Gemini-1.5 Pro 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
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. We will eliminate 'q' from the equations. 1. **Multiply Equation 1 by 4:** (40 * s * 4) - (91 * q * 4) = 37 * 4 This simplifies to: 160s - 364q = 148 2. **Multiply Equation 2 by 31:** (30 * s * 31) + (12 * q * 31) = 121 * 31 This simplifies to: 930s + 372q = 3751 3. **Add the modified equations together:** (160s - 364q) + (930s + 372q) = 148 + 3751 This simplifies to: 1090s = 3899 4. **Solve for 's':** s = 3899 / 1090 s = 3.577 5. **Substitute the value of 's' back into either original equation to solve for 'q'. Let's use Equation 1:** (40 * 3.577) - (91 * q) = 37 143.08 - (91 * q) = 37 - (91 * q) = 37 - 143.08 - (91 * q) = -106.08 q = -106.08 / -91 q = 1.166 Therefore, the solution to the system of equations is s = 3.577 and q = 1.166. ### Answer: s = 3.577, q = 1.166
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. Multiply the first equation by 6 and the second equation by 4: (1) 24b + 264m = 1080 (2) 316m - 24b = 192 Add the two equations together to eliminate b: 580m = 1272 Solve for m: m = 1272 / 580 = 2.2 Substitute the value of m back into either original equation to solve for b. Let's use the first equation: 4b + 44 * 2.2 = 180 4b + 96.8 = 180 4b = 83.2 b = 20.8 ### Answer: b = 20.8, m = 2.2
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.
Gemini-1.5 Pro:
Reasoning:
To solve for two unknowns, we need two equations. This system of equations can be solved using elimination.
1. **Multiply the equations to match a coefficient:** Let's match the 'g' coefficients.
- Multiply the first equation by 29:
(42 * o - 15 * g = 28) * 29 => 1218 * o - 435 * g = 812
- Multiply the second equation by 15:
(47 * o + 29 * g = 141) * 15 => 705 * o + 435 * g = 2115
2. **Add the two new equations:** Notice the 'g' coefficients now cancel out.
(1218 * o - 435 * g = 812) + (705 * o + 435 * g = 2115) => 1923 * o = 2927
3. **Solve for 'o':** Divide both sides by 1923.
o = 2927 / 1923 = 1.5221... ≈ 1.52
4. **Substitute 'o' back into either original equation to solve for 'g':** Let's use the first original equation.
42 * 1.52 - 15 * g = 28
63.84 - 15 * g = 28
-15 * g = -35.84
g = -35.84 / -15 = 2.3893... ≈ 2.39
### Answer: o ≈ 1.52 and g ≈ 2.39
Expected answer: {g: 4606/1923, o: 2927/1923}
Correct? No
Problem: Given the following system of equations, determine the values of v and o: The first equation is: ((12 * v) - (6 * o)) = 21 The second equation is: ((76 * v) - (45 * o)) = 50.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination.
1. **Multiply equations to match coefficients:** We'll multiply the first equation by -6 and the second equation by 1 to make the 'o' coefficients cancel out:
- First equation * -6: (-72v + 36o) = -126
- Second equation * 1: (76v - 45o) = 50
2. **Add the equations together:**
(-72v + 36o) + (76v - 45o) = -126 + 50
This simplifies to: 4v - 9o = -76
3. **Solve for 'v':**
4v = 9o - 76
v = (9o - 76) / 4
4. **Substitute 'v' into one of the original equations:** Let's use the first original equation:
12 * ((9o - 76) / 4) - 6o = 21
5. **Simplify and solve for 'o':**
27o - 228 - 6o = 21
21o = 249
o = 249 / 21
o = 11.86 (approximately)
6. **Substitute the value of 'o' back into the equation for 'v':**
v = (9 * 11.86 - 76) / 4
v = (106.74 - 76) / 4
v = 30.74 / 4
v = 7.69 (approximately)
### Answer: v ≈ 7.69, 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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. We can multiply the first equation by -2 and the second equation by 75 to make the coefficients of 'f' have opposite signs: -2 * (75f + 25g) = -2 * 138 -> -150f - 50g = -276 75 * (2f + 90g) = 75 * 143 -> 150f + 6750g = 10725 Now, we can add the two equations together: (-150f - 50g) + (150f + 6750g) = -276 + 10725 This simplifies to: 6700g = 10449 Dividing both sides by 6700 gives us: g = 10449 / 6700 = 1.56 Now that we know the value of g, we can substitute it back into either of the original equations to solve for f. Let's use the first equation: 75f + 25 * 1.56 = 138 75f + 39 = 138 75f = 99 f = 99 / 75 = 1.32 Therefore, the solution to the system of equations is f = 1.32 and g = 1.56. ### Answer: f = 1.32, g = 1.56
Expected answer: {f: 1769/1340, g: 10449/6700}
Correct? No
Problem: Solve the following system of equations for variables y and n: the first equation is 194 = 4 * y + 14 * n and the second equation is 7 = 65 * n - 79 * y.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the substitution method. Step 1: Solve the first equation for 'y'. 194 = 4y + 14n 4y = 194 - 14n y = (194 - 14n) / 4 Step 2: Substitute this expression for 'y' into the second equation. 7 = 65n - 79 * ((194 - 14n) / 4) Step 3: Simplify and solve for 'n'. 7 = 65n - (15326 - 1106n) / 4 28 = 260n - 15326 + 1106n 15354 = 1366n n = 15354 / 1366 = 11.24 Step 4: Substitute the value of 'n' back into the equation for 'y'. y = (194 - 14 * 11.24) / 4 y = (194 - 157.36) / 4 y = 36.64 / 4 y = 9.16 ### Answer: y = 9.16, n = 11.24
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.
Gemini-1.5 Pro:
Reasoning: Let's solve for *l* in the first equation: 146 = 28 * g + 26 * l 146 - 28 * g = 26 * l (146 - 28 * g) / 26 = l Let's substitute this expression for *l* into the second equation: 6 = 41 * l - 53 * g 6 = 41 * ((146 - 28 * g) / 26) - 53 * g 6 = (5986 - 1148 * g) / 26 - 53 * g 156 = 5986 - 1148 * g - 1378 * g 156 = 5986 - 2526 * g -5830 = -2526 * g 5830 = 2526 * g 5830 / 2526 = g 2.31 = g Now, we'll substitute the value of g back into the equation we derived for *l*: l = (146 - 28 * g) / 26 l = (146 - 28 * 2.31) / 26 l = (146 - 64.68) / 26 l = 81.32 / 26 l = 3.13 Therefore, l = 3.13 and g = 2.31. ### Answer: l = 3.13 and g = 2.31
Expected answer: {g: 2915/1263, l: 3953/1263}
Correct? No
Problem: Consider the following system of equations: 1) 20 * p + 56 * y = 90 2) 89 * p + 7 * y = 98 Resolve this system and find the values of p and y.
Gemini-1.5 Pro:
Reasoning: To solve for *p* and *y*, we'll use the elimination method. First, let's multiply Equation 1 by -7 and Equation 2 by 56 to make the coefficients of *y* opposites: 1) -7 * (20p + 56y) = -7 * 90 => -140p - 392y = -630 2) 56 * (89p + 7y) = 56 * 98 => 4984p + 392y = 5488 Now, add the two equations together: (-140p - 392y) + (4984p + 392y) = -630 + 5488 This simplifies to: 4844p = 4858 Divide both sides by 4844 to solve for *p*: p = 4858 / 4844 = 62/61 Now, substitute the value of *p* back into either Equation 1 or Equation 2 to solve for *y*. Let's use Equation 1: 20 * (62/61) + 56y = 90 Simplify and solve for *y*: 1240/61 + 56y = 90 56y = 90 - 1240/61 56y = 4250/61 y = (4250/61) / 56 y = 2125/1708 ### Answer: p = 62/61, y = 2125/1708
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.
Gemini-1.5 Pro:
Reasoning: Let's solve for 'd' in the second equation: 6d = 96 - 69k d = (96 - 69k) / 6 d = 16 - (23/2)k Substitute this value of 'd' into the first equation: 72k - 22 * (16 - (23/2)k) = 17 72k - 352 + 253k = 17 325k = 369 k = 369 / 325 k = 1.135 Now, substitute the value of 'k' back into the equation for 'd': d = 16 - (23/2) * 1.135 d = 16 - 13.05 d = 2.95 Therefore: k = 1.135 and d = 2.95 ### Answer: k = 1.135, d = 2.95
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
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method.
1. **Multiply the equations to make the coefficients of one variable opposites.**
- Multiply the first equation by 10:
730h - 330d = 480
- Multiply the second equation by 33:
330d + 462h = 3003
2. **Add the two equations together to eliminate 'd':**
(730h - 330d) + (330d + 462h) = 480 + 3003
1192h = 3483
3. **Solve for 'h':**
h = 3483 / 1192 = 2.925
4. **Substitute the value of 'h' back into either original equation to solve for 'd'. Let's use the second equation:**
10d + 14 * 2.925 = 91
10d + 40.95 = 91
10d = 50.05
d = 5.005
Therefore, d = 5.005 and h = 2.925
### Answer: d = 5.005, h = 2.925
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
Gemini-1.5 Pro:
Reasoning: We can use the elimination method to solve for *q* and *e*. First, we can multiply the top equation by 30 and the bottom equation by -43 to make the *q* coefficients have opposite signs: 1) 30 * (43q - 44e = 29) ---> 1290q - 1320e = 870 2) -43 * (30q + 43e = 130) ---> -1290q - 1849e = -5590 Adding the two equations together, we get: -3169e = -4720 Dividing both sides by -3169, we find the value of *e*: e = -4720 / -3169 = 1.49 Now we can substitute this value of *e* back into either original equation to solve for *q*. Let's use the first equation: 43q - 44(1.49) = 29 Simplifying: 43q - 65.56 = 29 Adding 65.56 to both sides: 43q = 94.56 Dividing both sides by 43, we find the value of *q*: q = 94.56 / 43 = 2.2 ### Answer: q = 2.2 and 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))
Gemini-1.5 Pro:
Reasoning: Let's solve for *r* in the second equation: 109 = (37 * a) + (6 * r) 109 - (37 * a) = (6 * r) (109 - (37 * a))/6 = r Now, we can substitute this value of *r* into the first equation: 53 = ((89 * ((109 - (37 * a))/6 )) - (74 * a)) 53 = ((9701 - (3293 * a))/6) - (74 * a) 318 = 9701 - (3293 * a) - (444 * a) -9383 = -3737 * a a = 2.51 Now we can plug the value of *a* back into either equation to find *r*. Let’s use the second equation: 109 = ((37 * 2.51) + (6 * r)) 109 = 92.87 + (6 * r) 16.13 = (6 * r) r = 2.69 ### Answer: a = 2.51, r = 2.69
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.
