 2) to solve for f. Finally, we'll substitute the value of f back into equation 1) to find the value of d. Step 1: Solve equation 1) for d in terms of f. 19d + 6f = 120 19d = 120 - 6f d = (120 - 6f) ÷ 19 d = (120 - 6f)/19 Step 2: Substitute the expression for d into equation 2) and solve for f. 90d - 42f = 69 90((120 - 6f)/19) - 42f = 69 (10800 - 540f)/19 - 42f = 69 10800/19 - 540f/19 - 42f = 69 (10800 - 540f - 798f)/19 = 69 (10800 - 1338f)/19 = 69 10800 - 1338f = 1311 -1338f = -9489 f = -9489/-1338 f = <<-9489/-1338=7.1>>7 Step 3: Substitute the value of f into equation 1) to find the value of d. 19d + 6f = 120 19d + 6(7) = 120 19d + 42 = 120 19d = 78 d = 78/19 d = <<78/19=4.1>>4 Therefore, the values of d and f are: d = 4 f = 7 ### Answer: d = 4, f = 7
Expected answer: {d: 909/223, f: 3163/446}
Correct? No
Problem: Here's a problem: A system of linear equations is given by the following formulas: 1) 42 * m + 55 * r = 127 2) 66 * m - 52 * r = 44 Can you solve this system to find the values of 'r' and 'm'?
Claude-3 Opus:
Sure, I can solve this system of linear equations using the substitution method. Let's solve for one of the variables in the first equation and substitute it into the second equation. Given equations: 1) 42 * m + 55 * r = 127 2) 66 * m - 52 * r = 44 Step 1: Solve equation 1 for m in terms of r. 42 * m + 55 * r = 127 42 * m = 127 - 55 * r m = (127 - 55 * r) / 42 Step 2: Substitute the expression for m into equation 2. 66 * ((127 - 55 * r) / 42) - 52 * r = 44 (66 * (127 - 55 * r)) / 42 - 52 * r = 44 (8382 - 3630 * r) / 42 - 52 * r = 44 199.5714 - 86.4286 * r - 52 * r = 44 199.5714 - 138.4286 * r = 44 -138.4286 * r = -155.5714 r = 1.1238 Step 3: Substitute the value of r into the expression for m. m = (127 - 55 * 1.1238) / 42 m = (127 - 61.8095) / 42 m = 65.1905 / 42 m = 1.5522 Therefore, the solution to the system of linear equations is: r ≈ 1.1238 m ≈ 1.5522 ### Answer: r ≈ 1.1238, m ≈ 1.5522
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 Opus:
To solve this system of linear equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, let's solve for n: 31n + 64w = 120 31n = 120 - 64w n = (120 - 64w) / 31 Now, substitute this expression for n into the second equation: 57w + 71n = 173 57w + 71((120 - 64w) / 31) = 173 Simplify: 57w + (8520 - 4544w) / 31 = 173 1767w + 8520 - 4544w = 5363 1767w - 4544w = 5363 - 8520 -2777w = -3157 w = 3157 / 2777 w = <<3157/2777=1.136693554>>1.136693554... Now that we have the value of w, we can substitute it back into the expression for n: n = (120 - 64w) / 31 n = (120 - 64(1.136693554)) / 31 n = (120 - 72.748387456) / 31 n = 47.251612544 / 31 n = <<47.251612544/31=1.524245243>>1.524245243... Therefore, the values of n and w that satisfy the system of linear equations simultaneously are: w ≈ 1.136693554 n ≈ 1.524245243 ### Answer: w ≈ 1.136693554, n ≈ 1.524245243
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 Opus:
To solve this system of linear equations, we can use the substitution method. Let's solve equation (1) for u in terms of a, then substitute the result into equation (2) to solve for a. Finally, we'll substitute the value of a back into equation (1) to find u. From equation (1), we have: 72 * a + 26 * u = 112 26 * u = 112 - 72 * a u = (112 - 72 * a) / 26 u = 56/13 - 36/13 * a Now, substitute this expression for u into equation (2): 76 * (56/13 - 36/13 * a) - 77 * a = 31 4256/13 - 2736/13 * a - 77 * a = 31 4256/13 - (2736/13 + 1001) * a = 31 4256/13 - (3737/13) * a = 31 4256 - 3737 * a = 403 519 - 3737 * a = 403 -3737 * a = -116 a = 116/3737 a = 31/1000 Now that we have the value of a, let's substitute it back into the equation for u: u = 56/13 - 36/13 * (31/1000) u = 56/13 - 1116/(13000) u = 728/169 - 93/1690 u = 635/169 Therefore, the values of u and a are: a = 31/1000 u = 635/169 ### Answer: a = 31/1000, u = 635/169
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 Opus:
To solve the system of equations, we can use the substitution method or the elimination method. Let's use the substitution method here. Given equations: 1. 54n + 18b = 143 2. 87n - 93b = 48 Step 1: Solve equation 1 for n in terms of b. 54n + 18b = 143 54n = 143 - 18b n = (143 - 18b) / 54 n = (143/54) - (1/3)b Step 2: Substitute the expression for n into equation 2. 87n - 93b = 48 87((143/54) - (1/3)b) - 93b = 48 (87 × 143/54) - (87/3)b - 93b = 48 (87 × 143/54) - (87/3 + 93)b = 48 (87 × 143/54) - (29 + 93)b = 48 (87 × 143/54) - 122b = 48 Step 3: Solve the equation for b. (87 × 143/54) - 122b = 48 -122b = 48 - (87 × 143/54) -122b = 48 - 231 -122b = -183 b = -183 / -122 b = 3/2 or 1.5 Step 4: Substitute the value of b into the expression for n from step 1. n = (143/54) - (1/3)b n = (143/54) - (1/3) × (3/2) n = (143/54) - (1/2) n = (143/54) - (27/54) n = (143 - 27)/54 n = 116/54 Therefore, the solution to the system of equations is: b = 3/2 or 1.5 n = 116/54 ### Answer: b = 3/2 or 1.5, n = 116/54
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. From the first equation, we can solve for p in terms of e: 53 * e + 7 * p = 103 7 * p = 103 - 53 * e p = (103 - 53 * e) / 7 Now, substitute this expression for p into the second equation: 69 * e - 88 * p = 6 69 * e - 88 * ((103 - 53 * e) / 7) = 6 69 * e - 88 * (103 / 7) + 88 * (53 * e / 7) = 6 69 * e - 1298 / 7 + 4664 * e / 7 = 6 483 * e / 7 + 4664 * e / 7 - 1298 / 7 = 6 5147 * e / 7 - 1298 / 7 = 6 5147 * e = 42 + 1298 5147 * e = 1340 e = 1340 / 5147 e = <<1340/5147=0.2603>>0.2603 Now that we have the value of e, we can substitute it back into the equation for p: p = (103 - 53 * e) / 7 p = (103 - 53 * 0.2603) / 7 p = (103 - 13.7959) / 7 p = 89.2041 / 7 p = <<89.2041/7=12.7434>>12.7434 ### Answer: The cost of electricity (e) is approximately 0.2603, and the cost of petroleum (p) is approximately 12.7434.
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 Opus:
To solve this system of linear equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. First equation: ((18 * p) - (7 * s)) = 21 18p - 7s = 21 18p = 21 + 7s p = (21 + 7s) / 18 p = (7s + 21) / 18 Now, substitute the expression for p into the second equation: Second equation: ((51 * s) - (24 * p)) = 14 51s - 24 * ((7s + 21) / 18) = 14 51s - (24 * (7s + 21)) / 18 = 14 51s - (168s + 504) / 18 = 14 51s - (168s/18 + 504/18) = 14 51s - (28s/3 + 84/3) = 14 153s/3 - 28s/3 - 84/3 = 14 (153s - 28s - 84) / 3 = 14 (125s - 84) / 3 = 14 125s - 84 = 42 125s = 126 s = 126 / 125 Now that we have the value of s, we can substitute it back into the equation for p: p = (7s + 21) / 18 p = (7 * (126/125) + 21) / 18 p = (882/125 + 21) / 18 p = (882/125 + 2625/125) / 18 p = 3507 / (125 * 18) p = 3507 / 2250 Therefore, the values of p and s that satisfy both equations are: s = 126 / 125 = 1.008 p = 3507 / 2250 ≈ 1.558 ### Answer: p ≈ 1.558, s = 1.008
Expected answer: {p: 1169/750, s: 126/125}
Correct? No
Problem: Solve the following system of equations for c and w: The first equation is 59 times c minus 34 times w is equal to 84. The second equation is 11 times c plus 42 times w is equal to 98.
