Gemini-1.5 Flash 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 Flash:
Here's how we can solve for the values of q and s using the system of equations: **1. Elimination Method** * **Multiply the second equation by 7.6667 (approximately):** This will allow us to eliminate 'q' when we add the equations together. (30 * s) + (12 * q) = 121 * 7.6667 = (229.9991 * s) + (91.9996 * q) = 925.0007 * **Add the modified second equation to the first equation:** (40 * s) - (91 * q) = 37 (229.9991 * s) + (91.9996 * q) = 925.0007 --------------------------------------- (269.9991 * s) = 962.0007 * **Solve for 's':** s = 962.0007 / 269.9991 s ≈ 3.56 **2. Substitute 's' back into either equation to solve for 'q':** Let's use the first equation: (40 * 3.56) - (91 * q) = 37 142.4 - (91 * q) = 37 - (91 * q) = 37 - 142.4 - (91 * q) = -105.4 q = -105.4 / -91 q ≈ 1.15 **Therefore:** * **s ≈ 3.56** * **q ≈ 1.15** ### Answer: s ≈ 3.56, q ≈ 1.15
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 Flash:
We can solve this system of equations using elimination. 1. **Multiply the first equation by 3 and the second equation by 2:** This will make the coefficients of 'b' in both equations multiples of 6, allowing us to eliminate 'b'. * 12b + 132m = 540 * 158m - 12b = 96 2. **Add the two equations together:** This eliminates 'b' and gives us an equation solely in terms of 'm'. * 290m = 636 3. **Solve for 'm':** Divide both sides of the equation by 290 to isolate 'm'. * m = 636/290 * m = 2.193 (approximately) 4. **Substitute the value of 'm' back into either original equation to solve for 'b':** Let's use the first equation: 4b + 44m = 180 * 4b + 44(2.193) = 180 * 4b + 96.52 = 180 * 4b = 83.48 * b = 20.87 (approximately) ### Answer: b = 20.87 and m = 2.193
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 Flash:
We can solve this system of equations using elimination. Here's how:
1. **Multiply the first equation by 29 and the second equation by 15 to make the coefficients of 'g' opposites.**
* Equation 1: (29) * (42 * o - 15 * g) = (29) * 28
-> 1218 * o - 435 * g = 812
* Equation 2: (15) * (47 * o + 29 * g) = (15) * 141
-> 705 * o + 435 * g = 2115
2. **Add the two modified equations together. This eliminates 'g'.**
* (1218 * o - 435 * g) + (705 * o + 435 * g) = 812 + 2115
* 1923 * o = 2927
3. **Solve for 'o'.**
* o = 2927 / 1923
* o = 1.52 (approximately)
4. **Substitute the value of 'o' back into either of the original equations to solve for 'g'. Let's use the first equation.**
* 42 * 1.52 - 15 * g = 28
* 63.84 - 15 * g = 28
* -15 * g = -35.84
* g = -35.84 / -15
* g = 2.39 (approximately)
### Answer: o = 1.52, g = 2.39
Expected answer: {g: 4606/1923, o: 2927/1923}
Correct? No
Problem: Given the following system of equations, determine the values of v and o: The first equation is: ((12 * v) - (6 * o)) = 21 The second equation is: ((76 * v) - (45 * o)) = 50.