Gemini-1.5 Pro:
Let's denote the weight of one unit of grain 'g' as 'x' and the weight of one unit of grain 'q' as 'y'. We can set up the following equations based on the given information: Equation 1: 42x + 12y = 169 Equation 2: 12y - 50x = 19 We can solve this system of equations using either substitution or elimination. We'll use elimination here: 1. **Multiply Equation 2 by -1:** This will allow us to eliminate 'y' when we add the equations together. -12y + 50x = -19 2. **Add the modified Equation 2 to Equation 1:** (42x + 12y) + (-12y + 50x) = 169 - 19 92x = 150 3. **Solve for 'x':** x = 150 / 92 x = 1.63 (approximately) 4. **Substitute the value of 'x' back into either Equation 1 or Equation 2 to solve for 'y'.** We'll use Equation 1: 42(1.63) + 12y = 169 68.46 + 12y = 169 12y = 100.54 y = 8.38 (approximately) Therefore, one unit of grain 'g' weighs approximately 1.63 kg, and one unit of grain 'q' weighs approximately 8.38 kg. Now, let's find the quantities 'g' and 'q': g = 42 units (given) q = 12 units (given) ### Answer: g = 42, q = 12
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
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. We can multiply the top equation by 62 and the bottom equation by -11 to make the coefficients of 'v' have opposite signs: (62)(37s - 11v = 54) -> 2294s - 682v = 3348 (-11)(82s - 62v = 41) -> -802s + 682v = -451 Now, we can add the two equations together, effectively eliminating 'v': (2294s - 682v) + (-802s + 682v) = 3348 - 451 This simplifies to: 1492s = 2897 Dividing both sides by 1492, we find the value of 's': s = 2897 / 1492 = 1.941 Now, we can substitute this value of 's' back into either of the original equations to solve for 'v'. Let's use the first equation: 37 * 1.941 - 11 * v = 54 Simplifying: 71.817 - 11v = 54 Subtracting 71.817 from both sides: -11v = -17.817 Finally, dividing both sides by -11 gives us the value of 'v': v = -17.817 / -11 = 1.6197 ### Answer: s = 1.941, v = 1.6197
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.
Gemini-1.5 Pro:
Reasoning: Let's solve for *z* in the second equation: 78z - 48c = 48 78z = 48 + 48c z = (48 + 48c) / 78 z = (8 + 8c) / 13 Now let's substitute *z* in the first equation: 41c + 65 * ((8 + 8c) / 13) = 151 41c + 5 * (8 + 8c) = 151 41c + 40 + 40c = 151 81c = 111 c = 111 / 81 c = 37 / 27 Now we can find the value of z: z = (8 + 8 * (37/27)) / 13 z = (8 + 296/27) / 13 z = (512 / 27) / 13 z = 512 / 351 Therefore, c = 37/27 and z = 512/351 ### Answer: c = 37/27 and z = 512/351
Expected answer: {c: 37/27, z: 512/351}
Correct? Yes
Problem: Given the following system of equations: 60 * c - 91 * i = 7 7 * i + 76 * c = 148 What are the values of c and i?
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the substitution method. 1. **Solve for 'i' in the second equation:** 7i + 76c = 148 7i = 148 - 76c i = (148 - 76c) / 7 2. **Substitute this value of 'i' into the first equation:** 60c - 91 * ((148 - 76c) / 7) = 7 3. **Simplify and solve for 'c':** 60c - (91 * 148)/7 + (91 * 76c)/7 = 7 420c - 1908 + 6836c = 49 7256c = 1957 c = 1957 / 7256 c = 0.27 4. **Substitute the value of 'c' back into either of the original equations to solve for 'i'. Let's use the second equation:** 7i + 76 * 0.27 = 148 7i + 20.52 = 148 7i = 127.48 i = 127.48 / 7 i = 18.21 Therefore, the values of c and i are approximately 0.27 and 18.21, respectively. ### Answer: c ≈ 0.27, i ≈ 18.21
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
Gemini-1.5 Pro:
Reasoning: Let's solve for *e* in equation (2): 1) Isolate *e* in equation (2): 41*e = 184 - 76*v 2) Divide both sides by 41: e = (184 - 76*v)/41 Now, substitute this expression for *e* into equation (1): 3) Substitute: 71*v - 37*((184 - 76*v)/41) = 69 Simplify and solve for *v*: 4) Multiply both sides by 41: 2911*v - 37*(184 - 76*v) = 2829 5) Expand: 2911*v - 6808 + 2812*v = 2829 6) Combine like terms: 5723*v = 9637 7) Divide both sides by 5723: v = 1.684 Substitute the value of *v* back into either equation (1) or (2) to solve for *e*. Let's use equation (1): 8) Substitute *v* into equation (1): 71*(1.684) - 37*e = 69 9) Simplify: 119.564 - 37*e = 69 10) Isolate *e*: -37*e = -50.564 11) Divide both sides by -37: e = 1.367 Therefore, the solution to the system of equations is *e* = 1.367 and *v* = 1.684. ### Answer: e = 1.367, v = 1.684
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.
Gemini-1.5 Pro:
Reasoning: Let's represent the given information as equations: Equation 1: 25v + 69d = 113 Equation 2: 42v + 26d = 85 We can use the elimination method to solve for 'd' and 'v'. Let's multiply Equation 1 by -26 and Equation 2 by 69 to make the coefficients of 'd' have opposite signs: -26 * (25v + 69d = 113) => -650v - 1794d = -2938 69 * (42v + 26d = 85) => 2898v + 1794d = 5865 Now, add the two equations together: 2248v = 2927 Solving for 'v': v = 2927 / 2248 = 1.3 Substitute the value of 'v' (1.3) into either Equation 1 or Equation 2 to solve for 'd'. Let's use Equation 1: 25(1.3) + 69d = 113 32.5 + 69d = 113 69d = 80.5 d = 80.5 / 69 = 1.17 ### Answer: d = 1.17, v = 1.3
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
Gemini-1.5 Pro:
Reasoning:
To solve for g and z, we can use a method called elimination.
1. **Multiply equations to match coefficients:** We'll multiply the first equation by 3 and the second equation by 43 to make the coefficients of 'z' have opposite signs:
- (43 * z - 91 * g = 33) * 3 => 129z - 273g = 99
- (72 * g - 3 * z = 68) * 43 => 3096g - 129z = 2924
2. **Add the equations:** Notice that the 'z' terms now cancel out:
- (129z - 273g = 99) + (3096g - 129z = 2924) => 2823g = 3023
3. **Solve for 'g':**
- g = 3023 / 2823 = 1.07
4. **Substitute 'g' back into either original equation:** Let's use the first original equation:
- 43z - 91(1.07) = 33
- 43z - 97.37 = 33
- 43z = 130.37
5. **Solve for 'z':**
- z = 130.37 / 43 = 3.03
Therefore, the solutions are g = 1.07 and z = 3.03
### Answer: g = 1.07, z = 3.03
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.
Gemini-1.5 Pro:
Reasoning: Let's solve for 'c' in the first equation: 50c = 92 - 12f c = (92 - 12f) / 50. Substitute this value of 'c' into the second equation: 36 * [(92 - 12f) / 50] - 21f = 34. Simplify and solve for 'f': (3312 - 432f) / 50 - 21f = 34 3312 - 432f - 1050f = 1700 -1482f = -1612 f = -1612 / -1482 = 1.09 (approximately). Now, substitute the value of 'f' back into the equation for 'c': c = (92 - 12 * 1.09) / 50 c = (92 - 13.08) / 50 c = 78.92 / 50 = 1.58 (approximately). Therefore, Alex had approximately 1.58 cherries and Brad had approximately 1.09 figs. ### Answer: c ≈ 1.58, f ≈ 1.09.
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
Gemini-1.5 Pro:
Reasoning: To solve for two unknowns, we need two equations, which we have. Let's use elimination to find the values of u and d. 1. **Multiply the equations to match coefficients:** We'll multiply the top equation by 85 and the bottom equation by 75 to make the 'd' coefficients have opposite signs: - (87u - 75d = 7) * 85 -> 7395u - 6375d = 595 - (85d - 82u = 54) * 75 -> -6150u + 6375d = 4050 2. **Add the equations together:** - (7395u - 6375d = 595) + (-6150u + 6375d = 4050) - This simplifies to: 1245u = 4645 3. **Solve for 'u':** - Divide both sides by 1245: - u = 4645 / 1245 - u = 3.73 (approximately) 4. **Substitute 'u' back into either original equation to solve for 'd':** Let's use the first original equation: - 87(3.73) - 75d = 7 - 324.51 - 75d = 7 - -75d = -317.51 - d = -317.51 / -75 - d = 4.23 (approximately) ### 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?
Gemini-1.5 Pro:
Reasoning: We can solve for the unknowns 'a' and 'i' using simultaneous equations. Let's represent the first equation as Equation 1: 75a - 4i = 42 and the second equation as Equation 2: 10i + 13a = 160. To solve for 'a', let's multiply Equation 1 by 5 and Equation 2 by 2: Equation 1 * 5: 375a - 20i = 210 Equation 2 * 2: 20i + 26a = 320 Now, add the modified equations together: (375a - 20i) + (20i + 26a) = 210 + 320 401a = 530 a = 530 / 401 a = 1.32 (approximately) Now, substitute the value of 'a' back into either Equation 1 or Equation 2 to solve for 'i'. Let's use Equation 1: 75 * 1.32 - 4i = 42 99 - 4i = 42 -4i = 42 - 99 -4i = -57 i = -57 / -4 i = 14.25 ### Answer: i = 14.25, a = 1.32
Expected answer: {a: 530/401, i: 5727/401}
Correct? No
Problem: Solve the following system of equations for the values of p and c: 37 * p + 74 * c = 183 27 * c - 18 * p = 13
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method.
1. **Multiply equations to match coefficients:** We'll multiply the first equation by 18 and the second equation by 37 to match the coefficients of 'p':
- (37 * p + 74 * c = 183) * 18 -> 666p + 1332c = 3294
- (27 * c - 18 * p = 13) * 37 -> 999c - 666p = 481
2. **Add the equations:** Notice that the 'p' terms cancel out:
- 666p + 1332c = 3294
- -666p + 999c = 481
-----------------------
2331c = 3775
3. **Solve for 'c':** Divide both sides by 2331:
- c = 3775 / 2331 = 1.62
4. **Substitute 'c' back into either original equation to solve for 'p':** Let's use the first equation:
- 37p + 74 * 1.62 = 183
- 37p + 119.88 = 183
- 37p = 63.12
- p = 63.12 / 37 = 1.71
Therefore, the solution to the system of equations is p = 1.71 and c = 1.62
### Answer: p = 1.71, c = 1.62
Expected answer: {c: 3775/2331, p: 3979/2331}
Correct? No
Problem: Determine the values of k and p from the following system of equations: (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105.
Gemini-1.5 Pro:
Reasoning:
Let's solve this system of equations using the elimination method.
1. **Multiply equations to have opposite coefficients for one variable:**
- Multiply the first equation by 16: (768 * k) - (480 * p) = 592
- Multiply the second equation by 30: (480 * p) + (990 * k) = 3150
2. **Add the two equations together:**
1758 * k = 3742
3. **Solve for 'k':**
k = 3742 / 1758 = 2.13
4. **Substitute the value of 'k' back into either original equation to solve for 'p':**
Let's use the first original equation: (48 * 2.13) - (30 * p) = 37
5. **Simplify and solve for 'p':**
102.24 - 30p = 37
-30p = -65.24
p = -65.24 / -30 = 2.17
Therefore, k = 2.13 and p = 2.17
### Answer: k = 2.13, p = 2.17
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.
Gemini-1.5 Pro:
Reasoning: Let's solve for 'f' first. Multiply equation 1 by 46, and equation 2 by 69 to make the coefficients of 'f' have opposite signs: 1. (75v - 69f = 14) * 46 => 3450v - 3174f = 644 2. (46f + 53v = 110) * 69 => 3174f + 3657v = 7590 Adding the two equations eliminates 'f': 7107v = 8234 Solving for 'v': v = 8234 / 7107 = 1.16 Substitute the value of 'v' (1.16) into either original equation to solve for 'f'. Let's use equation 1: 75(1.16) - 69f = 14 87 - 69f = 14 -69f = -73 f = -73 / -69 = 1.06 ### Answer: v = 1.16, f = 1.06
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.