Claude-3 Opus:
Let's solve this step by step using the substitution method. Given equations: 1. 59c - 34w = 84 2. 11c + 42w = 98 Step 1: Solve equation 1 for c in terms of w. 59c - 34w = 84 59c = 84 + 34w c = (84 + 34w) / 59 c = 84/59 + (34/59)w Step 2: Substitute the expression for c into equation 2. 11c + 42w = 98 11(84/59 + (34/59)w) + 42w = 98 (924/59 + (374/59)w) + 42w = 98 924/59 + (374/59)w + 42w = 98 924/59 + (374/59 + 42)w = 98 924/59 + (2478/59 + 42)w = 98 924/59 + (2520/59)w = 98 Step 3: Solve the equation for w. 924/59 + (2520/59)w = 98 (2520/59)w = 98 - 924/59 w = (98 - 924/59) / (2520/59) w = (5782 - 924) / 2520 w = 4858 / 2520 w = 1.927 Step 4: Substitute the value of w into the expression for c. c = 84/59 + (34/59)w c = 84/59 + (34/59)(1.927) c = 1.424 + 1.111 c = 2.535 Therefore, the solution to the system of equations is: c ≈ 2.535 w ≈ 1.927 ### Answer: c ≈ 2.535, w ≈ 1.927
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 1. 45f + 42z = 101 2. 60z - 42f = 13 Step 1: Solve for f in terms of z using the first equation. 45f + 42z = 101 45f = 101 - 42z f = (101 - 42z) ÷ 45 f = (101 - 42z) / 45 Step 2: Substitute the expression for f into the second equation. 60z - 42((101 - 42z) / 45) = 13 Step 3: Solve for z. 60z - 42(101 - 42z) / 45 = 13 60z - (42 × 101 - 42 × 42z) / 45 = 13 60z - (4242 - 1764z) / 45 = 13 2700z - 4242 + 1764z = 585 4464z - 4242 = 585 4464z = 4827 z = 4827 / 4464 z = 1.0813 (rounded to 4 decimal places) Step 4: Substitute the value of z into the expression for f from Step 1. f = (101 - 42 × 1.0813) / 45 f = (101 - 45.4146) / 45 f = 55.5854 / 45 f = 1.2352 (rounded to 4 decimal places) ### Answer: f = 1.2352, z = 1.0813
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 Opus:
To solve the system of equations, we can use the substitution method. Given equations: 26 * p - 21 * u = 6 15 * p + 58 * u = 83 Step 1: Solve the first equation for p in terms of u. 26 * p - 21 * u = 6 26 * p = 6 + 21 * u p = (6 + 21 * u) / 26 p = (3 + 21 * u) / 13 Step 2: Substitute the expression for p into the second equation. 15 * p + 58 * u = 83 15 * ((3 + 21 * u) / 13) + 58 * u = 83 (45 + 315 * u) / 13 + 58 * u = 83 Step 3: Solve the equation for u. (45 + 315 * u) / 13 + 58 * u = 83 45 / 13 + 315 * u / 13 + 58 * u = 83 45 / 13 + (315 / 13 + 58) * u = 83 45 / 13 + (315 / 13 + 754 / 13) * u = 83 45 / 13 + 1069 * u / 13 = 83 1069 * u / 13 = 83 - 45 / 13 1069 * u / 13 = 1034 / 13 u = 1034 / 1069 u = <<1034/1069=0.9672>>0.9672 Step 4: Substitute the value of u into the expression for p. p = (3 + 21 * u) / 13 p = (3 + 21 * 0.9672) / 13 p = 23.3112 / 13 p = <<23.3112/13=1.7932>>1.7932 Therefore, the values of p and u are: u = 0.9672 p = 1.7932 ### Answer: p = 1.7932, u = 0.9672
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 Opus:
To solve for the values of n and s using the given simultaneous equations, we can use the substitution method. Given equations: 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75 Step 1: Solve equation (2) for n in terms of s. 59 * n + 8 * s = 75 59 * n = 75 - 8 * s n = (75 - 8 * s) / 59 Step 2: Substitute the expression for n from step 1 into equation (1). 92 * ((75 - 8 * s) / 59) - 38 * s = 62 (92 * (75 - 8 * s)) / 59 - 38 * s = 62 (6900 - 736 * s) / 59 - 38 * s = 62 6900 / 59 - (736 * s) / 59 - 38 * s = 62 6900 / 59 - (736 * s + 2242 * s) / 59 = 62 6900 / 59 - (2978 * s) / 59 = 62 6900 - 2978 * s = 3658 3922 = 2978 * s s = 3922 / 2978 Step 3: Calculate the value of s. s = 3922 / 2978 ≈ 1.32 (rounded to two decimal places) Step 4: Substitute the value of s into the expression for n from step 1 to find the value of n. n = (75 - 8 * 1.32) / 59 n = (75 - 10.56) / 59 n = 64.44 / 59 ≈ 1.09 (rounded to two decimal places) Therefore, the values of n and s that satisfy the given simultaneous equations are approximately: n ≈ 1.09 s ≈ 1.32 ### Answer: n ≈ 1.09, s ≈ 1.32
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 Opus:
To solve for e and z in the given simultaneous equations, we can use the substitution method. Given equations: 87 * z - 35 * e = 63 ... (1) 81 * z + 58 * e = 154 ... (2) Step 1: Solve equation (1) for z in terms of e. 87 * z - 35 * e = 63 87 * z = 63 + 35 * e z = (63 + 35 * e) / 87 ... (3) Step 2: Substitute the expression for z from equation (3) into equation (2). 81 * ((63 + 35 * e) / 87) + 58 * e = 154 (81 * (63 + 35 * e)) / 87 + 58 * e = 154 (5103 + 2835 * e) / 87 + 58 * e = 154 5103 / 87 + (2835 * e) / 87 + 58 * e = 154 58.66 + 32.59 * e + 58 * e = 154 58.66 + 90.59 * e = 154 90.59 * e = 95.34 e = 95.34 / 90.59 e ≈ 1.05 Step 3: Substitute the value of e into equation (3) to find z. z = (63 + 35 * 1.05) / 87 z = (63 + 36.75) / 87 z = 99.75 / 87 z ≈ 1.15 Therefore, the values for e and z are approximately: e ≈ 1.05 z ≈ 1.15 ### Answer: e ≈ 1.05, z ≈ 1.15
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 Opus:
To solve the system of simultaneous linear equations, we can use the substitution method or the elimination method. Let's use the substitution method here. Given equations: (11 * t) + (44 * h) = 110 ... (1) (98 * t) - (36 * h) = 18 ... (2) Step 1: Solve equation (1) for t in terms of h. (11 * t) + (44 * h) = 110 11 * t = 110 - 44 * h t = (110 - 44 * h) / 11 t = 10 - 4 * h ... (3) Step 2: Substitute the expression for t from equation (3) into equation (2). (98 * t) - (36 * h) = 18 98 * (10 - 4 * h) - 36 * h = 18 980 - 392 * h - 36 * h = 18 980 - 428 * h = 18 -428 * h = -962 h = 962 / 428 h = 2.25 Step 3: Substitute the value of h into equation (3) to find the value of t. t = 10 - 4 * h t = 10 - 4 * 2.25 t = 10 - 9 t = 1 Therefore, the solution to the system of simultaneous linear equations is: 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 Opus:
To solve the system of linear equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. From the first equation: ((69 * b) - (50 * o)) = 55 69b - 50o = 55 69b = 55 + 50o b = (55 + 50o) / 69 Now, substitute the expression for b into the second equation: ((41 * o) + (18 * b)) = 85 41o + 18 * ((55 + 50o) / 69) = 85 41o + (990 + 900o) / 69 = 85 41o + (1890o + 990) / 69 = 85 2769o + 990 = 5865 2769o = 4875 o = 4875 / 2769 Now that we have the value of o, we can substitute it back into the expression for b: b = (55 + 50o) / 69 b = (55 + 50 * (4875 / 2769)) / 69 b = (55 + 243750 / 2769) / 69 b = (151745 / 2769) / 69 Simplifying the fractions: o = 4875 / 2769 ≈ 1.76 b = (151745 / 2769) / 69 ≈ 0.79 ### Answer: o ≈ 1.76, b ≈ 0.79
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the second equation and then substitute it into the first equation. From the second equation, 6 * q + 5 * p = 177, let's solve for p: 6 * q + 5 * p = 177 5 * p = 177 - 6 * q p = (177 - 6 * q) / 5 Now, substitute the expression for p into the first equation: 67 * q - 80 * p = 65 67 * q - 80 * ((177 - 6 * q) / 5) = 65 67 * q - 80 * (177 / 5 - 6 * q / 5) = 65 67 * q - 80 * (35.4 - 1.2 * q) = 65 67 * q - 2832 + 96 * q = 65 163 * q - 2832 = 65 163 * q = 2897 q = 2897 / 163 q = <<2897/163=17.77>>17.77 (rounded to two decimal places) Now that we have the value of q, we can substitute it into the equation for p: p = (177 - 6 * q) / 5 p = (177 - 6 * 17.77) / 5 p = (177 - 106.62) / 5 p = 70.38 / 5 p = <<70.38/5=14.08>>14.08 (rounded to two decimal places) ### Answer: p = 14.08, q = 17.77
Expected answer: {p: 11469/815, q: 2897/163}
Correct? No
Problem: Find the values of k and f that satisfy the following system of equations: 10 * k + 48 * f = 161 33 * k - 41 * f = 31.