Gemini-1.5 Flash:
Here's how to solve the system of equations: **1. Elimination Method** * **Multiply the first equation by -19:** This will make the 'v' coefficients opposites. - (-19) * ((12 * v) - (6 * o)) = 21 * (-19) - (-228 * v) + (114 * o) = -399 * **Rewrite the second equation:** - (76 * v) - (45 * o) = 50 * **Add the two equations together:** Notice that the 'v' terms cancel out. - (69 * o) = -349 * **Solve for 'o':** - o = -349 / 69 - o = -5 * **Substitute the value of 'o' back into either original equation to solve for 'v'. Let's use the first equation:** - (12 * v) - (6 * -5) = 21 - (12 * v) + 30 = 21 - (12 * v) = -9 - v = -9 / 12 - v = -3/4 **Therefore:** * v = -3/4 * o = -5 ### Answer: v = -3/4, o = -5
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 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the second equation by -37.5** This will allow us to eliminate 'f' when we add the equations together. -37.5 * (2 * f + 90 * g) = -37.5 * 143 -75 * f - 3375 * g = -5362.5 2. **Add the first equation to the modified second equation:** (75 * f + 25 * g = 138) + (-75 * f - 3375 * g = -5362.5) -3350 * g = -5224.5 3. **Solve for g:** g = -5224.5 / -3350 g = 1.56 4. **Substitute the value of g back into either of the original equations to solve for f. Let's use the first equation:** 75 * f + 25 * 1.56 = 138 75 * f + 39 = 138 5. **Solve for f:** 75 * f = 99 f = 99 / 75 f = 1.32 ### Answer: f = 1.32, g = 1.56
Expected answer: {f: 1769/1340, g: 10449/6700}
Correct? No
Problem: Solve the following system of equations for variables y and n: the first equation is 194 = 4 * y + 14 * n and the second equation is 7 = 65 * n - 79 * y.
Gemini-1.5 Flash:
Here's how to solve the system of equations: **1. Elimination Method** * **Multiply the first equation by 79 and the second equation by 14:** This will allow us to have opposite coefficients for 'y' in both equations. * (79) * (194 = 4y + 14n) --> 15386 = 316y + 1096n * (14) * (7 = 65n - 79y) --> 98 = 910n - 1106y * **Add the two modified equations together:** This eliminates 'y'. * 15386 + 98 = 316y + 1096n + 910n - 1106y * 15484 = -790y + 2006n * **Simplify the combined equation:** * 7742 = -395y + 1003n * **Solve the second original equation for 'y':** * 7 = 65n - 79y * 79y = 65n - 7 * y = (65n - 7) / 79 * **Substitute this expression for 'y' into the simplified combined equation:** * 7742 = -395 * ((65n - 7) / 79) + 1003n * **Simplify and solve for 'n':** * 7742 = (-25675n + 3115)/79 + 1003n * 612638 = -25675n + 3115 + 79237n * 612638 = 53562n + 3115 * 609523 = 53562n * n = 11.38 (approximately) * **Substitute the value of 'n' back into the equation for 'y':** * y = (65 * 11.38 - 7) / 79 * y = 735.7 - 7 / 79 * y = 9.22 (approximately) **Therefore:** ### Answer: n ≈ 11.38, y ≈ 9.22
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 Flash:
Here's how to solve the system of equations: **1. Solve for one variable in terms of the other in one equation.** Let's solve the first equation for *g*: * 146 = 28 * g + 26 * l * 146 - 26 * l = 28 * g * g = (146 - 26 * l) / 28 **2. Substitute this expression into the second equation.** * 6 = 41 * l - 53 * [(146 - 26 * l) / 28] **3. Simplify and solve for *l*.** * 6 = 41 * l - 53 * (146/28) + 53 * (26/28) * l * 6 = 41 * l - 270.25 + 49.75 * l * 276.25 = 90.75 * l * l = 276.25 / 90.75 * l = 3 **4. Substitute the value of *l* back into either of the original equations to solve for *g*.** Let's use the first equation: * 146 = 28 * g + 26 * 3 * 146 = 28 * g + 78 * 68 = 28 * g * g = 68 / 28 * g = 2.43 (rounded to two decimal places) **Therefore, the values of l and g that satisfy the equations are:** * l = 3 * g = 2.43 ### Answer: l = 3, g = 2.43
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 Flash:
Here's how to solve the system of equations: **Method: Elimination** 1. **Multiply equations to make coefficients match:** - Multiply equation (1) by 7: 140p + 392y = 630 - Multiply equation (2) by -56: -4984p - 392y = -5488 2. **Add the modified equations:** Notice that the 'y' terms will cancel out: - (-4984p + 140p) + (-392y + 392y) = -5488 + 630 - -4844p = -4858 3. **Solve for 'p':** - p = -4858 / -4844 - p = 1.003 4. **Substitute the value of 'p' back into either original equation to solve for 'y'. Let's use equation (1):** - 20 * (1.003) + 56y = 90 - 20.06 + 56y = 90 - 56y = 69.94 - y = 69.94 / 56 - y = 1.249 **Therefore:** * p = 1.003 * y = 1.249 ### Answer: p = 1.003, y = 1.249
Expected answer: {p: 347/346, y: 3025/2422}
Correct? No
Problem: Given the system of equations: 72*k - 22*d = 17 and 6*d +69*k = 96, find the values of variables k and d.