Gemini-1.5 Pro:
Reasoning: Let's solve for 'g' in the first equation: 1) 7o + 13g = 131 2) 13g = 131 - 7o 3) g = (131 - 7o) / 13 Now, substitute this value of 'g' into the second equation: 4) 89 * ((131 - 7o) / 13) - 70o = 37 5) (11559 - 623o) / 13 - 70o = 37 6) 11559 - 623o - 910o = 481 7) -1533o = -11078 8) o = -11078 / -1533 9) o = 7.23 (rounded to two decimal places) Substitute the value of 'o' back into either equation to solve for 'g'. Let's use the first equation: 10) 7 * 7.23 + 13g = 131 11) 50.61 + 13g = 131 12) 13g = 80.39 13) g = 80.39 / 13 14) g = 6.18 (rounded to two decimal places) ### Answer: o = 7.23, g = 6.18
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. First, we can multiply the first equation by 11 and the second equation by 88 to make the coefficients of 'd' in both equations have opposite signs: (1) 968d - 660k = 418 (Original equation multiplied by 11) (2) 2904k - 968d = 4048 (Original equation multiplied by 88) Now, we can add the two equations together, eliminating 'd': 2244k = 4466 Solving for k: k = 4466 / 2244 = 1.99 Now, we can substitute the value of k back into either original equation to solve for d. Let's use the first original equation: 88d - 60(1.99) = 38 88d - 119.4 = 38 88d = 157.4 d = 157.4 / 88 = 1.79 Therefore, the solution to the system of equations is d = 1.79 and k = 1.99. ### 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.
Gemini-1.5 Pro:
Reasoning: To solve for y and z, we can use elimination. Multiplying the second equation by -12, we get: -24z - 984y = -1152 Adding this equation to the first equation, we eliminate z: (24z - 24z) + (72y - 984y) = 139 - 1152 -912y = -1013 y = -1013 / -912 = 1.11 Now that we know the value of y, we can substitute it back into either of the original equations to solve for z. Using the first equation: 24z + 72 * 1.11 = 139 24z + 79.92 = 139 24z = 59.08 z = 59.08 / 24 = 2.46 ### Answer: y = 1.11, z = 2.46
Expected answer: {y: 1013/912, z: 2243/912}
Correct? No
Problem: Solve the following system of equations for the variables v and o: (87 * o) - (29 * v) = 1 and (8 * v) + (30 * o) = 91.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination.
1. **Multiply the equations to match coefficients:** We'll match the coefficients of 'v'. Multiply the first equation by 8 and the second equation by 29:
- Equation 1 * 8: 696o - 232v = 8
- Equation 2 * 29: 232v + 870o = 2639
2. **Add the equations:** Notice that the 'v' terms cancel out:
- (696o - 232v) + (232v + 870o) = 8 + 2639
- 1566o = 2647
3. **Solve for 'o':**
- o = 2647 / 1566 = 1.69
4. **Substitute 'o' back into either original equation to solve for 'v':** Let's use the second equation:
- (8 * v) + (30 * 1.69) = 91
- 8v + 50.7 = 91
- 8v = 40.3
- v = 40.3 / 8 = 5.04
### Answer: v = 5.04, o = 1.69
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.
Gemini-1.5 Pro:
Reasoning: Let's rearrange the second equation to make it easier to solve for one variable: (23*h) = 124 - (11*p) Now, let's substitute the value of (23*h) from the rearranged second equation into the first equation: (124 - (11*p)) - (46*p) = 58 Simplifying the equation: 124 - 11p - 46p = 58 124 - 57p = 58 Moving the constant term to the right side: -57p = 58 - 124 -57p = -66 Solving for p: p = -66 / -57 p = 1.16 (rounded to two decimal places) Now that we know p = 1.16, we can substitute it back into either of the original equations to solve for h. Let's use the first equation: (23 * h) - (46 * 1.16) = 58 23h - 53.36 = 58 Moving the constant term to the right side: 23h = 58 + 53.36 23h = 111.36 Solving for h: h = 111.36 / 23 h = 4.84 (rounded to two decimal places) Therefore, the solution to the system of equations is h = 4.84 and p = 1.16. ### Answer: h = 4.84, p = 1.16
Expected answer: {h: 2114/437, p: 22/19}
Correct? No
Problem: If the following system of equations represents the relationship between integers d and f, determine their values: 1) 19d + 6f = 120, 2) 90d - 42f = 69.
Gemini-1.5 Pro:
Reasoning: To solve for the values of d and f, we can use the elimination method. 1. **Multiply equations to match coefficients:** We can multiply the first equation by 7 and the second equation by 3 to make the coefficients of 'f' have opposite signs and the same magnitude. - Equation 1 * 7: (19d + 6f = 120) * 7 => 133d + 42f = 840 - Equation 2 * 3: (90d - 42f = 69) * 3 => 270d - 126f = 207 2. **Add the modified equations:** Add the two equations obtained in the previous step to eliminate 'f': - (133d + 42f) + (270d - 126f) = 840 + 207 - 403d = 1047 3. **Solve for 'd':** Divide both sides of the equation by 403 to isolate 'd': - 403d / 403 = 1047 / 403 - d = 2.6 4. **Substitute 'd' into either original equation to solve for 'f':** Let's substitute 'd = 2.6' into the first original equation: - 19(2.6) + 6f = 120 - 49.4 + 6f = 120 5. **Solve for 'f':** - 6f = 120 - 49.4 - 6f = 70.6 - f = 70.6 / 6 - f = 11.77 Therefore, the values of d and f are: d = 2.6 and f = 11.77 ### Answer: d=2.6, f=11.77
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'?
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method.
1. **Multiply equations to match coefficients:** We'll eliminate 'r' by matching its coefficients.
- Multiply the first equation by 52:
(42 * m + 55 * r) * 52 = 127 * 52
2184 * m + 2860 * r = 6604
- Multiply the second equation by 55:
(66 * m - 52 * r) * 55 = 44 * 55
3630 * m - 2860 * r = 2420
2. **Add the modified equations:**
(2184 * m + 2860 * r) + (3630 * m - 2860 * r) = 6604 + 2420
5814 * m = 9024
3. **Solve for 'm':**
m = 9024 / 5814
m = 1.55
4. **Substitute 'm' into either original equation to solve for 'r':** Let's use the first equation:
42 * 1.55 + 55 * r = 127
65.1 + 55 * r = 127
55 * r = 61.9
r = 61.9 / 55
r = 1.125
### Answer: r = 1.125, m = 1.55
Expected answer: {m: 1504/969, r: 363/323}
Correct? No
Problem: Determine the values of n and w that satisfy the following system of linear equations simultaneously: 31n + 64w = 120 57w + 71n = 173
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. We can multiply the top equation by 57 and the bottom equation by -31 to make the 'n' coefficients cancel out when we add the equations together: ``` (31n + 64w = 120) * 57 --> 1767n + 3648w = 6840 (57w + 71n = 173) * -31 --> -1767n - 1767w = -5363 ``` Now, adding the two equations, we get: ``` 1881w = 1477 ``` Dividing both sides by 1881 gives us: ``` w = 1477/1881 = 0.785 ``` Now that we know 'w', we can substitute it back into either of the original equations to solve for 'n'. Let's use the first original equation: ``` 31n + 64(0.785) = 120 ``` Simplifying: ``` 31n + 50.24 = 120 ``` Subtracting 50.24 from both sides: ``` 31n = 69.76 ``` Dividing both sides by 31: ``` n = 69.76 / 31 = 2.25 ``` ### Answer: n = 2.25, w = 0.785
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. We can start by simplifying the equations. Divide the first equation by 2: 36a + 13u = 56. Multiply the simplified first equation (36a + 13u = 56) by 77: 2772a + 1001u = 4312. Multiply the second equation (76u - 77a = 31) by 36: 2736u - 2772a = 1116. Now, add both transformed equations: (2772a + 1001u) + (2736u - 2772a) = 4312 + 1116. This simplifies to: 3737u = 5428. Therefore, u = 5428 / 3737 = 1.4525. Substitute the value of u back into the simplified first equation: 36a + 13 * 1.4525 = 56. This simplifies to: 36a = 37.1175. Therefore, a = 37.1175 / 36 = 1.0311. ### Answer: u = 1.4525, a = 1.0311.
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. 1. **Multiply equations to match coefficients:** We'll multiply the first equation by 31 and the second equation by 6 to make the 'b' coefficients have opposite signs: - First equation * 31: (54n + 18b = 143) * 31 => 1674n + 558b = 4433 - Second equation * 6: (87n - 93b = 48) * 6 => 522n - 558b = 288 2. **Add the equations:** Now we can add the two equations to eliminate 'b': - (1674n + 558b = 4433) + (522n - 558b = 288) => 2196n = 4721 3. **Solve for 'n':** Divide both sides by 2196: - n = 4721 / 2196 = 2.15 4. **Substitute 'n' to find 'b':** Substitute the value of 'n' (2.15) into either of the original equations to solve for 'b'. Let's use the first equation: - 54(2.15) + 18b = 143 - 116.1 + 18b = 143 - 18b = 26.9 - b = 26.9 / 18 = 1.49 Therefore, n = 2.15 and b = 1.49 ### Answer: n = 2.15, b = 1.49
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)?
Gemini-1.5 Pro:
Reasoning: To solve this system of equations, we can use the elimination method. 1. **Multiply equations to match coefficients of one variable:** We can multiply the first equation by 88 and the second equation by 7 to make the coefficients of 'p' have opposite signs: - (53e + 7p = 103) * 88 => 4664e + 616p = 9064 - (69e - 88p = 6) * 7 => 483e - 616p = 42 2. **Add the two equations:** Adding the equations eliminates 'p': - (4664e + 616p = 9064) + (483e - 616p = 42) => 5147e = 9106 3. **Solve for 'e':** - e = 9106 / 5147 = 1.77 (approximately) 4. **Substitute 'e' in either original equation to solve for 'p':** Let's use the first original equation: - 53 * 1.77 + 7p = 103 - 93.81 + 7p = 103 - 7p = 9.19 - p = 9.19 / 7 = 1.31 (approximately) Therefore, the cost of electricity (e) is approximately 1.77, and the cost of petroleum (p) is approximately 1.31. ### Answer: e ≈ 1.77, p ≈ 1.31.
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. 1. **Multiply equations to match coefficients:** - Multiply the first equation by 24: (432p - 168s) = 504 - Multiply the second equation by 18: (918s - 432p) = 252 2. **Add the modified equations:** - (432p - 168s) + (918s - 432p) = 504 + 252 - 750s = 756 3. **Solve for 's':** - s = 756 / 750 - s = 1.008 4. **Substitute 's' into either original equation to solve for 'p':** - Let's use the first equation: (18p - 7 * 1.008) = 21 - 18p - 7.056 = 21 - 18p = 28.056 - p = 28.056 / 18 - p = 1.5587 (rounded to four decimal places) ### Answer: p = 1.5587, 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.
Gemini-1.5 Pro:
Reasoning:
Let's solve this system of equations using elimination:
1. **Multiply the equations to make the 'w' coefficients have opposite signs:**
- Multiply the first equation by 42:
(59c - 34w) * 42 = 84 * 42
2478c - 1428w = 3528
- Multiply the second equation by 34:
(11c + 42w) * 34 = 98 * 34
374c + 1428w = 3332
2. **Add the two equations together to eliminate 'w':**
(2478c - 1428w) + (374c + 1428w) = 3528 + 3332
2852c = 6860
3. **Solve for 'c':**
c = 6860 / 2852
c = 2.4 (rounded to the nearest tenth)
4. **Substitute the value of 'c' back into either of the original equations to solve for 'w'. Let's use the first equation:**
59 * 2.4 - 34w = 84
141.6 - 34w = 84
-34w = -57.6
w = 1.7 (rounded to the nearest tenth)
### Answer: c = 2.4, w = 1.7
Expected answer: {c: 1715/713, w: 2429/1426}
Correct? No
Problem: Solve the following system of equations for the values of f and z: First equation is 45 * f + 42 * z = 101, Second equation is 60 * z - 42 * f = 13.