Claude-3 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for k in the second equation and substitute it into the first equation to solve for f. From the second equation: 33 * k - 41 * f = 31 33 * k = 31 + 41 * f k = (31 + 41 * f) / 33 Substitute this value of k into the first equation: 10 * k + 48 * f = 161 10 * ((31 + 41 * f) / 33) + 48 * f = 161 (310 + 410 * f) / 33 + 48 * f = 161 310 / 33 + 410 * f / 33 + 48 * f = 161 310 / 33 + (410 / 33 + 48) * f = 161 310 / 33 + (410 / 33 + 1584 / 33) * f = 161 310 / 33 + 1994 * f / 33 = 161 310 + 1994 * f = 161 * 33 310 + 1994 * f = 5313 1994 * f = 5313 - 310 1994 * f = 5003 f = 5003 / 1994 f = <<5003/1994=2.51>>2.51 Now, substitute the value of f into the second equation to find k: 33 * k - 41 * f = 31 33 * k - 41 * 2.51 = 31 33 * k - 102.91 = 31 33 * k = 31 + 102.91 33 * k = 133.91 k = 133.91 / 33 k = <<133.91/33=4.06>>4.06 ### Answer: k = 4.06, f = 2.51
Expected answer: {f: 5003/1994, k: 8089/1994}
Correct? No
Problem: Solve the following system of equations for the values of l and n: The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l
Claude-3 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for n in the first equation and substitute it into the second equation to solve for l. 1. Solve the first equation for n: 31 = 58 * l - n 31 - 58 * l = -n n = 58 * l - 31 2. Substitute the expression for n into the second equation: 103 = (58 * l - 31) + 73 * l 103 = 58 * l - 31 + 73 * l 103 = 131 * l - 31 3. Solve the equation for l: 103 + 31 = 131 * l 134 = 131 * l l = 134 / 131 <<134/131=1.022900763358779>>l ≈ 1.023 4. Substitute the value of l into the expression for n: n = 58 * l - 31 n = 58 * 1.023 - 31 <<58*1.023-31=28.334>>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 Opus:
To solve this system of equations, we can use the substitution method. Step 1: Solve the first equation for y in terms of j. 29y - 28j = 39 29y = 39 + 28j y = (39 + 28j) / 29 Step 2: Substitute the expression for y into the second equation. 25y + 37j = 116 25((39 + 28j) / 29) + 37j = 116 Step 3: Simplify the equation and solve for j. 25(39 + 28j) + 1073j = 3364 975 + 700j + 1073j = 3364 1773j = 2389 j = 2389 / 1773 j = 1.347 (rounded to three decimal places) Step 4: Substitute the value of j into the expression for y from Step 1. y = (39 + 28j) / 29 y = (39 + 28(1.347)) / 29 y = 76.716 / 29 y = 2.645 (rounded to three decimal places) Therefore, the values of y and j that satisfy the system of equations are: y ≈ 2.645 j ≈ 1.347 ### Answer: 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 Opus:
To solve for u and a, we can use the substitution method or the elimination method. Let's use the substitution method here. Step 1: Solve the first equation for u in terms of a. 93u - 85a = 74 93u = 74 + 85a u = (74 + 85a) ÷ 93 u = (74 + 85a) / 93 Step 2: Substitute the expression for u into the second equation. 96u + 17a = 186 96((74 + 85a) / 93) + 17a = 186 Step 3: Simplify the equation and solve for a. (96(74 + 85a) / 93) + 17a = 186 (7104 + 8160a) / 93 + 17a = 186 7104 / 93 + 8160a / 93 + 17a = 186 76.387 + 87.742a + 17a = 186 104.742a = 109.613 a = 109.613 / 104.742 a ≈ 1.046 Step 4: Substitute the value of a into the expression for u from Step 1. u = (74 + 85a) / 93 u = (74 + 85(1.046)) / 93 u = (74 + 88.91) / 93 u = 162.91 / 93 u ≈ 1.752 Therefore, the solution to the system of equations is: a ≈ 1.046 u ≈ 1.752 ### Answer: u ≈ 1.752, a ≈ 1.046
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 Opus:
To solve for u and g, we can use the substitution method. Let's solve for u in the first equation and substitute it into the second equation to solve for g. From the first equation, we have: 97 * u - 61 * g = 15 97 * u = 61 * g + 15 u = (61 * g + 15) / 97 Now, let's substitute this expression for u into the second equation: 85 * u + 23 * g = 186 85 * ((61 * g + 15) / 97) + 23 * g = 186 Simplify: (5185 * g + 1275) / 97 + 23 * g = 186 5185 * g + 1275 + 2231 * g = 18062 7416 * g + 1275 = 18062 7416 * g = 16787 g = 16787 / 7416 g = <<16787/7416=2.26>>2.26 (rounded to two decimal places) Now that we have the value of g, we can substitute it back into the equation for u: u = (61 * g + 15) / 97 u = (61 * 2.26 + 15) / 97 u = 152.86 / 97 u = <<152.86/97=1.58>>1.58 (rounded to two decimal places) ### Answer: u = 1.58, g = 2.26
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one of the variables in the first equation and substitute it into the second equation. From the first equation, let's solve for r: (17 * r) + (67 * b) = 103 17r = 103 - 67b r = (103 - 67b) / 17 Now, substitute this expression for r into the second equation: (85 * r) + (67 * b) = 187 85 * ((103 - 67b) / 17) + 67b = 187 Simplify: (8755 - 5695b) / 17 + 67b = 187 8755 - 5695b + 1139b = 187 * 17 8755 - 4556b = 3179 -4556b = -5576 b = -5576 / -4556 b = <<-5576/-4556=1.224>>1.224 Now that we have the value of b, we can substitute it back into the expression for r: r = (103 - 67b) / 17 r = (103 - 67 * 1.224) / 17 r = (103 - 82.008) / 17 r = 20.992 / 17 r = <<20.992/17=1.235>>1.235 ### Answer: r = 1.235, b = 1.224
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 Opus:
To solve for the values of t and n, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. From equation 1, solve for n in terms of t: (85 * n) - (54 * t) = 41 (85 * n) = (54 * t) + 41 n = ((54 * t) + 41) / 85 Now, substitute the expression for n into equation 2: (38 * t) + (30 * (((54 * t) + 41) / 85)) = 126 Simplify: (38 * t) + ((1620 * t) + 1230) / 85 = 126 (3230 * t) + 1230 = 10710 3230t = 9480 t = 9480 / 3230 t ≈ 2.9350931677018633540372670807453 <<38*2.9350931677018633540372670807453+30*((54*2.9350931677018633540372670807453+41)/85)≈126>> Now that we have the value of t, we can substitute it back into the equation for n: n = ((54 * 2.9350931677018633540372670807453) + 41) / 85 n ≈ 2.0117647058823529411764705882353 <<85*2.0117647058823529411764705882353-54*2.9350931677018633540372670807453≈41>> Therefore, the solution to the system of equations is: t ≈ 2.9350931677018633540372670807453 n ≈ 2.0117647058823529411764705882353 ### Answer: t ≈ 2.9350931677018633540372670807453, n ≈ 2.0117647058823529411764705882353
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 Opus:
Let's solve this system of equations step by step: 1) 75u - 69q = 5 2) 48u + 25q = 140 Step 1: Multiply equation (1) by 48 and equation (2) by 75. 48(75u - 69q = 5) → 3600u - 3312q = 240 75(48u + 25q = 140) → 3600u + 1875q = 10500 Step 2: Subtract the new equation (1) from the new equation (2) to eliminate the variable u. (3600u + 1875q = 10500) - (3600u - 3312q = 240) 1875q + 3312q = 10500 - 240 5187q = 10260 Step 3: Solve for q by dividing both sides by 5187. q = 10260 ÷ 5187 q = 1.9779... ≈ 1.98 Step 4: Substitute the value of q into equation (1) to solve for u. 75u - 69(1.98) = 5 75u - 136.62 = 5 75u = 141.62 u = 141.62 ÷ 75 u = 1.8882... ≈ 1.89 Therefore, the solution to the system of equations is: u ≈ 1.89 q ≈ 1.98 ### Answer: u ≈ 1.89, q ≈ 1.98
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for v in the first equation and substitute it into the second equation to solve for x. 1. Solve for v in the first equation: ((20 * v) + (17 * x)) = 149 (20 * v) = 149 - (17 * x) v = (149 - (17 * x)) / 20 2. Substitute the expression for v into the second equation: ((24 * x) - (49 * ((149 - (17 * x)) / 20))) = 22 ((24 * x) - (49 * (149 - (17 * x)) / 20)) = 22 ((24 * x) - ((7301 - (833 * x)) / 20)) = 22 (480 * x) - (7301 - (833 * x)) = 440 (480 * x) - 7301 + (833 * x) = 440 (1313 * x) - 7301 = 440 (1313 * x) = 7741 x = 7741 / 1313 x = 5.896 (rounded to three decimal places) 3. Substitute the value of x into the expression for v: v = (149 - (17 * 5.896)) / 20 v = (149 - 100.232) / 20 v = 48.768 / 20 v = 2.438 (rounded to three decimal places) Therefore, the values of v and x are approximately: v ≈ 2.438 x ≈ 5.896 ### Answer: v ≈ 2.438, x ≈ 5.896
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 Opus:
To solve the system of equations, we can use the substitution method or the elimination method. Let's use the substitution method here. Given equations: 39 * t + 41 * g = 147 ... (1) 94 * t - 91 * g = 42 ... (2) Step 1: Solve equation (1) for t in terms of g. 39 * t + 41 * g = 147 39 * t = 147 - 41 * g t = (147 - 41 * g) / 39 Step 2: Substitute the expression for t into equation (2). 94 * ((147 - 41 * g) / 39) - 91 * g = 42 Step 3: Simplify the equation and solve for g. (94 * (147 - 41 * g)) / 39 - 91 * g = 42 (13818 - 3854 * g) / 39 - 91 * g = 42 354.3076923 - 98.82051282 * g - 91 * g = 42 354.3076923 - 189.8205128 * g = 42 -189.8205128 * g = -312.3076923 g = 1.645390071 Step 4: Substitute the value of g into the expression for t from Step 1. t = (147 - 41 * 1.645390071) / 39 t = (147 - 67.46099291) / 39 t = 79.53900709 / 39 t = 2.039974797 Therefore, the solution to the system of equations is: g = 1.645390071 t = 2.039974797 ### Answer: t = 2.039974797, g = 1.645390071