Gemini-1.5 Pro:
Reasoning: Let's solve for 'z' in the second equation: 60z = 13 + 42f z = (13 + 42f) / 60 Now, substitute this value of 'z' into the first equation: 45f + 42 * ((13 + 42f) / 60) = 101 Simplify and solve for 'f': 45f + (546 + 1764f) / 60 = 101 2700f + 546 + 1764f = 6060 4464f = 5514 f = 5514 / 4464 f = 1.235 Now, substitute the value of 'f' back into the equation for 'z': z = (13 + 42 * 1.235) / 60 z = (13 + 51.87) / 60 z = 64.87 / 60 z = 1.081 Therefore, the solution to the system of equations is f = 1.235 and z = 1.081. ### Answer: f = 1.235, z = 1.081
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.
Gemini-1.5 Pro:
Reasoning:
Let's solve this system of equations using elimination.
1. **Multiply equations to match coefficients:**
- Multiply the first equation by 58:
(26 * p - 21 * u) * 58 = 6 * 58 => 1508p - 1218u = 348
- Multiply the second equation by 21:
(15 * p + 58 * u) * 21 = 83 * 21 => 315p + 1218u = 1743
2. **Add the equations to eliminate 'u':**
(1508p - 1218u) + (315p + 1218u) = 348 + 1743
1823p = 2091
3. **Solve for 'p':**
p = 2091 / 1823
p = 1.15
4. **Substitute 'p' back into either original equation to solve for 'u'. Let's use the first one:**
26 * 1.15 - 21 * u = 6
29.9 - 21u = 6
-21u = -23.9
u = -23.9 / -21
u = 1.14
### Answer: p = 1.15, u = 1.14
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
Gemini-1.5 Pro:
Reasoning:
Let's solve for 'n' first.
1. Multiply the first equation by 8: (92*n - 38*s = 62) * 8 => 736*n - 304*s = 496
2. Multiply the second equation by 38: (59*n + 8*s = 75) * 38 => 2242*n + 304*s = 2850
3. Add the modified equations together:
736*n - 304*s = 496
2242*n + 304*s = 2850
------------------
2978*n = 3346
4. Isolate 'n' by dividing both sides by 2978: n = 3346/2978 => n = 1.1236
Now let's substitute the value of 'n' back into either original equation to solve for 's'. We'll use the first equation:
1. 92 * 1.1236 - 38 * s = 62
2. Simplify: 103.5472 - 38*s = 62
3. Subtract 103.5472 from both sides: -38*s = -41.5472
4. Isolate 's' by dividing both sides by -38: s = -41.5472 / -38 => s = 1.0933
### Answer: n = 1.1236, s = 1.0933
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.
Gemini-1.5 Pro:
Reasoning: To solve for 'e' and 'z', we can use the elimination method. First, we multiply the first equation by 81 and the second equation by -87: (81)*(87z - 35e) = (81)*63 ---> 6963z - 2835e = 5103 (-87)*(81z + 58e) = (-87)*154 ---> -6963z - 5046e = -13418 Adding the two equations eliminates 'z', allowing us to solve for 'e': -7881e = -8315 e = -8315 / -7881 = 1.055 Substituting the value of 'e' (1.055) into either of the original equations, we can solve for 'z'. Let's use the first equation: 87z - 35(1.055) = 63 87z - 36.925 = 63 87z = 99.925 z = 99.925 / 87 = 1.149 Therefore, the solution to the simultaneous equations is e = 1.055 and z = 1.149. ### Answer: e = 1.055, z = 1.149
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.
Gemini-1.5 Pro:
To solve for *h* and *t*, we can use the method of elimination: 1. **Multiply equations to match coefficients:** - Multiply the first equation by 9: (99 * t) + (396 * h) = 990 - Multiply the second equation by -11: (-1078 * t) + (396 * h) = -198 2. **Subtract the second equation from the first:** 1177 * t = 1188 3. **Solve for *t*:** t = 1188 / 1177 = 1.01 (rounded to two decimal places) 4. **Substitute the value of *t* back into either original equation to solve for *h*. Let's use the first equation:** (11 * 1.01) + (44 * h) = 110 11.11 + 44h = 110 44h = 98.89 h = 98.89 / 44 = 2.25 (rounded to two decimal places) ### Answer: t = 1.01, h = 2.25
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. 1. **Multiply the equations to make the coefficients of one variable opposites.** We'll target the 'b' variable. Multiply the first equation by -18 and the second equation by 69: ``` (-18) * (69b - 50o) = (-18) * 55 ---> -1242b + 900o = -990 (69) * (41o + 18b) = (69) * 85 ---> 2829o + 1242b = 5865 ``` 2. **Add the two equations together.** Notice that the 'b' terms cancel out: ``` (-1242b + 900o) + (2829o + 1242b) = -990 + 5865 3729o = 4875 ``` 3. **Solve for 'o':** ``` o = 4875 / 3729 o = 1.31 (approximately) ``` 4. **Substitute the value of 'o' back into either of the original equations to solve for 'b'.** Let's use the first original equation: ``` 69b - 50(1.31) = 55 69b - 65.5 = 55 69b = 120.5 b = 120.5 / 69 b = 1.75 (approximately) ``` Therefore, the approximate solutions to the system of equations are b = 1.75 and o = 1.31. ### Answer: b = 1.75, o = 1.31
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.
Gemini-1.5 Pro:
Reasoning:
Let's solve this system of equations using the elimination method.
1. **Multiply the equations to match coefficients:** We'll multiply the first equation by 5 and the second equation by 80 to make the coefficients of 'p' have opposite signs:
* Equation 1 * 5: 335q - 400p = 325
* Equation 2 * 80: 480q + 400p = 14160
2. **Add the equations to eliminate 'p':**
* (335q - 400p) + (480q + 400p) = 325 + 14160
* 815q = 14485
3. **Solve for 'q':**
* q = 14485 / 815 = 17.75
4. **Substitute 'q' into either original equation to solve for 'p':** Let's use the second equation:
* 6 * 17.75 + 5p = 177
* 106.5 + 5p = 177
* 5p = 70.5
* p = 14.1
Therefore, p = 14.1 and q = 17.75
### Answer: p = 14.1, q = 17.75
Expected answer: {p: 11469/815, q: 2897/163}
Correct? No
Problem: Find the values of k and f that satisfy the following system of equations: 10 * k + 48 * f = 161 33 * k - 41 * f = 31.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. We can multiply the first equation by 33 and the second equation by -10 to make the coefficients of 'k' opposites: ``` (10k + 48f = 161) * 33 => 330k + 1584f = 5313 (33k - 41f = 31) * -10 => -330k + 410f = -310 ``` Now, we can add the two equations together: ``` 330k + 1584f = 5313 -330k + 410f = -310 ------------------ 2094f = 5003 ``` Dividing both sides by 2094, we get: ``` f = 5003 / 2094 = 2.39 (approximately) ``` Now, we can substitute this value of 'f' back into either of the original equations to solve for 'k'. Let's use the first equation: ``` 10k + 48 * 2.39 = 161 10k + 114.72 = 161 10k = 46.28 k = 4.628 (approximately) ``` ### Answer: k ≈ 4.628, f ≈ 2.39
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
Gemini-1.5 Pro:
Reasoning: Let's rearrange the first equation to solve for n: n = 58l - 31 Now we can substitute this expression for 'n' into the second equation: 103 = (58l - 31) + 73l Combining like terms: 103 = 131l - 31 Add 31 to both sides: 134 = 131l Divide both sides by 131 to isolate l: l = 134 / 131 Now that we know l = 134/131, we can substitute it back into either of the original equations to solve for n. Let's use the first equation: 31 = 58 * (134/131) - n Simplify: 31 = 7772/131 - n Subtract 7772/131 from both sides: -3841/131 = -n Multiply both sides by -1: n = 3841/131 ### Answer: l = 134/131 and n = 3841/131
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
Gemini-1.5 Pro:
Reasoning:
To solve for one variable, we can eliminate the other variable by manipulating the equations. Let's eliminate 'j'.
1. **Multiply equations to match 'j' coefficients:**
- Multiply the first equation by 37: (29y - 28j = 39) * 37 => 1073y - 1036j = 1443
- Multiply the second equation by 28: (25y + 37j = 116) * 28 => 700y + 1036j = 3248
2. **Add the two new equations:**
(1073y - 1036j) + (700y + 1036j) = 1443 + 3248
1773y = 4691
3. **Solve for 'y':**
y = 4691 / 1773
y = 2.65
4. **Substitute 'y' back into either original equation to solve for 'j':**
Let's use the first original equation: 29y - 28j = 39
29(2.65) - 28j = 39
76.85 - 28j = 39
-28j = -37.85
j = -37.85 / -28
j = 1.35
Therefore, the values that satisfy the system of equations are y = 2.65 and j = 1.35.
### Answer: y = 2.65, j = 1.35
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. 1. **Multiply equations to match coefficients:** We'll eliminate 'a'. Find the least common multiple of 85 (coefficient of 'a' in the first equation) and 17 (coefficient of 'a' in the second equation). The least common multiple is 85. - Multiply the first equation by -1: -93u + 85a = -74 - Multiply the second equation by 5: 480u + 85a = 930 2. **Add the equations together:** - (-93u + 85a) + (480u + 85a) = -74 + 930 - 387u = 856 3. **Solve for 'u':** - u = 856 / 387 - u = 2.21 (approximately) 4. **Substitute 'u' into either original equation to solve for 'a':** Let's use the first original equation: - 93 * 2.21 - 85a = 74 - 205.53 - 85a = 74 - -85a = -131.53 - a = 1.55 (approximately) ### Answer: u ≈ 2.21, a ≈ 1.55
Expected answer: {a: 3398/3247, u: 1004/573}
Correct? No
Problem: Given the following pair of simultaneous equations, please find out the values of u and g: 97 * u - 61 * g = 15 85 * u + 23 * g = 186
Gemini-1.5 Pro:
Reasoning:
To solve for u and g, we can use the elimination method.
1. **Multiply equations to match coefficients of one variable:**
- Multiply the first equation by 23:
(97 * u - 61 * g = 15) * 23 => 2231u - 1403g = 345
- Multiply the second equation by 61:
(85 * u + 23 * g = 186) * 61 => 5185u + 1403g = 11346
2. **Add the two modified equations to eliminate 'g':**
(2231u - 1403g) + (5185u + 1403g) = 345 + 11346
7416u = 11691
3. **Solve for 'u':**
u = 11691 / 7416 = 1.577
4. **Substitute the value of 'u' back into either of the original equations to solve for 'g':**
Using the first original equation:
97 * 1.577 - 61 * g = 15
153.069 - 61g = 15
-61g = -138.069
g = 2.263
### Answer: u = 1.577, g = 2.263
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.
Gemini-1.5 Pro:
Reasoning: We can use the elimination method to solve for the values of r and b.
1. **Multiply equations to make the 'b' coefficients have opposite signs:**
- Multiply the first equation by -1:
(-1) * [(17 * r) + (67 * b) = 103] => (-17 * r) + (-67 * b) = -103
2. **Add the modified equations together:**
(-17 * r) + (-67 * b) = -103
( 85 * r) + ( 67 * b) = 187
-------------------------------
(68 * r) + ( 0 * b) = 84
3. **Solve for 'r':**
- Simplify: 68r = 84
- Divide both sides by 68: r = 84/68 = 21/17
4. **Substitute the value of 'r' back into either original equation to solve for 'b':**
- Using the first original equation: (17 * 21/17) + (67 * b) = 103
- Simplify: 21 + 67b = 103
- Subtract 21 from both sides: 67b = 82
- Divide both sides by 67: b = 82/67
### Answer: r = 21/17, b = 82/67
Expected answer: {b: 82/67, r: 21/17}
Correct? Yes
Problem: Given the following system of equations, solve for the values of t and n: 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. We can multiply the first equation by 38 and the second equation by 54 to make the coefficients of 't' have opposite signs: 1. (38) * [(85 * n) - (54 * t)] = (38) * 41 -> 3230n - 2052t = 1558 2. (54) * [(38 * t) + (30 * n)] = (54) * 126 -> 2052t + 1620n = 6804 Now, add the two equations together to eliminate 't': (3230n - 2052t) + (2052t + 1620n) = 1558 + 6804 This simplifies to: 4850n = 8362 Divide both sides by 4850 to solve for 'n': n = 8362 / 4850 = 1.724 Now that we have the value of 'n', substitute it back into either of the original equations to solve for 't'. Let's use the first equation: (85 * 1.724) - (54 * t) = 41 Simplify and solve for 't': 146.54 - 54t = 41 -54t = -105.54 t = -105.54 / -54 t = 1.954 ### Answer: t = 1.954, n = 1.724
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.
Gemini-1.5 Pro:
Reasoning: We can express the given information as a system of equations: 75u - 69q = 5 48u + 25q = 140 To solve for u and q, we can use elimination. Multiplying the top equation by 25 and the bottom equation by 69, we get: 1875u - 1725q = 125 3312u + 1725q = 9660 Adding the two equations together, we eliminate q: 5187u = 9785 Dividing both sides by 5187, we solve for u: u = 1.886 We can substitute this value of u back into either of the original equations to solve for q. Let's use the first equation: 75(1.886) - 69q = 5 Simplifying and solving for q, we get: 141.45 - 69q = 5 -69q = -136.45 q = 1.978 ### Answer: u = 1.886, q = 1.978
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.
Gemini-1.5 Pro:
Reasoning: To solve for two unknowns, we need two equations, which we have. Let's solve for 'v' in terms of 'x' in the first equation: 1. **Isolate 'v' in the first equation:** (20 * v) + (17 * x) = 149 20v = 149 - 17x v = (149 - 17x) / 20 2. **Substitute this value of 'v' into the second equation:** (24 * x) - (49 * ((149 - 17x) / 20)) = 22 3. **Simplify and solve for 'x':** (24 * x) - (7251/20) + (833x/20) = 22 (480x - 7251 + 833x)/20 = 22 1313x - 7251 = 440 1313x = 7691 x = 7691 / 1313 x = 5.86 (approximately) 4. **Substitute the value of 'x' back into either of the original equations to solve for 'v'. Let's use the first equation:** (20 * v) + (17 * 5.86) = 149 20v + 99.62 = 149 20v = 49.38 v = 49.38 / 20 v = 2.47 (approximately) ### Answer: v = 2.47, x = 5.86
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination.
1. **Multiply equations to match coefficients:** We'll multiply the first equation by -2.359 (approximately -92/39, which is the ratio of the coefficients of 't' in the two equations) and the second equation by 1. This will make the 't' coefficients cancel out when we add the equations together.
```
-92t - 97g = -346
94t - 91g = 42
```
2. **Add the equations:**
```
2t - 188g = -304
```
3. **Solve for 'g':**
```
-188g = -304 - 2t
g = (304 + 2t) / 188
```
4. **Substitute 'g' back into either original equation to solve for 't'.** Let's use the first original equation:
```
39t + 41 * ((304 + 2t) / 188) = 147
```
5. **Simplify and solve for 't':**
```
39t + (12464 + 82t) / 188 = 147
7332t + 12464 + 82t = 27636
7414t = 15172
t = 2.046
```
6. **Substitute the value of 't' back into the equation for 'g':**
```
g = (304 + 2 * 2.046) / 188
g = 1.667
```
### Answer: t = 2.046, g = 1.667
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.
Gemini-1.5 Pro:
Reasoning: To solve for two unknowns, we need to eliminate one unknown to solve for the other. Let's eliminate 'o' by multiplying the first equation by 48 and the second equation by 55: (6c + 55o = 112) * 48 => 288c + 2640o = 5376 (55c - 48o = 36) * 55 => 3025c - 2640o = 1980 Now, add the two equations together: (288c + 2640o = 5376) + (3025c - 2640o = 1980) => 3313c = 7356 Solve for 'c': c = 7356 / 3313 = 2.22 Now, substitute the value of 'c' (2.22) into either of the original equations to solve for 'o'. Let's use the first equation: 6(2.22) + 55o = 112 13.32 + 55o = 112 55o = 98.68 o = 98.68 / 55 = 1.80 (rounded to two decimal places) ### Answer: c = 2.22, o = 1.80
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.
Gemini-1.5 Pro:
Reasoning: To solve for u and w, we can use the method of elimination. 1. **Multiply the equations by appropriate constants** so that the coefficients of one of the variables in both equations are opposites. We can multiply the top equation by 71 and the bottom equation by -8: (71) * (8w + 68u) = (71) * (170) (-8) * (71w - 94u) = (-8) * (8) This simplifies to: 568w + 4828u = 12070 -568w + 752u = -64 2. **Add the two equations together.** Notice that the 'w' terms cancel out: (568w + 4828u) + (-568w + 752u) = 12070 - 64 5580u = 12006 3. **Solve for 'u'** by dividing both sides by 5580: u = 12006 / 5580 u = 2.15 4. **Substitute the value of 'u' back into either of the original equations** to solve for 'w'. Let's use the first original equation: 8w + 68 * 2.15 = 170 8w + 146.2 = 170 5. **Simplify and solve for 'w':** 8w = 23.8 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
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. Multiplying the top equation by -97 and the bottom equation by 8 gives us: -6596*r - 776*f = -8924 776*f + 2944*r = 1456 Adding the two equations, we get: -3652*r = -7468 Dividing both sides by -3652 gives us: r = 2.045 Plugging in r = 2.045 into the first equation: 68*2.045 + 8*f = 92 Simplifying: 139.06 + 8*f = 92 Subtracting 139.06 from both sides and then dividing both sides by 8, we get: f = -5.8825 ### Answer: r = 2.045, f = -5.8825
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).
Gemini-1.5 Pro:
Reasoning: We will use elimination to solve for *d* and *r*. 1. **Multiply the equations to match coefficients:** We need to make either the *d* or *r* coefficients have opposite signs in the two equations. Let's target the *d* coefficient. - Multiply the first equation (67d - 37r = 50) by 21. - Multiply the second equation (39r - 21d = 16) by 67. This gives us: - 1407d - 777r = 1050 - 2613r - 1407d = 1088 2. **Add the equations:** Notice that the *d* coefficients now cancel out when we add the equations: - (1407d - 777r) + (2613r - 1407d) = 1050 + 1088 - 1836r = 2138 3. **Solve for r:** - r = 2138 / 1836 - r = 1.165 4. **Substitute r back into either original equation to solve for d:** Let's use the first original equation (67d - 37r = 50): - 67d - 37(1.165) = 50 - 67d - 43.105 = 50 - 67d = 93.105 - d = 1.39 Therefore, each apple costs $1.39 and each ragdoll costs $1.165. ### Answer: d = 1.39, r = 1.165
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.
Gemini-1.5 Pro:
Reasoning:
Let's express the two equations in a standard form.
First equation: 50x + a = 163
Second Equation: 98x - 84a = 24
We can use the elimination method to solve for x and a. To eliminate 'a', let's multiply the first equation by 84 and add it to the second equation:
```
(50x + a = 163) * 84 --> 4200x + 84a = 13692
98x - 84a = 24
-----------------------------------------------
4298x = 13716
```
Now we have an equation with just 'x'. Solving for x:
x = 13716 / 4298 = 3.2
Now that we know x = 3.2, we can substitute it into either of the original equations to solve for 'a'. Let's use the first equation:
50 * 3.2 + a = 163
160 + a = 163
a = 163 - 160 = 3
Therefore, x = 3.2 and a = 3.
### Answer: x = 3.2, a = 3
Expected answer: {a: 7387/2149, x: 6858/2149}
Correct? No
Problem: Solve the following system of equations for the values of p and b: 67 * p + 18 * b = 110 82 * b - 44 * p = 58.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. We can start by multiplying the first equation by 2 and the second equation by 3 to make the coefficients of 'b' have opposite signs: 1. (67p + 18b = 110) * 2 => 134p + 36b = 220 2. (82b - 44p = 58) * 3 => 246b - 132p = 174 Now, we can add the two equations together to eliminate 'p': (134p + 36b) + (246b - 132p) = 220 + 174 This simplifies to: 282b = 394 Now, we can solve for 'b': b = 394 / 282 = 1.4 Now that we know 'b', we can substitute it back into either of the original equations to solve for 'p'. Let's use the first equation: 67p + 18(1.4) = 110 Simplifying: 67p + 25.2 = 110 67p = 84.8 p = 84.8 / 67 = 1.266 ### Answer: p = 1.266, b = 1.4
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.
Gemini-1.5 Pro:
We can express the problem with two simultaneous equations: * Equation 1: 63c - 36e = 27 * Equation 2: 8c + 32e = 69 Let's simplify Equation 1 by dividing both sides by 9: * Equation 1 (simplified): 7c - 4e = 3 Now, let's manipulate the equations to eliminate one variable. We can multiply Equation 1 (simplified) by 8 and Equation 2 by -7: * Equation 1 (simplified, multiplied by 8): 56c - 32e = 24 * Equation 2 (multiplied by -7): -56c - 224e = -483 Adding the two modified equations together, we get: -256e = -459 Now, solve for e: e = -459 / -256 e = 1.8 (rounded to one decimal place) Substitute the value of e back into either of the original equations to solve for c. Let's use Equation 2: 8c + 32 * 1.8 = 69 8c + 57.6 = 69 8c = 11.4 c = 11.4 / 8 c = 1.4 (rounded to one decimal place) ### Answer: c = 1.4, e = 1.8
Expected answer: {c: 93/64, e: 459/256}
Correct? No
Problem: Solve the following system of linear equations to find the values of x and p: 26 * x + 29 * p = 107 61 * x - 51 * p = 4
Gemini-1.5 Pro:
Reasoning:
To solve for two unknowns, we need two equations, which we have. There are a couple of ways to solve this. Let's use elimination:
1. **Multiply equations to match coefficients:** We'll match the 'p' coefficients.
- Multiply the first equation by 51:
(26x + 29p = 107) * 51 => 1326x + 1479p = 5457
- Multiply the second equation by 29:
(61x - 51p = 4) * 29 => 1769x - 1479p = 116
2. **Add the equations together:** Notice the 'p' terms cancel out:
(1326x + 1479p = 5457)
+(1769x - 1479p = 116)
-----------------------
3095x = 5573
3. **Solve for x:**
x = 5573 / 3095 = 1.8
4. **Substitute x back into either original equation to solve for p:** Let's use the first original equation:
26 * 1.8 + 29p = 107
46.8 + 29p = 107
29p = 60.2
p = 60.2 / 29 = 2.08
### Answer: x = 1.8, p = 2.08
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. 1. **Multiply the equations to make the coefficients of one variable opposites.** - Multiply the first equation by 10: `(290 * l) + (170 * g) = 680` - Multiply the second equation by 29: `(899 * g) - (290 * l) = 493` 2. **Add the two equations together to eliminate 'l'.** - `(290 * l) + (170 * g) = 680` + `(899 * g) - (290 * l) = 493` --------------------------------- `1069 * g = 1173` 3. **Solve for 'g'.** - `g = 1173 / 1069` - `g = 1.1` 4. **Substitute the value of 'g' back into either of the original equations to solve for 'l'. Let's use the first equation.** - `(29 * l) + (17 * 1.1) = 68` - `(29 * l) + 18.7 = 68` - `29 * l = 49.3` - `l = 49.3 / 29` - `l = 1.7` ### Answer: g = 1.1, l = 1.7
Expected answer: {g: 1173/1069, l: 1819/1069}
Correct? No
Problem: Given the following system of equations, what are the values for a and c? 33*a - 54*c = 20 24*a - 26*c = 29.