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other in the first equation, then substitute it into the second equation. 1. Solve the first equation for c in terms of o: 6c + 55o = 112 6c = 112 - 55o c = (112 - 55o) / 6 c = 56/3 - 55o/6 2. Substitute the expression for c into the second equation: 55c - 48o = 36 55(56/3 - 55o/6) - 48o = 36 (55 × 56/3) - (55 × 55o/6) - 48o = 36 1026.67 - 504.17o - 48o = 36 1026.67 - 552.17o = 36 -552.17o = -990.67 o = 990.67/552.17 o = 1.79 (rounded to two decimal places) 3. Substitute o = 1.79 into the expression for c: c = 56/3 - 55o/6 c = 56/3 - 55 × 1.79/6 c = 18.67 - 16.41 c = 2.26 (rounded to two decimal places) Therefore, the values that satisfy the system of equations are approximately: c = 2.26 o = 1.79 ### Answer: c = 2.26, o = 1.79
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 8w + 68u = 170 ... (1) 71w - 94u = 8 ... (2) Step 1: Solve for w in equation (1): 8w + 68u = 170 8w = 170 - 68u w = (170 - 68u) ÷ 8 w = 21.25 - 8.5u ... (3) Step 2: Substitute w from equation (3) into equation (2): 71(21.25 - 8.5u) - 94u = 8 1508.75 - 603.5u - 94u = 8 1508.75 - 697.5u = 8 -697.5u = 8 - 1508.75 -697.5u = -1500.75 u = -1500.75 ÷ -697.5 u = 2.15 Step 3: Substitute u = 2.15 into equation (3) to find w: w = 21.25 - 8.5(2.15) w = 21.25 - 18.275 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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, solve for f: 68*r + 8*f = 92 8*f = 92 - 68*r f = (92 - 68*r) / 8 f = (92/8) - (68/8)*r f = 11.5 - 8.5*r Now, substitute the expression for f into the second equation: 97*f + 38*r = 182 97*(11.5 - 8.5*r) + 38*r = 182 1115.5 - 824.5*r + 38*r = 182 1115.5 - 786.5*r = 182 -786.5*r = -933.5 r = 933.5 / 786.5 r = 1.1869436201780415 ≈ 1.19 Now that we have the value of r, we can substitute it back into the expression for f: f = 11.5 - 8.5*r f = 11.5 - 8.5*1.1869436201780415 f = 11.5 - 10.089020771513353 f = 1.4109792284866471 ≈ 1.41 Therefore, the solution to the system of equations is: r ≈ 1.19 f ≈ 1.41 ### Answer: r ≈ 1.19, f ≈ 1.41
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 Opus:
Let's solve this problem step by step using the given equations: Equation 1: 67d - 37r = 50 Equation 2: 39r - 21d = 16 Step 1: Multiply equation 2 by 2 to eliminate the decimal fraction when adding the equations later. 2 × (39r - 21d = 16) 78r - 42d = 32 Step 2: Add equation 1 and the modified equation 2 to eliminate the variable d. (67d - 37r = 50) + (78r - 42d = 32) 25d + 41r = 82 Step 3: Substitute the value of 25d from equation 1 into the equation from step 2. Equation 1: 67d - 37r = 50 25d = 50 + 37r Substituting in the equation from step 2: (50 + 37r) + 41r = 82 50 + 78r = 82 Step 4: Solve for r. 78r = 82 - 50 78r = 32 r = 32 ÷ 78 r = 0.4102564... Rounded to the nearest cent, r ≈ $0.41 Step 5: Substitute the value of r in equation 1 to solve for d. 67d - 37 × 0.41 = 50 67d - 15.17 = 50 67d = 65.17 d = 65.17 ÷ 67 d = 0.9726865... Rounded to the nearest cent, d ≈ $0.97 Therefore, the cost of each ragdoll (r) is approximately $0.41, and the cost of each apple (d) is approximately $0.97. ### Answer: r ≈ $0.41, d ≈ $0.97
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 Opus:
To solve this system of simultaneous linear equations, we can use the substitution method. Let's solve for x in the second equation and substitute it into the first equation to find the value of a. Then, we can use the value of a to find the value of x. 1. Solve the second equation for x: 24 = 98x - 84a 24 + 84a = 98x (24 + 84a) / 98 = x x = (24 + 84a) / 98 2. Substitute the expression for x into the first equation: 163 = 50x + a 163 = 50((24 + 84a) / 98) + a 163 = (1200 + 4200a) / 98 + a 163 = (1200 / 98) + (4200a / 98) + a 163 = 12.24 + 42.86a + a 163 = 12.24 + 43.86a 163 - 12.24 = 43.86a 150.76 = 43.86a a = 150.76 / 43.86 a ≈ 3.44 3. Substitute the value of a into the expression for x: x = (24 + 84a) / 98 x = (24 + 84 × 3.44) / 98 x = (24 + 288.96) / 98 x = 312.96 / 98 x ≈ 3.19 Therefore, the values of x and a that satisfy the given system of simultaneous linear equations are: ### Answer: x ≈ 3.19 and 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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 67 * p + 18 * b = 110 ... (1) 82 * b - 44 * p = 58 ... (2) Step 1: Solve for b in terms of p using equation (1). 67 * p + 18 * b = 110 18 * b = 110 - 67 * p b = (110 - 67 * p) / 18 ... (3) Step 2: Substitute the expression for b from equation (3) into equation (2). 82 * ((110 - 67 * p) / 18) - 44 * p = 58 (9020 - 5494 * p) / 18 - 44 * p = 58 501.11 - 305.22 * p - 44 * p = 58 501.11 - 349.22 * p = 58 -349.22 * p = -443.11 p = -443.11 / -349.22 p = <<-443.11/-349.22=1.27>>1.27 Step 3: Substitute the value of p into equation (3) to solve for b. b = (110 - 67 * 1.27) / 18 b = (110 - 85.09) / 18 b = 24.91 / 18 b = <<24.91/18=1.38>>1.38 Therefore, the solution to the system of equations is: p = 1.27 b = 1.38 ### Answer: p = 1.27, b = 1.38
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 Opus:
Let's solve this step by step: 1. Let's write the equations based on the given information: - 63c - 36e = 27 - 8c + 32e = 69 2. Multiply the second equation by 2 to make the coefficients of e equal: - 63c - 36e = 27 - 16c + 64e = 138 3. Add the two equations to eliminate e: - 79c = 165 4. Solve for c: - c = 165 ÷ 79 - c = 165/79 ≈ 2.089 (rounded to 3 decimal places) 5. Substitute the value of c in one of the original equations to solve for e. Let's use the first equation: - 63 × 2.089 - 36e = 27 - 131.607 - 36e = 27 - -36e = -104.607 - e = 104.607 ÷ 36 - e = 104.607/36 ≈ 2.906 (rounded to 3 decimal places) Therefore, the values of c and e are: - c ≈ 2.089 - e ≈ 2.906 ### Answer: c ≈ 2.089, e ≈ 2.906
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 Opus:
To solve the system of linear equations, we can use the substitution method or the elimination method. Let's use the substitution method here. Given equations: 26 * x + 29 * p = 107 ... (1) 61 * x - 51 * p = 4 ... (2) Step 1: Solve equation (1) for x in terms of p. 26 * x + 29 * p = 107 26 * x = 107 - 29 * p x = (107 - 29 * p) / 26 Step 2: Substitute the expression for x into equation (2). 61 * ((107 - 29 * p) / 26) - 51 * p = 4 (6527 - 1769 * p) / 26 - 51 * p = 4 6527 - 1769 * p - 1326 * p = 104 6527 - 3095 * p = 104 -3095 * p = -6423 p = 2.076 ... (rounded to 3 decimal places) Step 3: Substitute the value of p into the expression for x from Step 1. x = (107 - 29 * 2.076) / 26 x = (107 - 60.204) / 26 x = 46.796 / 26 x = 1.800 ... (rounded to 3 decimal places) Therefore, the solution to the system of linear equations is: x ≈ 1.800 p ≈ 2.076 ### Answer: x ≈ 1.800, p ≈ 2.076
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, let's solve for l: ((29 * l) + (17 * g)) = 68 29l + 17g = 68 29l = 68 - 17g l = (68 - 17g) / 29 Now, substitute the expression for l into the second equation: ((31 * g) - (10 * ((68 - 17g) / 29))) = 17 Simplify: 31g - (10 * (68 - 17g) / 29) = 17 31g - ((680 - 170g) / 29) = 17 31g - (680/29 - 170g/29) = 17 31g - 680/29 + 170g/29 = 17 (31g + 170g/29) - 680/29 = 17 (899g/29) - 680/29 = 17 899g/29 = 17 + 680/29 899g/29 = 697/29 899g = 697 g = 697/899 g = <<697/899=0.7753059>>0.7753059 Now, substitute the value of g back into the equation for l: l = (68 - 17g) / 29 l = (68 - 17 * 0.7753059) / 29 l = (68 - 13.1802) / 29 l = 54.8198 / 29 l = <<54.8198/29=1.8903379>>1.8903379 Therefore, the values for g and l are: g ≈ 0.7753059 l ≈ 1.8903379 ### Answer: g ≈ 0.7753059, l ≈ 1.8903379
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for 'a' in the first equation and substitute it into the second equation to solve for 'c'. Then, we can use the value of 'c' to find 'a' in the first equation. 1. Solve for 'a' in the first equation: 33*a - 54*c = 20 33*a = 54*c + 20 a = (54*c + 20) / 33 2. Substitute 'a' into the second equation: 24*a - 26*c = 29 24*((54*c + 20) / 33) - 26*c = 29 3. Simplify the equation: 24*(54*c + 20) / 33 - 26*c = 29 (24*54*c + 24*20) / 33 - 26*c = 29 (1296*c + 480) / 33 - 26*c = 29 4. Multiply both sides by 33 to eliminate fractions: 1296*c + 480 - 33*26*c = 29*33 1296*c + 480 - 858*c = 957 438*c + 480 = 957 5. Subtract 480 from both sides: 438*c = 477 6. Divide both sides by 438 to solve for 'c': c = 477 / 438 c ≈ 1.089 (rounded to three decimal places) 7. Substitute 'c' into the first equation to solve for 'a': 33*a - 54*1.089 = 20 33*a - 58.806 = 20 33*a = 78.806 a = 78.806 / 33 a ≈ 2.388 (rounded to three decimal places) ### Answer: a ≈ 2.388, c ≈ 1.089
Expected answer: {a: 523/219, c: 159/146}
Correct? No
Problem: Kat found herself stuck solving the following system of equations with two variables, d and j. Can you help her resolve it? 71d + 9j = 132 43j - 84d = 91.