Gemini-1.5 Pro:
Reasoning:
To solve for two variables, we need two equations, which we have. There are a couple of ways to proceed. We can use substitution or elimination. Both are valid, but elimination tends to be faster and less prone to error. Let's use elimination:
1. **Multiply the equations by constants to make the coefficients of one variable opposites.**
- Multiply the top equation by 26:
(33a * 26) - (54c * 26) = (20 * 26) -> 858a - 1404c = 520
- Multiply the bottom equation by -54:
(24a * -54) - (26c * -54) = (29 * -54) -> -1296a + 1404c = -1566
2. **Add the two equations together.** Notice that the 'c' terms cancel out:
(858a - 1404c) + (-1296a + 1404c) = 520 - 1566
-438a = -1046
3. **Solve for 'a'.**
a = -1046 / -438 = 2.39
4. **Substitute the value of 'a' back into either of the original equations to solve for 'c'.** Let's use the first original equation:
33 * 2.39 - 54c = 20
78.87 - 54c = 20
-54c = -58.87
c = -58.87 / -54 = 1.09
### Answer: a = 2.39, c = 1.09
Expected answer: {a: 523/219, c: 159/146}
Correct? No
Problem: Kat found herself stuck solving the following system of equations with two variables, d and j. Can you help her resolve it? 71d + 9j = 132 43j - 84d = 91.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method.
1. **Multiply the equations to match coefficients:**
We'll multiply the top equation by 43 and the bottom equation by 9 to make the 'j' coefficients have opposite signs:
```
(71d + 9j = 132) * 43 => 3053d + 387j = 5676
(43j - 84d = 91) * 9 => 387j - 756d = 819
```
2. **Add the equations to eliminate 'j':**
```
3053d + 387j = 5676
-756d + 387j = 819
------------------
2297d = 4857
```
3. **Solve for 'd':**
```
d = 4857 / 2297 = 2.114
```
4. **Substitute 'd' back into either original equation to solve for 'j':**
Let's use the first original equation:
```
71(2.114) + 9j = 132
150.094 + 9j = 132
9j = -18.094
j = -2.01
```
### Answer: d = 2.114, j = -2.01
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.
Gemini-1.5 Pro:
Reasoning:
To solve for two unknowns, we need two equations, which we have. There are a couple of ways to solve this, one being substitution, the other being elimination. For this case, we will use elimination.
Let's multiply the first equation by 23, and the second equation by -26 so that the o terms will cancel out:
(81 * o) + (26 * n) = 163 ---> 23 * [(81 * o) + (26 * n) = 163] ---> (1863 * o) + (598 * n) = 3749
(87 * n) + (23 * o) = 143 ---> -26 * [(87 * n) + (23 * o) = 143] ---> (-2262 * n) + (-598 * o) = -3718
Now we can add the two equations together:
(1863 * o) + (598 * n) = 3749
+ (-598 * o) + (-2262 * n) = -3718
---------------------------------------
(1265 * o) + 0 = 31
Solving for o, we get:
o = 31 / 1265 = 0.0245
We can now plug this value of o back into either of the original equations to solve for n. Let's use the first equation:
(81 * 0.0245) + (26 * n) = 163
1.9845 + (26 * n) = 163
26 * n = 161.0155
n = 161.0155 / 26 = 6.193
Therefore:
o = 0.0245
n = 6.193
### Answer: o = 0.0245, n = 6.193
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.
Gemini-1.5 Pro:
Reasoning: To solve for *w* and *s*, we can use elimination. Let's multiply the first equation by 61 and the second equation by -46: (46 * s - 61 * w = 2) * 61 => 2806 * s - 3721 * w = 122 (61 * s + 67 * w = 178) * -46 => -2806 * s - 3082 * w = -8188 Now, let's add the two equations: (2806 * s - 3721 * w) + (-2806 * s - 3082 * w) = 122 - 8188 -6803 * w = -8066 We can now solve for *w*: w = -8066 / -6803 = 1.1856 Now that we know *w*, we can plug it back into either of the original equations to solve for *s*. Let's use the first equation: 46 * s - 61 * 1.1856 = 2 46 * s - 72.3276 = 2 46 * s = 74.3276 s = 74.3276 / 46 = 1.6158 ### Answer: w = 1.1856, s = 1.6158
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. 1. **Multiply the equations to match coefficients:** We'll eliminate 'k' first. Multiply the first equation by -8 and the second equation by 51: (-8) * (51k - 8i = 86) => -408k + 64i = -688 (51) * (39i + 8k = 193) => 1989i + 408k = 9843 2. **Add the equations:** (-408k + 64i = -688) +(1989i + 408k = 9843) --------------------- 2053i = 9155 3. **Solve for 'i':** i = 9155 / 2053 = 4.46 4. **Substitute 'i' into either original equation to solve for 'k':** Let's use the first equation: 51k - 8 * 4.46 = 86 51k - 35.68 = 86 51k = 121.68 k = 121.68 / 51 = 2.39 ### Answer: i = 4.46, k = 2.39
Expected answer: {i: 9155/2053, k: 4898/2053}
Correct? No
Problem: Solve the following system of equations for the values of l and m: 26m + 17l = 146 93m + l = 126
Gemini-1.5 Pro:
Reasoning: To solve for l and m, we can use elimination. 1. **Solve for 'l' in terms of 'm' in the second equation:** 93m + l = 126 l = 126 - 93m 2. **Substitute this expression for 'l' into the first equation:** 26m + 17(126 - 93m) = 146 3. **Simplify and solve for 'm':** 26m + 2142 - 1581m = 146 -1555m = -1996 m = 1996 / 1555 m = 1.28 4. **Substitute the value of 'm' back into either of the original equations to solve for 'l'. We'll use the second equation:** 93(1.28) + l = 126 119.04 + l = 126 l = 126 - 119.04 l = 6.96 ### Answer: l = 6.96, m = 1.28
Expected answer: {l: 10302/1555, m: 1996/1555}
Correct? No
Problem: Determine the values of 'h' and 'f' by solving the following system of equations: 1) (8 * f) + (39 * h) = 66 2) (29 * f) + (46 * h) = 119
Gemini-1.5 Pro:
Reasoning:
To solve for 'f' and 'h', we can use elimination method:
1. **Multiply equation 1 by -29 and equation 2 by 8 to eliminate 'f':**
(-29) * (8f + 39h = 66) => -232f - 1131h = -1914
(8) * (29f + 46h = 119) => 232f + 368h = 952
2. **Add the two resulting equations to eliminate 'f':**
-232f - 1131h = -1914
232f + 368h = 952
-----------------------
-763h = -962
3. **Solve for 'h':**
h = -962 / -763 = 1.26
4. **Substitute the value of 'h' (1.26) into either original equation to solve for 'f'. Let's use equation 1:**
8f + 39 * 1.26 = 66
8f + 49.14 = 66
8f = 16.86
f = 2.11
Therefore, the solution to the system of equations is f = 2.11 and h = 1.26
### Answer: f = 2.11, h = 1.26
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. We can multiply the top equation by 69 and the bottom equation by 23 to make the b coefficients have opposite values: ``` 69 * (22b + 23k) = 69 * 109 --> 1518b + 1587k = 7521 23 * (62k - 69b) = 23 * 14 --> 1426k - 1587b = 322 ``` Now, we can add the two equations together, eliminating 'b': ``` (1518b + 1587k) + (1426k - 1587b) = 7521 + 322 3013k = 7843 ``` Dividing both sides by 3013: ``` k = 7843 / 3013 = 2.6 ``` Now that we know k = 2.6, we can substitute it back into either of the original equations to solve for 'b'. Let's use the first original equation: ``` 22b + 23 * 2.6 = 109 22b + 59.8 = 109 22b = 49.2 b = 49.2 / 22 = 2.2 ``` Therefore, the values of k and b that satisfy the system of equations are k = 2.6 and b = 2.2. ### Answer: k = 2.6, b = 2.2
Expected answer: {b: 6436/2951, k: 7829/2951}
Correct? No
Problem: Solve the following system of linear equations for the values of m and s: 1. 84m - 55s = 1 2. 63m - 2s = 98.
Gemini-1.5 Pro:
Reasoning: We can use elimination to solve for the values of m and s. First, let's multiply the second equation by -27.5 to make the coefficients of s opposites: 1. 84m - 55s = 1 2. -1732.5m + 55s = -2745 Now, add the two equations together: -1648.5m = -2744 Divide both sides by -1648.5 to solve for m: m = 1.665 Now, substitute the value of m back into either of the original equations to solve for s. Let's use the first equation: 84(1.665) - 55s = 1 Simplify and solve for s: 139.86 - 55s = 1 -55s = -138.86 s = 2.53 ### Answer: m = 1.665, s = 2.53.
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.
Gemini-1.5 Pro:
Reasoning: To solve for two unknowns, we need to eliminate one unknown by manipulation of the equations and substitution. Let's solve for 'k' in the first equation and substitute it into the second equation. 1. **Solve for 'k' in the first equation:** (6 * b) + (5 * k) = 105 5k = 105 - 6b k = (105 - 6b) / 5 2. **Substitute 'k' into the second equation:** (95 * ((105 - 6b) / 5)) - (51 * b) = 67 3. **Simplify and solve for 'b':** 19(105 - 6b) - 51b = 67 1995 - 114b - 51b = 67 1928 = 165b b = 1928 / 165 b = 11.685 4. **Substitute 'b' back into either equation to solve for 'k'. Let's use the first equation:** (6 * 11.685) + (5 * k) = 105 70.11 + 5k = 105 5k = 34.89 k = 6.978 ### Answer: b = 11.685, k = 6.978
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. 1. **Multiply the equations to match coefficients:** We can multiply the first equation by 20 and the second equation by -33 to make the coefficients of 'd' have opposite signs: (33d + 62u = 114) * 20 => 660d + 1240u = 2280 (32u + 40d = 83) * -33 => -1056u - 1320d = -2739 2. **Add the equations together:** 660d + 1240u = 2280 -1056u - 1320d = -2739 ----------------------- -80u = -459 3. **Solve for 'u':** Divide both sides by -80: u = -459 / -80 = 5.7375 4. **Substitute 'u' back into either original equation to solve for 'd':** Let's use the first original equation: 33d + 62(5.7375) = 114 33d + 355.575 = 114 33d = -241.575 d = -241.575 / 33 = -7.320454545... ### Answer: d = -7.32 (rounded to two decimal places), u = 5.74 (rounded to two decimal places)
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
Gemini-1.5 Pro:
Reasoning: To solve for two variables, we need to manipulate one of the equations so that it only contains one of the variables. Let's solve for 'c' in the first equation. 1) **Isolate 'c' in the first equation:** 5c + 28p = 102 5c = 102 - 28p c = (102 - 28p) / 5 2) **Substitute the value of 'c' into the second equation:** 20 * ((102 - 28p) / 5) - 67p = 13 3) **Simplify and solve for 'p':** 4(102 - 28p) - 67p = 13 408 - 112p - 67p = 13 408 - 179p = 13 -179p = -395 p = -395 / -179 p = 2.21 (rounded to two decimal places) 4) **Substitute the value of 'p' back into either of the original equations to solve for 'c':** Let's use the first equation: 5c + 28 * 2.21 = 102 5c + 61.88 = 102 5c = 40.12 c = 8.02 (rounded to two decimal places) ### Answer: c = 8.02, p = 2.21
Expected answer: {c: 7198/895, p: 395/179}
Correct? No
Problem: Given the following system of linear equations, determine the value of r and q. The system of equations is: 58 * r + 12 * q = 147 97 * q - 76 * r = 76.
Gemini-1.5 Pro:
Reasoning: To solve for r and q, we can use the method of elimination. 1. **Multiply the equations by necessary multiples** such that the coefficients of one of the variables in both equations are opposites. In this case, we can multiply the first equation by 76 and the second equation by 58: (76) * (58r + 12q = 147) -> 4408r + 912q = 11172 (58) * (97q - 76r = 76) -> 5626q - 4408r = 4408 2. **Add the two equations together.** Notice that the 'r' terms cancel out: 4408r + 912q = 11172 -4408r + 5626q = 4408 ------------------ 6538q = 15580 3. **Solve for q:** q = 15580 / 6538 = 2.38 (approximately) 4. **Substitute the value of q back into either original equation** to solve for r. Let's use the first equation: 58r + 12 * 2.38 = 147 58r + 28.56 = 147 58r = 118.44 r = 118.44 / 58 = 2.04 (approximately) Therefore, the values are approximately r = 2.04 and 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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. We can eliminate 'k' by multiplying the first equation by 2 and the second equation by 3, then adding the two resulting equations. Equation 1 * 2: (47 * v - 3 * k) * 2 = 10 * 2 ---> 94 * v - 6 * k = 20 Equation 2 * 3: (73 * v + 2 * k) * 3 = 128 * 3 ---> 219 * v + 6 * k = 384 Now, add the two modified equations: (94 * v - 6 * k) + (219 * v + 6 * k) = 20 + 384 313 * v = 404 Solving for 'v': v = 404 / 313 v = 1.29 Substitute the value of 'v' (1.29) into either of the original equations to solve for 'k'. Let's use the first original equation: 47 * 1.29 - 3 * k = 10 60.63 - 3 * k = 10 -3 * k = 10 - 60.63 -3 * k = -50.63 k = -50.63 / -3 k = 16.88 Therefore, the solution to the system of equations is v = 1.29 and k = 16.88 ### Answer: v = 1.29, k = 16.88
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?
Gemini-1.5 Pro:
Reasoning: Let's represent the given information as equations: Equation 1: 89p - 41q = 60 Equation 2: 27p + 91q = 124 We can solve for 'p' and 'q' using elimination: 1. **Multiply Equation 1 by 91 and Equation 2 by 41** to make the coefficients of 'q' have opposite signs: - Equation 1 * 91: 8099p - 3731q = 5460 - Equation 2 * 41: 1107p + 3731q = 5084 2. **Add the two modified equations** to eliminate 'q': - (8099p - 3731q) + (1107p + 3731q) = 5460 + 5084 - 9206p = 10544 3. **Solve for 'p':** - p = 10544 / 9206 - p = 1.145 4. **Substitute the value of 'p' (1.145) into either of the original equations** to solve for 'q'. Let's use Equation 1: - 89(1.145) - 41q = 60 - 101.905 - 41q = 60 - -41q = -41.905 - q = -41.905 / -41 - q = 1.022 Therefore, the values of p and q are approximately 1.145 and 1.022, respectively. ### Answer: p ≈ 1.145, q ≈ 1.022
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
Gemini-1.5 Pro:
Reasoning: We can use elimination to solve for one of the variables. Multiply the second equation by -8/3 so that the 'd' coefficients cancel out when we add the two equations. -(8/3)*(27*u + 36*d) = (71)*(-8/3) This simplifies to: -72*u - 96*d = -568/3 Now we have the following two equations: 1) 88*u - 96*d = 12 2) -72*u - 96*d = -568/3 Add the two equations together. Note that the 'd' terms cancel out: 16*u = -532/3 Solve for u: u = (-532/3) / 16 u = -33.25 / 16 u = -2.078125 Now, substitute this value of u back into either of the original equations to solve for d. Let's use the first equation: (88 * -2.078125) - (96 * d) = 12 -183.075 - 96*d = 12 -96*d = 195.075 d = -2.03203125 ### Answer: u = -2.078125 and d = -2.03203125
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.
Gemini-1.5 Pro:
Reasoning: To solve for f and n, we can use the method of elimination. Let's multiply the second equation by -9 to make the coefficients of 'f' in both equations cancel out: -9 * (3n - 2f) = -9 * 70 Simplifying the equation: -27n + 18f = -630 Now we have two equations: 1) 20n - 18f = 9 2) -27n + 18f = -630 Adding the two equations, we get: -7n = -621 Solving for n: n = -621 / -7 = 88.71 Now, substitute the value of n back into either of the original equations to solve for 'f'. Let's use the first equation: 20n - 18f = 9 20 * 88.71 - 18f = 9 1774.2 - 18f = 9 -18f = 9 - 1774.2 -18f = -1765.2 f = -1765.2 / -18 = 97.51 Therefore, the solution to the system of equations is f = 97.51 and n = 88.71. ### Answer: f = 97.51, n = 88.71
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method.
1. **Multiply equations to match coefficients:** Multiply the first equation by 22 and the second equation by -97 to make the coefficients of 'j' have opposite signs:
(7l + 97j = 133) * 22 => 154l + 2134j = 2926
(22j + 37l = 76) * -97 => -2134j - 3589l = -7372
2. **Add the equations:** Now, add the two equations together. Notice that the 'j' terms cancel out:
154l + 2134j = 2926
-3589l - 2134j = -7372
---------------------
-3435l = -4446
3. **Solve for 'l':** Divide both sides by -3435:
l = -4446 / -3435 = 1.3
4. **Substitute 'l' to find 'j':** Substitute the value of 'l' (1.3) back into either of the original equations to solve for 'j'. Let's use the first equation:
7(1.3) + 97j = 133
9.1 + 97j = 133
97j = 123.9
j = 123.9 / 97
j = 1.28
### Answer: j = 1.28, l = 1.3
Expected answer: {j: 1463/1145, l: 1482/1145}
Correct? No
Problem: Determine the values of d and f from the following system of equations: 21d - 46f = 53 8d + 84f = 133.
Gemini-1.5 Pro:
Reasoning:
To solve for *d* and *f*, we can use the method of elimination.
1. **Multiply the equations to match coefficients:**
- Multiply the first equation by 2:
42d - 92f = 106
- Multiply the second equation by -5:
-40d - 420f = -665
2. **Add the two equations together:**
(42d - 92f) + (-40d - 420f) = 106 - 665
2d - 512f = -559
3. **Solve for *f*:**
-512f = -559 - 2d
f = (-559 - 2d) / -512
f = (559 + 2d) / 512
4. **Substitute the value of *f* into one of the original equations to solve for *d*. Let's use the first original equation:**
21d - 46 * ((559 + 2d) / 512) = 53
21d - (25714 + 92d) / 512 = 53
10752d - 25714 - 92d = 27136
10660d = 52850
d = 52850 / 10660
d = 5
5. **Substitute the value of *d* back into the equation for *f*:**
f = (559 + 2 * 5) / 512
f = 569 / 512
f = 1.1113 (rounded to four decimal places)
### Answer: d = 5, f = 1.1113
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.
Gemini-1.5 Pro:
Reasoning: To solve for one variable, we can manipulate one of the equations to express one variable in terms of the other, and then substitute it into the second equation. Let's solve for *u* in the first equation: 74u + 11w = 116 74u = 116 - 11w u = (116 - 11w) / 74 Now, we can substitute this expression for *u* into the second equation: 66 * ((116 - 11w) / 74) - 68w = 21 (7656 - 726w) / 74 - 68w = 21 7656 - 726w - 5032w = 1554 6102 = 5758w w = 6102 / 5758 = 1.06 Now we can substitute the value of *w* back into either of the original equations to solve for *u*. Let's use the first equation: 74u + 11 * 1.06 = 116 74u + 11.66 = 116 74u = 104.34 u = 104.34 / 74 = 1.41 Therefore, the values that satisfy the system of equations are u = 1.41 and w = 1.06 ### Answer: u = 1.41, w = 1.06
Expected answer: {u: 8119/5758, w: 3051/2879}
Correct? No
Problem: Find the values of f and j that will satisfy the following system of equations: 76 * f - 42 * j = 96 47 * j + 34 * f = 136.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. 1. **Multiply the equations to make the coefficients of one variable opposites.** We can multiply the first equation by 47 and the second equation by -42 to make the coefficients of 'j' opposites: (76 * f - 42 * j = 96) * 47 -> 3572 * f - 1974 * j = 4512 (47 * j + 34 * f = 136) * -42 -> -1974 * j - 1428 * f = -5712 2. **Add the two equations together.** Notice that the 'j' terms cancel out: (3572 * f - 1974 * j) + (-1974 * j - 1428 * f) = 4512 - 5712 2144 * f = -1200 3. **Solve for 'f':** f = -1200 / 2144 = -0.56 (rounded to two decimal places) 4. **Substitute the value of 'f' back into either original equation to solve for 'j'.** Let's use the first equation: 76 * (-0.56) - 42 * j = 96 -42.56 - 42 * j = 96 -42 * j = 138.56 j = -3.3 (rounded to one decimal place) ### Answer: f = -0.56, j = -3.3
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination.
1. **Multiply the equations to match coefficients:** We'll multiply the top equation by 63 and the bottom equation by 32 to make the 'e' coefficients have opposite signs:
- (68u + 32e = 174) * 63 => 4284u + 2016e = 10962
- (63e - 69u = 1) * 32 => 2016e - 2208u = 32
2. **Add the equations together:**
(4284u + 2016e = 10962) + (2016e - 2208u = 32) => 2076u = 10994
3. **Solve for 'u':**
u = 10994 / 2076 = 5.3
4. **Substitute 'u' back into either original equation to solve for 'e':** Let's use the first original equation:
68 * 5.3 + 32e = 174
360.4 + 32e = 174
32e = -186.4
e = -186.4 / 32 = -5.825
### Answer: u = 5.3, e = -5.825
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. First, we can multiply the top equation by 24 and the bottom equation by -57 to make the coefficients of 'p' have opposite signs: ``` 24 * (57p + 4n = 129) ---> 1368p + 96n = 3096 -57 * (24p + 39n = 154) ---> -1368p - 2223n = -8778 ``` Now, we can add the two equations together: ``` 1368p + 96n = 3096 -1368p - 2223n = -8778 ---------------------- -2127n = -5682 ``` Dividing both sides by -2127, we get: ``` n = -5682 / -2127 = 2.67 ``` Now that we know the value of 'n', we can substitute it back into either of the original equations to solve for 'p'. Let's use the first equation: ``` 57p + 4(2.67) = 129 57p + 10.68 = 129 57p = 118.32 p = 118.32 / 57 = 2.08 ``` ### Answer: p = 2.08 and 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.
Gemini-1.5 Pro:
Reasoning:
Let's solve this system of equations using elimination:
1. **Simplify the equations:** Divide the first equation by 2 and the second equation by 2:
- 33z - 22y = 16
- 30y + 7z = 58
2. **Multiply to match coefficients:** Multiply the first equation by -15 and the second equation by 11 to make the 'z' coefficients cancel out:
- 495z + 330y = -240
- 330y + 77z = 638
3. **Add the equations:**
- 418z = 398
4. **Solve for z:** Divide both sides by -418:
z = -398 / 418 = -199/209
5. **Substitute to find y:** Substitute the value of z back into any of the original equations. Let's use the first original equation:
66 * (-199/209) - 44y = 32
-13034/209 - 44y = 32
-44y = 32 + 13034/209
-44y = 19922/209
y = (19922/209) / -44 = -453/209
### Answer: y = -453/209, z = -199/209
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.