Claude-3 Opus:
Sure, I can help Kat solve this system of equations using the substitution method. Let's solve for one of the variables in the first equation and substitute it into the second equation. Given equations: 71d + 9j = 132 ... (1) 43j - 84d = 91 ... (2) Step 1: Solve equation (1) for j in terms of d. 71d + 9j = 132 9j = 132 - 71d j = (132 - 71d) ÷ 9 j = (132 ÷ 9) - (71 ÷ 9)d j = 14 - (71 ÷ 9)d ... (3) Step 2: Substitute the expression for j from equation (3) into equation (2). 43(14 - (71 ÷ 9)d) - 84d = 91 602 - (1591 ÷ 3)d - 84d = 91 602 - 530.33d - 84d = 91 602 - 614.33d = 91 Step 3: Solve the resulting equation for d. 602 - 614.33d = 91 -614.33d = 91 - 602 -614.33d = -511 d = -511 ÷ -614.33 d ≈ 0.832 Step 4: Substitute the value of d into equation (3) to find j. j = 14 - (71 ÷ 9)(0.832) j = 14 - 6.554 j ≈ 7.446 Therefore, the solution to the system of equations is: d ≈ 0.832 and j ≈ 7.446 ### Answer: d ≈ 0.832, j ≈ 7.446
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 Opus:
To solve this system of equations, we can use the substitution method. Given equations: (81 * o) + (26 * n) = 163 ... (1) (87 * n) + (23 * o) = 143 ... (2) Step 1: Solve equation (1) for o in terms of n. (81 * o) + (26 * n) = 163 81 * o = 163 - 26 * n o = (163 - 26 * n) / 81 ... (3) Step 2: Substitute the expression for o from equation (3) into equation (2). (87 * n) + (23 * ((163 - 26 * n) / 81)) = 143 Step 3: Simplify the equation and solve for n. (87 * n) + ((3749 - 598 * n) / 81) = 143 7047 * n + 3749 - 598 * n = 11583 6449 * n = 7834 n = 7834 / 6449 n ≈ 1.215 ... (4) Step 4: Substitute the value of n from equation (4) into equation (3) to find o. o = (163 - 26 * 1.215) / 81 o = (163 - 31.59) / 81 o = 131.41 / 81 o ≈ 1.622 Therefore, the approximate values of o and n are: o ≈ 1.622 n ≈ 1.215 ### Answer: o ≈ 1.622, n ≈ 1.215
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, let's solve for s: 46 * s - 61 * w = 2 46 * s = 61 * w + 2 s = (61 * w + 2) / 46 Now, substitute the expression for s into the second equation: 61 * s + 67 * w = 178 61 * ((61 * w + 2) / 46) + 67 * w = 178 Simplify: (61 * (61 * w + 2)) / 46 + 67 * w = 178 (3721 * w + 122) / 46 + 67 * w = 178 Multiply both sides by 46: 3721 * w + 122 + 3082 * w = 8188 6803 * w + 122 = 8188 Subtract 122 from both sides: 6803 * w = 8066 Divide both sides by 6803: w = 8066 / 6803 w ≈ 1.1857 Now that we have the value of w, we can substitute it back into the expression for s: s = (61 * w + 2) / 46 s = (61 * 1.1857 + 2) / 46 s ≈ 1.5857 Therefore, the solution to the system of equations is: w ≈ 1.1857 s ≈ 1.5857 ### Answer: w ≈ 1.1857, s ≈ 1.5857
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, we can solve for k: 51*k - 8*i = 86 51*k = 86 + 8*i k = (86 + 8*i) / 51 k = (86/51) + (8/51)*i k = (86 + 8*i) / 51 Now, substitute the expression for k into the second equation: 39*i + 8*k = 193 39*i + 8*((86 + 8*i) / 51) = 193 39*i + (688 + 64*i) / 51 = 193 39*i + (688/51) + (64/51)*i = 193 39*i + (64/51)*i = 193 - (688/51) (1989/51)*i = 9073/51 i = 9073/1989 i = 91/21 Now that we have the value of i, we can substitute it back into the expression for k: k = (86 + 8*i) / 51 k = (86 + 8*(91/21)) / 51 k = (86 + 364/21) / 51 k = (1806 + 364) / (21*51) k = 2170 / 1071 k = 10/49 Therefore, the values of i and k are: i = 91/21 k = 10/49 ### Answer: i = 91/21, k = 10/49
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for l in the second equation and substitute it into the first equation to find the value of m. Then, we can use the value of m to find the value of l. 1. Solve the second equation for l: 93m + l = 126 l = 126 - 93m 2. Substitute the expression for l into the first equation: 26m + 17l = 146 26m + 17(126 - 93m) = 146 26m + 2142 - 1581m = 146 -1555m + 2142 = 146 -1555m = -1996 m = -1996 ÷ (-1555) m = 128/111 3. Substitute the value of m into the second equation to find l: 93m + l = 126 93(128/111) + l = 126 (93 × 128) ÷ 111 + l = 126 11904 ÷ 111 + l = 126 107 + l = 126 l = 126 - 107 l = 19 Therefore, the solution to the system of equations is: m = 128/111 l = 19 ### Answer: m = 128/111, l = 19
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve equation (1) for f in terms of h, then substitute the expression for f into equation (2) to solve for h. Finally, we'll substitute the value of h back into equation (1) to find f. Step 1: Solve equation (1) for f in terms of h. (8 * f) + (39 * h) = 66 8f = 66 - 39h f = (66 - 39h) / 8 Step 2: Substitute the expression for f into equation (2) and solve for h. (29 * f) + (46 * h) = 119 29 * ((66 - 39h) / 8) + 46h = 119 (29 * (66 - 39h)) / 8 + 46h = 119 (1914 - 1131h) / 8 + 46h = 119 1914 - 1131h + 368h = 952 1914 - 763h = 952 -763h = -962 h = 962 / 763 h = 1.26079947651663 Step 3: Substitute the value of h into the equation for f from Step 1. f = (66 - 39h) / 8 f = (66 - 39 * 1.26079947651663) / 8 f = (66 - 49.1711790841485) / 8 f = 16.8288209158515 / 8 f = 2.10360261448144 Therefore, the values of h and f are: h ≈ 1.26079947651663 f ≈ 2.10360261448144 ### Answer: h ≈ 1.26079947651663, f ≈ 2.10360261448144
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 Opus:
To solve for k and b in the given system of equations, we can use the substitution method. Given equations: 1. 22 * b + 23 * k = 109 2. 62 * k - 69 * b = 14 Step 1: Solve equation 1 for b in terms of k. 22 * b + 23 * k = 109 22 * b = 109 - 23 * k b = (109 - 23 * k) / 22 Step 2: Substitute the expression for b into equation 2. 62 * k - 69 * ((109 - 23 * k) / 22) = 14 Step 3: Simplify the equation and solve for k. 62 * k - (3021 - 1587 * k) / 22 = 14 1364 * k - 3021 + 1587 * k = 308 2951 * k - 3021 = 308 2951 * k = 3329 k = 3329 / 2951 <<3329/2951=1.128>>k ≈ 1.128 Step 4: Substitute the value of k into the expression for b from step 1. b = (109 - 23 * 1.128) / 22 b = (109 - 25.944) / 22 b = 83.056 / 22 <<83.056/22=3.775>>b ≈ 3.775 Therefore, the values of k and b that satisfy the given system of equations are: ### Answer: k ≈ 1.128 and b ≈ 3.775
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 Opus:
To solve the system of linear equations, we can use the substitution method. Given equations: 1. 84m - 55s = 1 2. 63m - 2s = 98 Step 1: Solve equation 2 for s in terms of m. 63m - 2s = 98 -2s = -63m + 98 s = (63m - 98) / 2 Step 2: Substitute the expression for s into equation 1. 84m - 55((63m - 98) / 2) = 1 84m - (3465m - 5390) / 2 = 1 168m - 3465m + 5390 = 2 -3297m + 5390 = 2 -3297m = -5388 m = -5388 / -3297 m = 1.634 (rounded to three decimal places) Step 3: Substitute the value of m into the expression for s from step 1. s = (63(1.634) - 98) / 2 s = (102.942 - 98) / 2 s = 4.942 / 2 s = 2.471 (rounded to three decimal places) Therefore, the solution to the system of linear equations is: m ≈ 1.634 s ≈ 2.471 ### Answer: m ≈ 1.634, s ≈ 2.471
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From equation 1, let's solve for b: (6 * b) + (5 * k) = 105 6b = 105 - 5k b = (105 - 5k) / 6 b = 17.5 - (5/6)k Now, substitute this expression for b into equation 2: (95 * k) - (51 * (17.5 - (5/6)k)) = 67 95k - 892.5 + (255/6)k = 67 95k + (255/6)k - 892.5 = 67 (570/6)k + (255/6)k = 67 + 892.5 (825/6)k = 959.5 k = 959.5 / (825/6) k = 959.5 * (6/825) k = 6.98 Now that we have the value of k, we can substitute it back into the expression for b: b = 17.5 - (5/6)k b = 17.5 - (5/6) * 6.98 b = 17.5 - 5.82 b = 11.68 Therefore, the values of b and k are: b ≈ 11.68 k ≈ 6.98 ### Answer: b ≈ 11.68, k ≈ 6.98
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 Opus:
To solve for d and u, we can use the substitution method. Let's solve for u in the second equation and substitute it into the first equation to solve for d. From the second equation, 32u + 40d = 83, we get: 32u = 83 - 40d u = (83 - 40d) / 32 u = (83/32) - (40/32)d u = 2.59375 - 1.25d Substituting this expression for u into the first equation: 33d + 62(2.59375 - 1.25d) = 114 33d + 160.8125 - 77.5d = 114 -44.5d + 160.8125 = 114 -44.5d = -46.8125 d = (-46.8125) / (-44.5) d = 1.052 Now that we have the value of d, we can substitute it back into the expression for u: u = 2.59375 - 1.25d u = 2.59375 - 1.25(1.052) u = 2.59375 - 1.315 u = 1.27875 Therefore, the values of d and u are: d = 1.052 u = 1.27875 ### Answer: d = 1.052, u = 1.27875
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, let's solve for c: 5c + 28p = 102 5c = 102 - 28p c = (102 - 28p) / 5 Now, substitute this expression for c into the second equation: 20 * ((102 - 28p) / 5) - 67p = 13 Simplify: (2040 - 560p) / 5 - 67p = 13 2040/5 - 560p/5 - 67p = 13 408 - 112p - 67p = 13 408 - 179p = 13 Solve for p: -179p = 13 - 408 -179p = -395 p = -395 / -179 p = 395/179 Now that we have the value of p, we can substitute it back into the equation for c: c = (102 - 28p) / 5 c = (102 - 28 * (395/179)) / 5 c = (102 - 61.8435754) / 5 c = 40.1564246 / 5 c = 8.0312849 Therefore, the solutions are: p = 395/179 ≈ 2.2067039 c = 40.1564246 / 5 ≈ 8.0312849 ### Answer: p ≈ 2.2067039, c ≈ 8.0312849
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 Opus:
To solve this system of linear equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, let's solve for r: 58 * r + 12 * q = 147 58 * r = 147 - 12 * q r = (147 - 12 * q) / 58 Now, substitute the expression for r into the second equation: 97 * q - 76 * r = 76 97 * q - 76 * ((147 - 12 * q) / 58) = 76 Simplify the equation: 97 * q - 76 * (147 - 12 * q) / 58 = 76 97 * q - (76 * 147 - 76 * 12 * q) / 58 = 76 97 * q - (11172 - 912 * q) / 58 = 76 Multiply both sides by 58: 97 * 58 * q - 11172 + 912 * q = 76 * 58 5626 * q - 11172 + 912 * q = 4408 6538 * q - 11172 = 4408 6538 * q = 15580 q = 15580 / 6538 q = 2.38 Now that we have the value of q, we can substitute it back into the expression for r: r = (147 - 12 * q) / 58 r = (147 - 12 * 2.38) / 58 r = (147 - 28.56) / 58 r = 118.44 / 58 r = 2.04 Therefore, the values of r and q are: r = 2.04 q = 2.38 ### Answer: r = 2.04, q = 2.38
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 Opus:
To solve the given system of simultaneous linear equations, we can use the substitution method or the elimination method. Let's use the substitution method here. Given equations: 47 * v - 3 * k = 10 ... (1) 73 * v + 2 * k = 128 ... (2) Step 1: Solve equation (1) for k in terms of v. 47 * v - 3 * k = 10 -3 * k = -47 * v + 10 k = (47 * v - 10) / 3 ... (3) Step 2: Substitute the expression for k from equation (3) into equation (2). 73 * v + 2 * ((47 * v - 10) / 3) = 128 73 * v + (94 * v - 20) / 3 = 128 219 * v + 94 * v - 20 = 384 313 * v - 20 = 384 313 * v = 404 v = 404 / 313 ... (4) Step 3: Substitute the value of v from equation (4) into equation (3) to find k. k = (47 * (404 / 313) - 10) / 3 k = (18988 / 313 - 10) / 3 k = (18978 / 313) / 3 k = 6058 / 313 Step 4: Simplify the values of v and k. v = 404 / 313 ≈ 1.29 ... (5) k = 6058 / 313 ≈ 19.35 ... (6) Therefore, the solution to the given system of simultaneous linear equations is: v ≈ 1.29 k ≈ 19.35 ### Answer: v ≈ 1.29, k ≈ 19.35
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 Opus:
Let's solve this system of equations step by step using the substitution method. Given equations: 1. 89p - 41q = 60 2. 27p + 91q = 124 Step 1: Solve equation 1 for p in terms of q. 89p - 41q = 60 89p = 60 + 41q p = (60 + 41q) / 89 Step 2: Substitute the expression for p into equation 2. 27p + 91q = 124 27((60 + 41q) / 89) + 91q = 124 Step 3: Simplify the equation and solve for q. (1620 + 1107q) / 89 + 91q = 124 1620 + 1107q + 8099q = 11036 9206q = 9416 q = 9416 / 9206 q = <<9416/9206=1.0228>>1.0228 Step 4: Substitute the value of q into the expression for p from step 1. p = (60 + 41q) / 89 p = (60 + 41 × 1.0228) / 89 p = 101.9348 / 89 p = <<101.9348/89=1.1453>>1.1453 Therefore, the values of p and q are: ### Answer: p = 1.1453, q = 1.0228
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 Opus:
To solve for u and d, we can use the substitution method. Let's solve for u in the first equation and substitute it into the second equation to solve for d. First equation: (88 * u) - (96 * d) = 12 88u - 96d = 12 88u = 96d + 12 u = (96d + 12) / 88 u = (12d + 1.5) Now, substitute the expression for u into the second equation: Second equation: (27 * u) + (36 * d) = 71 27 * (12d + 1.5) + 36d = 71 324d + 40.5 + 36d = 71 360d + 40.5 = 71 360d = 30.5 d = 30.5 / 360 d = 0.0847222... Now that we have the value of d, we can substitute it back into the expression for u: u = (12d + 1.5) u = (12 * 0.0847222... + 1.5) u = 2.5166666... Therefore, the values of u and d that make both equations true are approximately: d ≈ 0.0847 u ≈ 2.5167 ### Answer: u ≈ 2.5167, d ≈ 0.0847
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 Opus:
To solve the system of equations, we can use the substitution method. Given equations: 1. 20 * n - 18 * f = 9 2. 3 * n - 2 * f = 70 Step 1: Solve equation 2 for n in terms of f. 3 * n - 2 * f = 70 3 * n = 70 + 2 * f n = (70 + 2 * f) / 3 Step 2: Substitute the expression for n into equation 1. 20 * ((70 + 2 * f) / 3) - 18 * f = 9 (1400 + 40 * f) / 3 - 18 * f = 9 1400 / 3 + 40 * f / 3 - 18 * f = 9 1400 / 3 + 40 * f / 3 - 54 * f / 3 = 9 1400 / 3 - 14 * f = 9 1400 / 3 - 9 = 14 * f 1400 / 3 - 9 = 14 * f (1400 - 27) / 3 = 14 * f 1373 / 3 = 14 * f 1373 / 42 = f f ≈ 32.69 Step 3: Substitute the value of f into the expression for n. n = (70 + 2 * 32.69) / 3 n = (70 + 65.38) / 3 n = 135.38 / 3 n ≈ 45.13 ### Answer: f ≈ 32.69, n ≈ 45.13
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, let's solve for l: 7 * l + 97 * j = 133 7 * l = 133 - 97 * j l = (133 - 97 * j) / 7 Now, substitute the expression for l into the second equation: 22 * j + 37 * ((133 - 97 * j) / 7) = 76 Simplify: 22 * j + (4921 - 3589 * j) / 7 = 76 154 * j + 4921 - 3589 * j = 532 4921 - 3435 * j = 532 -3435 * j = -4389 j = -4389 / -3435 j = 1.278 Now that we have the value of j, we can substitute it back into the equation for l: l = (133 - 97 * 1.278) / 7 l = (133 - 124) / 7 l = 9 / 7 l = 1.286 Therefore, the values of j and l are: j ≈ 1.278 l ≈ 1.286 ### Answer: j ≈ 1.278, l ≈ 1.286
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for d in the first equation and substitute it into the second equation to solve for f. 1. Solve for d in the first equation: 21d - 46f = 53 21d = 53 + 46f d = (53 + 46f) / 21 d = (53 + 46f) ÷ 21 2. Substitute the expression for d into the second equation: 8d + 84f = 133 8((53 + 46f) ÷ 21) + 84f = 133 (8(53 + 46f) ÷ 21) + 84f = 133 (424 + 368f) ÷ 21 + 84f = 133 (424 ÷ 21) + (368f ÷ 21) + 84f = 133 20.19 + 17.52f + 84f = 133 20.19 + 101.52f = 133 101.52f = 112.81 f = 112.81 ÷ 101.52 f ≈ 1.11 3. Substitute the value of f into the expression for d: d = (53 + 46f) ÷ 21 d = (53 + 46(1.11)) ÷ 21 d = (53 + 51.06) ÷ 21 d = 104.06 ÷ 21 d ≈ 4.96 Therefore, the values of d and f are approximately: 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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, 74u + 11w = 116, let's solve for u: 74u = 116 - 11w u = (116 - 11w) / 74 Now, substitute this expression for u into the second equation, 66u - 68w = 21: 66 * ((116 - 11w) / 74) - 68w = 21 Simplify: (66 * (116 - 11w)) / 74 - 68w = 21 (7656 - 726w) / 74 - 68w = 21 103.4594594594595 - 9.810810810810813w - 68w = 21 103.4594594594595 - 77.81081081081081w = 21 Subtract 103.4594594594595 from both sides: -77.81081081081081w = -82.4594594594595 Divide both sides by -77.81081081081081: w = 1.06 Now that we have the value of w, we can substitute it back into the equation for u: u = (116 - 11w) / 74 u = (116 - 11 * 1.06) / 74 u = 104.34 / 74 u = 1.41 Therefore, the values of u and w that satisfy the 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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 76 * f - 42 * j = 96 ... (1) 47 * j + 34 * f = 136 ... (2) Step 1: Solve equation (1) for f in terms of j. 76 * f - 42 * j = 96 76 * f = 96 + 42 * j f = (96 + 42 * j) / 76 f = (4 + 21 * j) / 38 ... (3) Step 2: Substitute the expression for f from equation (3) into equation (2). 47 * j + 34 * ((4 + 21 * j) / 38) = 136 Step 3: Solve the resulting equation for j. 47 * j + (136 + 714 * j) / 38 = 136 1786 * j + 136 + 714 * j = 5168 2500 * j = 5032 j = 5032 / 2500 j = <<5032/2500=2.0128>>2.0128 Step 4: Substitute the value of j into equation (3) to find f. f = (4 + 21 * 2.0128) / 38 f = 46.2688 / 38 f = <<46.2688/38=1.2176>>1.2176 Therefore, the values of f and j that satisfy the given system of equations are: f = 1.2176 j = 2.0128 ### Answer: f = 1.2176, j = 2.0128
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 Opus:
To solve the system of equations, we can use the substitution method or the elimination method. Let's use the substitution method here. From the first equation, we can express u in terms of e: 68u + 32e = 174 68u = 174 - 32e u = (174 - 32e) / 68 Now, substitute this expression for u into the second equation: 63e - 69u = 1 63e - 69((174 - 32e) / 68) = 1 Simplify the equation: 63e - (69 × 174 - 69 × 32e) / 68 = 1 63e - (12006 - 2208e) / 68 = 1 4284e - 12006 + 2208e = 68 6492e - 12006 = 68 6492e = 12074 e = 12074 / 6492 Simplify the fraction: e = <<12074/6492=1.86>>1.86 Now that we have the value of e, we can substitute it back into the expression for u: u = (174 - 32e) / 68 u = (174 - 32 × 1.86) / 68 u = (174 - 59.52) / 68 u = 114.48 / 68 Simplify the fraction: u = <<114.48/68=1.68>>1.68 Therefore, the values of u and e are: u = 1.68 e = 1.86 ### Answer: u = 1.68, e = 1.86
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 Opus:
To solve this system of linear equations, we can use the substitution method. 1. Solve the first equation for p in terms of n: 57p + 4n = 129 57p = 129 - 4n p = (129 - 4n) / 57 2. Substitute the expression for p into the second equation: 24p + 39n = 154 24((129 - 4n) / 57) + 39n = 154 3. Simplify the equation: (24(129 - 4n) / 57) + 39n = 154 (3096 - 96n) / 57 + 39n = 154 3096/57 - 96n/57 + 39n = 154 54.3157895 - 1.6842105n + 39n = 154 37.3157895n + 54.3157895 = 154 37.3157895n = 99.6842105 n = 99.6842105 / 37.3157895 n = 2.67 4. Substitute n = 2.67 into the equation for p from step 1: p = (129 - 4n) / 57 p = (129 - 4(2.67)) / 57 p = (129 - 10.68) / 57 p = 118.32 / 57 p = 2.075 Therefore, the values that satisfy the system of linear equations are: p ≈ 2.075 n ≈ 2.67 ### Answer: p ≈ 2.075, n ≈ 2.67
Expected answer: {n: 1894/709, p: 4415/2127}
Correct? No
Problem: Determine the values of y and z from the given system of equations: 66 * z - 44 * y = 32 60 * y + 14 * z = 116.
Claude-3 Opus:
To solve the given system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 66 * z - 44 * y = 32 ... (1) 60 * y + 14 * z = 116 ... (2) Step 1: Solve equation (1) for y in terms of z. 66 * z - 44 * y = 32 -44 * y = -66 * z + 32 y = (66 * z - 32) / 44 y = (33 * z - 16) / 22 ... (3) Step 2: Substitute equation (3) into equation (2) and solve for z. 60 * ((33 * z - 16) / 22) + 14 * z = 116 (1980 * z - 960) / 22 + 14 * z = 116 1980 * z - 960 + 308 * z = 2552 2288 * z - 960 = 2552 2288 * z = 3512 z = 3512 / 2288 z = <<3512/2288=1.535>>1.535 Step 3: Substitute the value of z into equation (3) to solve for y. y = (33 * 1.535 - 16) / 22 y = (50.655 - 16) / 22 y = 34.655 / 22 y = <<34.655/22=1.575>>1.575 Therefore, the values of y and z are: z = 1.535 y = 1.575 ### Answer: y = 1.575, z = 1.535
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 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 1. 83t - 58h = 45 2. 69h + 11t = 124 Step 1: Solve the first equation for t in terms of h. 83t - 58h = 45 83t = 58h + 45 t = (58h + 45) / 83 Step 2: Substitute the expression for t into the second equation. 69h + 11((58h + 45) / 83) = 124 Step 3: Simplify the equation and solve for h. 69h + (638h + 495) / 83 = 124 5727h + 495 = 10292 5727h = 9797 h = 9797 / 5727 h = 1.71 Step 4: Substitute the value of h into the expression for t from Step 1. t = (58(1.71) + 45) / 83 t = 143.98 / 83 t = 1.73 Therefore, the solution to the system of equations is: h = 1.71 t = 1.73 ### Answer: h = 1.71, t = 1.73
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 Opus:
Let's solve this system of equations step by step using the substitution method. Given equations: 1. ((79 * h) - (61 * o)) = 9 2. ((16 * h) - (12 * o)) = 9 Step 1: Simplify equation 2 to isolate h in terms of o. ((16 * h) - (12 * o)) = 9 16h - 12o = 9 16h = 12o + 9 h = (12o + 9) / 16 h = (3o + 9/4) / 4 h = (3/4)o + 9/16 Step 2: Substitute the expression for h into equation 1. ((79 * h) - (61 * o)) = 9 (79 * ((3/4)o + 9/16)) - (61 * o) = 9 (237/4)o + 711/16 - 61o = 9 Step 3: Simplify the equation and solve for o. (237/4)o - 61o + 711/16 = 9 (237/4 - 61)o + 711/16 = 9 (237/4 - 244/4)o + 711/16 = 9 (-7/4)o + 711/16 = 9 -7o/4 = 9 - 711/16 -7o/4 = (144 - 711)/16 -7o/4 = -567/16 -7o = -567/4 o = 81/7 Step 4: Substitute the value of o into the expression for h to find its value. h = (3/4)o + 9/16 h = (3/4)(81/7) + 9/16 h = 243/28 + 9/16 h = (243/28 + 63/112) h = (243 + 63)/28 h = 306/28 h = 153/14 Therefore, the values of o and h are: o = 81/7 h = 153/14 ### Answer: o = 81/7, h = 153/14
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 Opus:
To solve for variables s and y in the given system of equations, we can use the substitution method or the elimination method. Let's use the substitution method here. Given equations: 4y + 56s = 139 ... (1) 65y - 90s = 16 ... (2) Step 1: Solve equation (1) for y in terms of s. 4y + 56s = 139 4y = 139 - 56s y = (139 - 56s) ÷ 4 y = 34.75 - 14s ... (3) Step 2: Substitute the expression for y from equation (3) into equation (2). 65(34.75 - 14s) - 90s = 16 2258.75 - 910s - 90s = 16 2258.75 - 1000s = 16 -1000s = 16 - 2258.75 -1000s = -2242.75 s = 2.24275 Step 3: Substitute the value of s into equation (3) to find y. y = 34.75 - 14(2.24275) y = 34.75 - 31.3985 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 Opus:
To solve the pair of simultaneous equations, we can use the substitution method. Given equations: 42 * d - 4 * z = 75 ... (1) 40 * d + 45 * z = 162 ... (2) Step 1: Solve equation (1) for d in terms of z. 42 * d - 4 * z = 75 42 * d = 75 + 4 * z d = (75 + 4 * z) / 42 d = 75/42 + (4/42) * z d = 25/14 + (1/21) * z ... (3) Step 2: Substitute the expression for d from equation (3) into equation (2). 40 * (25/14 + (1/21) * z) + 45 * z = 162 1000/14 + 40/21 * z + 45 * z = 162 1000/14 + (40/21 + 45) * z = 162 1000/14 + (200/21 + 945/21) * z = 162 1000/14 + 1145/21 * z = 162 Step 3: Solve the resulting equation for z. 1000/14 + 1145/21 * z = 162 1145/21 * z = 162 - 1000/14 1145/21 * z = (3402 - 1500)/42 1145/21 * z = 1902/42 z = (1902/42) / (1145/21) z = (1902/42) * (21/1145) z = 1902 / (42 * 1145) z = 1902 / 48090 z = 1/253 Step 4: Substitute the value of z into equation (3) to find the value of d. d = 25/14 + (1/21) * (1/253) d = 25/14 + 1/5313 d = (132825 + 14) / (14 * 5313) d = 132839 / 74382 d = 33/18579 Therefore, the values of d and z are: z = 1/253 d = 33/18579 ### Answer: z = 1/253, d = 33/18579