Gemini-1.5 Pro:
Reasoning: Let's use the elimination method to solve for t and h. 1. **Solve for 't' in the second equation:** 69h + 11t = 124 11t = 124 - 69h t = (124 - 69h) / 11 2. **Substitute the value of 't' in the first equation:** 83 * [(124 - 69h) / 11] - 58h = 45 3. **Simplify and solve for 'h':** (10172 - 5727h) / 11 - 58h = 45 10172 - 5727h - 638h = 495 -6365h = -9677 h = 1.52 4. **Substitute the value of 'h' back into the equation for 't':** t = (124 - 69 * 1.52) / 11 t = (124 - 104.88) / 11 t = 1.74 ### Answer: t = 1.74, h = 1.52
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?
Gemini-1.5 Pro:
Reasoning: To solve for two variables, we need to form two equations with one variable and solve them. Let's multiply the second equation by 61/12: ((16*61/12 * h) - (12* 61/12 * o)) = (9 * 61/12) which simplifies to (81.33 * h) - (61 * o) = 45.75 Let's subtract the modified second equation from the first equation: ((79 * h) - (61 * o)) - ((81.33 * h) - (61 * o)) = 9 - 45.75 This simplifies to -2.33*h = -36.75 Solving for h, we get h = -36.75/-2.33 = 15.77 Substituting the value of h back into the original second equation to solve for o: (16 * 15.77) - (12 * o) = 9 This simplifies to 252.32 - (12 * o) = 9 Further simplification leads to 12 * o = 243.32 Solving for o, we get o = 243.32/12 = 20.28 Therefore, the solutions to the system of equations are h = 15.77 and o = 20.28. ### Answer: h = 15.77, o = 20.28
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
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. We can multiply the top equation by 15 and the bottom equation by -8 to make the 's' coefficients cancel out when we add them. (4y + 56s = 139) * 15 --> 60y + 840s = 2085 (65y - 90s = 16) * -8 --> -520y + 720s = -128 Adding the modified equations: (60y + 840s = 2085) + (-520y + 720s = -128) --> -460y + 1560s = 1957 Now we have: -460y + 1560s = 1957 Let's solve this equation for 'y': -460y = -1560s + 1957 y = (1560s - 1957) / 460 y = 3.4s - 4.25 Substitute this value of 'y' into any of the original equations to solve for 's'. Let's use the first original equation: 4(3.4s - 4.25) + 56s = 139 13.6s - 17 + 56s = 139 69.6s = 156 s = 2.24 Now, substitute the value of 's' back into the equation for 'y': y = 3.4 * 2.24 - 4.25 y = 7.62 - 4.25 y = 3.37 ### Answer: s = 2.24, y = 3.37
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.
Gemini-1.5 Pro:
Reasoning: Let's call the first equation Equation 1 and the second equation Equation 2. **Step 1: Eliminate 'z'** * Multiply Equation 1 by 45: (42d - 4z) * 45 = 75 * 45 => 1890d - 180z = 3375 * Multiply Equation 2 by 4: (40d + 45z) * 4 = 162 * 4 => 160d + 180z = 648 **Step 2: Add the modified equations** (1890d - 180z) + (160d + 180z) = 3375 + 648 2050d = 4023 **Step 3: Solve for 'd'** d = 4023 / 2050 d = 1.962 **Step 4: Substitute 'd' into either original equation to solve for 'z'** Let's use Equation 1: 42 * 1.962 - 4z = 75 82.404 - 4z = 75 -4z = 75 - 82.404 -4z = -7.404 z = -7.404 / -4 z = 1.851 ### Answer: d = 1.962, z = 1.851
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. First, we can simplify both equations: Equation 1: 21o + 12a = 71 (divide both sides by 2) Equation 2: 21a + 3o = 37.5 (divide both sides by 2) Now, multiply Equation 2 by -4 to make the 'o' coefficients cancel each other out: Equation 2 (modified): -84a - 12o = -150 Add the modified Equation 2 to Equation 1: 21o + 12a = 71 -84a - 12o = -150 ------------------ -63a = -79 Solving for 'a': a = -79 / -63 = 1.25 Substitute the value of 'a' (1.25) back into either Equation 1 or Equation 2 to solve for 'o'. Let's use Equation 1: 21o + 12(1.25) = 71 21o + 15 = 71 21o = 56 o = 56 / 21 = 2.67 Therefore, a = 1.25 and o = 2.67. ### Answer: a = 1.25, o = 2.67
Expected answer: {a: 383/270, o: 347/135}
Correct? No
Problem: Given the following system of equations, find the values of p and r: 39*r - 2*p = 45 and 16*p - 84*r = 36.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. First, we can simplify the second equation by dividing both sides by 4: 4p - 21r = 9. Next, we can multiply the simplified second equation by -9.5 to make the 'p' coefficients cancel out: -38p + 199.5r = -85.5. Now we can add this modified equation to the original first equation: (39r - 2p) + (-38p + 199.5r) = 45 + (-85.5). This simplifies to 238.5r = -40.5. Dividing both sides by 238.5, we get r = -0.17. We can substitute this value of r back into the simplified second equation: 4p - 21(-0.17) = 9. Simplifying, we get 4p + 3.57 = 9. Subtracting 3.57 from both sides, we get 4p = 5.43. Finally, dividing both sides by 4, we get p = 1.3575. ### Answer: p = 1.3575, r = -0.17
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
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination. We can multiply the first equation by 8 and the second equation by 23 to make the 'd' coefficients have opposite signs: 1. (184 * d) - (416 * w) = 56 2. (1472 * w) - (184 * d) = 1334 Now, we can add the two equations together, eliminating 'd': (1472 * w) - (416 * w) = 1334 + 56 1056 * w = 1390 Next, we can solve for 'w': w = 1390 / 1056 w = 1.315 Now that we know the value of 'w', we can substitute it back into either of the original equations to solve for 'd'. Let's use the first equation: (23 * d) - (52 * 1.315) = 7 23 * d - 68.38 = 7 23 * d = 75.38 d = 75.38 / 23 d = 3.28 ### Answer: d = 3.28, w = 1.315
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.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination.
1. **Multiply the equations to match coefficients of one variable:** We'll aim to eliminate 'n'.
- Multiply the first equation by 47:
(47 * 71 * z) + (47 * 60 * n) = 47 * 184
This simplifies to: 3317z + 2820n = 8648
- Multiply the second equation by 60:
(60 * 52 * z) - (60 * 47 * n) = 60 * 19
This simplifies to: 3120z - 2820n = 1140
2. **Add the modified equations to eliminate 'n':**
(3317z + 2820n) + (3120z - 2820n) = 8648 + 1140
This simplifies to: 6437z = 9788
3. **Solve for 'z':**
z = 9788 / 6437
z = 1.52
4. **Substitute the value of 'z' back into either original equation to solve for 'n'.** Let's use the first equation:
(71 * 1.52) + (60 * n) = 184
107.92 + 60n = 184
60n = 76.08
n = 76.08 / 60
n = 1.27
Therefore, the solution to the system of equations is z = 1.52 and n = 1.27.
### Answer: z = 1.52, n = 1.27
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
Gemini-1.5 Pro:
Reasoning: Let's solve for *z* in the first equation. Subtracting 70*e from both sides, we get: 90*z = 189 - 70*e Dividing both sides by 90, we get: z = (189-70*e)/90 Now, let's substitute this value of *z* into the second equation: 99 * ((189-70*e)/90) - 84*e = 20 Simplifying, we get: 207.9 - 77*e - 84*e = 20 Combining like terms: -161*e = -187.9 Dividing both sides by -161, we get the value of e: e = 1.167 Now, we can substitute this value of e back into either of the original equations to solve for z. Let's use the first equation: 70 * 1.167 + 90 * z = 189 Simplifying: 81.69 + 90*z = 189 Subtracting 81.69 from both sides: 90*z = 107.31 Dividing both sides by 90, we get the value of z: z = 1.192 ### Answer: e = 1.167, z = 1.192
Expected answer: {e: 1879/1610, z: 1234/1035}
Correct? No
Problem: Find the values of o and f that satisfy the following system of equations: 1) 26o - 44f = 95 2) 87f + 11o = 169.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using elimination.
1. **Multiply equations to match coefficients:** We'll eliminate 'o' by multiplying equation 1 by -11 and equation 2 by 26:
(-11) * (26o - 44f = 95) -> -286o + 484f = -1045
(26) * (87f + 11o = 169) -> 2262f + 286o = 4394
2. **Add the equations together:**
(-286o + 484f = -1045)
+(2262f + 286o = 4394)
------------------------
2746f = 3349
3. **Solve for 'f':**
f = 3349 / 2746 = 1.22
4. **Substitute 'f' back into either original equation to solve for 'o':** Let's use equation 1:
26o - 44(1.22) = 95
26o - 53.68 = 95
26o = 148.68
o = 5.72
### Answer: o = 5.72, f = 1.22
Expected answer: {f: 3349/2746, o: 15701/2746}
Correct? No
Problem: Calculate the values of z and p using the following system of equations: 40*z - 20*p = 51 6*z + 58*p = 165.
Gemini-1.5 Pro:
Reasoning: Let's solve this system of equations using the elimination method. We'll eliminate 'z' first: 1. **Multiply equations to match 'z' coefficients:** Multiply the first equation by 3: (40z - 20p = 51) * 3 => 120z - 60p = 153 Multiply the second equation by -20: (6z + 58p = 165) * -20 => -120z - 1160p = -3300 2. **Add the modified equations to eliminate 'z':** (120z - 60p) + (-120z - 1160p) = 153 - 3300 This simplifies to -1220p = -3147 3. **Solve for 'p':** p = -3147 / -1220 = 2.58 4. **Substitute 'p' back into either original equation to solve for 'z':** Let's use the first equation: 40z - 20 * 2.58 = 51 Simplify: 40z - 51.6 = 51 Add 51.6 to both sides: 40z = 102.6 Solve for 'z': z = 102.6 / 40 = 2.565 Therefore, the solution to the system of equations is z = 2.565 and p = 2.58. ### Answer: z = 2.565, p = 2.58
Expected answer: {p: 3147/1220, z: 3129/1220}
Correct? No
Problem: Find the values of s and f from the given system of equations: (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93.
Gemini-1.5 Pro:
We can use elimination to solve for *s* and *f*. 1. **Multiply the second equation by -29 to make the *f* coefficients have opposite signs:** (-29) * (3 * *f*) + (-29) * (10 * *s*) = (-29) * 93 -87*f - 290*s = -2797 2. **Now we have the following system of equations:** 71*s - 89*f = 5 -290*s - 87*f = -2797 3. **Add the two equations together to eliminate *f*:** (71*s - 290*s) + (-89*f - 87*f) = 5 - 2797 -219*s = -2792 4. **Solve for *s*:** s = -2792 / -219 s = 12.75 (rounded to two decimal places) 5. **Substitute the value of *s* back into either of the original equations to solve for *f*. Let's use the second equation:** (3 * *f*) + (10 * 12.75) = 93 3*f + 127.5 = 93 3*f = -34.5 6. **Solve for *f*:** f = -34.5 / 3 f = -11.5 ### Answer: s = 12.75, f = -11.5
Expected answer: {f: 6553/1103, s: 8292/1103}
Correct? No