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 Opus:
Let's solve this step by step using the substitution method. 1. From the first equation, express 'o' in terms of 'a': (42 * o) + (24 * a) = 142 42o = 142 - 24a o = (142 - 24a) / 42 o = 142/42 - 24a/42 o = 3.380952381 - 0.5714285714a 2. Substitute the expression for 'o' into the second equation: (42 * a) + (6 * (3.380952381 - 0.5714285714a)) = 75 42a + 20.28571429 - 3.428571429a = 75 38.57142857a + 20.28571429 = 75 38.57142857a = 54.71428571 a = 54.71428571 / 38.57142857 a = 1.419354839 3. Substitute the value of 'a' back into the expression for 'o': o = 3.380952381 - 0.5714285714a o = 3.380952381 - 0.5714285714 * 1.419354839 o = 3.380952381 - 0.8110749185 o = 2.569877463 Therefore, the values of 'a' and 'o' are: a ≈ 1.419354839 o ≈ 2.569877463 ### Answer: a ≈ 1.419354839, o ≈ 2.569877463
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation. From the first equation, 39*r - 2*p = 45, let's solve for p: 39*r - 2*p = 45 -2*p = -39*r + 45 p = (39*r - 45) / 2 Now, substitute this expression for p into the second equation: 16*p - 84*r = 36 16*((39*r - 45) / 2) - 84*r = 36 (16*(39*r - 45)) / 2 - 84*r = 36 (624*r - 720) / 2 - 84*r = 36 312*r - 360 - 84*r = 36 228*r - 360 = 36 228*r = 396 r = 396 / 228 r = <<396/228=1.7368421052631578>>1.7368421052631578 Now that we have the value of r, we can substitute it back into the expression for p: p = (39*r - 45) / 2 p = (39*1.7368421052631578 - 45) / 2 p = (67.73684210526316 - 45) / 2 p = 22.73684210526316 / 2 p = <<22.73684210526316/2=11.36842105263158>>11.36842105263158 ### Answer: p = 11.36842105263158, r = 1.7368421052631578
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 Opus:
To solve for the values of 'd' and 'w', we can use the substitution method. Let's solve for 'd' in terms of 'w' using the first equation, then substitute this expression into the second equation to solve for 'w'. From equation 1: (23 * d) - (52 * w) = 7 23d - 52w = 7 23d = 52w + 7 d = (52w + 7) / 23 d = (52/23)w + (7/23) Now, substitute this expression for 'd' into equation 2: 64w - 8((52/23)w + (7/23)) = 58 64w - (416/23)w - (56/23) = 58 (1472/23)w - (416/23)w = 58 + (56/23) (1056/23)w = (1398/23) w = (1398/1056) w = <<1398/1056=1.323863636363636>>1.32 Now that we have the value for 'w', we can substitute it back into the expression for 'd': d = (52/23)w + (7/23) d = (52/23)(1.32) + (7/23) d = 2.98 + 0.30 d = <<2.98+0.30=3.28>>3.28 ### Answer: d = 3.28, w = 1.32
Expected answer: {d: 433/132, w: 695/528}
Correct? No
Problem: Solve the following system of equations for the values of n and z: (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19.
Claude-3 Opus:
To solve the system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation. From the first equation, let's solve for n: (71 * z) + (60 * n) = 184 60 * n = 184 - (71 * z) n = (184 - (71 * z)) / 60 Now, substitute the expression for n into the second equation: (52 * z) - (47 * ((184 - (71 * z)) / 60)) = 19 Simplify: (52 * z) - (47 * (184 - (71 * z))) / 60 = 19 (52 * z) - ((8648 - (3337 * z)) / 60) = 19 (52 * z) - (144.1333... - (55.6166... * z)) = 19 (52 * z) - 144.1333... + (55.6166... * z) = 19 (107.6166... * z) - 144.1333... = 19 (107.6166... * z) = 163.1333... z = 163.1333... / 107.6166... z = 1.5161290322580645 Now that we have the value of z, we can substitute it back into the expression for n: n = (184 - (71 * 1.5161290322580645)) / 60 n = (184 - 107.64516129032258) / 60 n = 76.35483870967742 / 60 n = 1.2725806451612903 Therefore, the solution to the system of equations is: z ≈ 1.5161290322580645 n ≈ 1.2725806451612903 ### Answer: z ≈ 1.5161290322580645, n ≈ 1.2725806451612903
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 Opus:
To solve the given system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable. Given equations: 70 * e + 90 * z = 189 ... (1) 99 * z - 84 * e = 20 ... (2) Step 1: Solve equation (1) for z in terms of e. 70 * e + 90 * z = 189 90 * z = 189 - 70 * e z = (189 - 70 * e) / 90 z = 2.1 - (7/9) * e ... (3) Step 2: Substitute the expression for z from equation (3) into equation (2). 99 * (2.1 - (7/9) * e) - 84 * e = 20 207.9 - 77 * e - 84 * e = 20 207.9 - 161 * e = 20 -161 * e = -187.9 e = 187.9 / 161 e ≈ 1.167 ... (4) Step 3: Substitute the value of e from equation (4) into equation (3) to find z. z = 2.1 - (7/9) * 1.167 z = 2.1 - 0.907 z ≈ 1.193 ... (5) Therefore, the values of e and z are approximately: e ≈ 1.167 z ≈ 1.193 ### Answer: e ≈ 1.167, z ≈ 1.193
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 Opus:
To solve this system of equations, we can use the substitution method. Let's solve equation (1) for o in terms of f, then substitute the expression for o into equation (2) to solve for f. Finally, we'll substitute the value of f back into equation (1) to find o. From equation (1), we have: 26o - 44f = 95 26o = 44f + 95 o = (44f + 95) / 26 o = (44/26)f + 95/26 Now, let's substitute this expression for o into equation (2): 87f + 11((44/26)f + 95/26) = 169 87f + (484/26)f + 1045/26 = 169 (87 + 484/26)f = 169 - 1045/26 (2262/26 + 484/26)f = 169 - 1045/26 (2746/26)f = 169 - 1045/26 f = (169 - 1045/26) / (2746/26) f = (4394 - 1045) / 2746 f = 3349 / 2746 f = <<3349/2746=1.22>>1.22 Now that we have the value of f, let's substitute it back into the equation for o: o = (44/26)f + 95/26 o = (44/26)(1.22) + 95/26 o = <<(44/26)*(1.22)+(95/26)=5.32>>5.32 Therefore, the values that satisfy the system of equations are: f = 1.22 o = 5.32 ### Answer: o = 5.32, 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 Opus:
To solve for z and p, we can use the substitution method. Let's solve for z in the first equation and substitute it into the second equation to solve for p. 1. Solve for z in the first equation: 40*z - 20*p = 51 40*z = 51 + 20*p z = (51 + 20*p) / 40 z = 51/40 + (20/40)*p z = 1.275 + 0.5*p 2. Substitute z into the second equation: 6*z + 58*p = 165 6*(1.275 + 0.5*p) + 58*p = 165 7.65 + 3*p + 58*p = 165 7.65 + 61*p = 165 61*p = 165 - 7.65 61*p = 157.35 p = 157.35 / 61 p = 2.58 3. Substitute p back into the equation for z: z = 1.275 + 0.5*p z = 1.275 + 0.5*2.58 z = 1.275 + 1.29 z = 2.565 Therefore, the values of z and p are: p = 2.58 z = 2.565 ### Answer: z = 2.565, p = 2.58
Expected answer: {p: 3147/1220, z: 3129/1220}
Correct? No
Problem: Find the values of s and f from the given system of equations: (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93.
Claude-3 Opus:
To solve for s and f, we can use the substitution method. Let's solve for one variable in terms of the other using the second equation and then substitute it into the first equation. From the second equation, we can solve for f in terms of s: (3 * f) + (10 * s) = 93 3f = 93 - 10s f = (93 - 10s) / 3 f = 31 - (10s / 3) Now, let's substitute this expression for f into the first equation: (71 * s) - (89 * f) = 5 (71 * s) - (89 * (31 - (10s / 3))) = 5 (71 * s) - (2759 - (890s / 3)) = 5 (71 * s) - 2759 + (890s / 3) = 5 71s + (890s / 3) = 2764 213s + 890s = 8292 1103s = 8292 s = 8292 / 1103 s = <<8292/1103=752/103>>752/103 Now that we have the value of s, we can substitute it back into the expression for f: f = 31 - (10s / 3) f = 31 - (10 * (752/103) / 3) f = 31 - (7520/309) f = (9579/309) - (7520/309) f = 2059/309 Therefore, the values of s and f are: s = 752/103 f = 2059/309 ### Answer: s = 752/103, f = 2059/309
Expected answer: {f: 6553/1103, s: 8292/1103}
Correct? No