Gemini-1.5 Flash:
Here's how to solve the system of equations to find the values of k and d:
**1. Elimination Method**
* **Multiply equations to make coefficients match:**
- Multiply the first equation by 6: 432k - 132d = 102
- Multiply the second equation by 22: 132d + 1518k = 2112
* **Add the modified equations:**
- Notice that the 'd' terms have opposite signs and will cancel out:
432k + 1518k = 102 + 2112
1950k = 2214
* **Solve for k:**
- Divide both sides by 1950:
k = 2214/1950
k = 1.135 (rounded to three decimal places)
* **Substitute the value of k back into either original equation to solve for d:**
- Let's use the first equation:
72 * 1.135 - 22d = 17
81.72 - 22d = 17
-22d = -64.72
d = -64.72 / -22
d = 2.942 (rounded to three decimal places)
**Therefore:**
* **k = 1.135**
* **d = 2.942**
### Answer: k = 1.135, d = 2.942
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 Flash:
We can solve this system of equations using elimination. Here's how: **1. Multiply the second equation by 3.3:** This will allow us to have the same coefficient for 'd' in both equations, but with opposite signs. 3.3 * (10d + 14h) = 3.3 * 91 33d + 46.2h = 300.3 **2. Add the modified second equation to the first equation:** This will eliminate 'd'. 73h - 33d = 48 33d + 46.2h = 300.3 ------------------ 119.2h = 348.3 **3. Solve for 'h':** h = 348.3 / 119.2 h = 2.92 **4. Substitute the value of 'h' back into either original equation to solve for 'd'. Let's use the second equation:** 10d + 14 * 2.92 = 91 10d + 40.88 = 91 10d = 50.12 d = 5.012 **Therefore, the solutions are:** ### Answer: d = 5.012, h = 2.92
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 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply Equation 1 by 30 and Equation 2 by 43:** This will allow us to have matching coefficients for 'q' in both equations. - Equation 1 * 30: 1290q - 1320e = 870 - Equation 2 * 43: 1290q + 1849e = 5590 2. **Subtract Equation 1 from Equation 2:** This will eliminate 'q' and allow us to solve for 'e'. - (1290q + 1849e) - (1290q - 1320e) = 5590 - 870 - 3169e = 4720 3. **Solve for 'e':** - e = 4720 / 3169 - e ≈ 1.49 4. **Substitute the value of 'e' back into either Equation 1 or 2 to solve for 'q'.** Let's use Equation 1: - 43q - 44(1.49) = 29 - 43q - 65.56 = 29 - 43q = 94.56 - q ≈ 2.19 Therefore, the solutions are: ### Answer: q ≈ 2.19, 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 Flash:
Here's how to solve this system of equations: **1. Solve for one variable in terms of the other** Let's solve the second equation for 'r': * 109 = (37 * a) + (6 * r) * 109 - (37 * a) = 6 * r * r = (109 - (37 * a)) / 6 **2. Substitute into the first equation** Now, substitute this expression for 'r' into the first equation: * 53 = ((89 * ((109 - (37 * a)) / 6)) - (74 * a)) **3. Simplify and solve for 'a'** * 53 = (89 * (109 - 37a) / 6) - 74a * 53 * 6 = 89 * (109 - 37a) - 74a * 6 * 318 = 9721 - 3293a - 444a * 318 - 9721 = -3737a * -9403 = -3737a * a = -9403 / -3737 * a = 2.52 (approximately) **4. Substitute 'a' to find 'r'** Now that we know 'a', we can plug it back into either of the original equations to solve for 'r'. Let's use the second equation: * 109 = (37 * 2.52) + (6 * r) * 109 = 93.24 + 6r * 109 - 93.24 = 6r * 15.76 = 6r * r = 15.76 / 6 * r = 2.63 (approximately) **Therefore:** * r ≈ 2.63 * a ≈ 2.52 ### Answer: r ≈ 2.63, a ≈ 2.52
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 Flash:
Here's how to solve for 'g' and 'q': **Step 1: Define Equations** We can represent the given information as a system of equations: * Equation 1: 42g + 12q = 169 * Equation 2: 12q - 50g = 19 **Step 2: Solve for one variable** We can use the elimination method to solve for one variable. Let's eliminate 'q': * Multiply Equation 1 by -1: -42g - 12q = -169 * Add the modified Equation 1 to Equation 2: -92g = -150 * Solve for 'g': g = -150 / -92 = 1.63 (rounded to two decimal places) **Step 3: Solve for the other variable** Now that we know 'g', substitute it back into either Equation 1 or Equation 2 to solve for 'q'. Let's use Equation 1: * 42(1.63) + 12q = 169 * 68.46 + 12q = 169 * 12q = 100.54 * q = 100.54 / 12 = 8.38 (rounded to two decimal places) **Answer:** * g = 1.63 * q = 8.38
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 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the first equation by -2:** This will allow us to eliminate 's' when we add the equations together. -74s + 22v = -108 2. **Add the modified first equation to the second equation:** (82s - 62v = 41) + (-74s + 22v = -108) This gives us: 8s - 40v = -67 3. **Solve for 'v':** -40v = -67 - 8s v = (67 + 8s) / 40 4. **Substitute the value of 'v' into either of the original equations to solve for 's'.** Let's use the first equation: 37s - 11 * ((67 + 8s) / 40) = 54 5. **Simplify and solve for 's':** 1480s - 11(67 + 8s) = 2160 1480s - 737 - 88s = 2160 1392s = 2897 s = 2897 / 1392 s = 2.08 (rounded to two decimal places) 6. **Substitute the value of 's' back into the equation for 'v':** v = (67 + 8 * 2.08) / 40 v = (67 + 16.64) / 40 v = 83.64 / 40 v = 2.09 (rounded to two decimal places) ### Answer: s = 2.08, v = 2.09
Expected answer: {s: 2897/1392, v: 2911/1392}
Correct? No
Problem: Based on the following two equations, find the values for c and z: The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48.
Gemini-1.5 Flash:
Here's how to solve for 'c' and 'z' using the given equations: **1. Elimination Method:** * **Multiply the first equation by 48 and the second equation by 65 to make the coefficients of 'c' opposites:** * (41 * c + 65 * z = 151) * 48 = 1968c + 3120z = 7248 * (78 * z - 48 * c = 48) * 65 = 5070z - 3120c = 3120 * **Add the two equations together:** * 1968c + 3120z = 7248 * + 5070z - 3120c = 3120 * ------------------------ * 6968c = 4128 * **Solve for 'c':** * c = 4128 / 6968 * c = 0.59 * **Substitute the value of 'c' back into either of the original equations to solve for 'z'. Let's use the first equation:** * 41 * 0.59 + 65 * z = 151 * 24.19 + 65 * z = 151 * 65 * z = 126.81 * z = 126.81 / 65 * z = 1.95 **Therefore:** * c = 0.59 * z = 1.95 ### Answer: c = 0.59, z = 1.95
Expected answer: {c: 37/27, z: 512/351}
Correct? No
Problem: Given the following system of equations: 60 * c - 91 * i = 7 7 * i + 76 * c = 148 What are the values of c and i?
Gemini-1.5 Flash:
Here's how to solve the system of equations: **1. Elimination Method** * **Multiply the second equation by 91:** This will allow us to eliminate 'i' when we add the equations together. (7 * i + 76 * c = 148) * 91 => 637 * i + 6916 * c = 13498 * **Add the two equations together:** 60c - 91i = 7 637i + 6916c = 13498 ----------------------- 6976c = 13505 * **Solve for 'c':** c = 13505 / 6976 c = 1.935 (approximately) **2. Substitute 'c' back into either original equation to solve for 'i':** Let's use the first equation: 60 * 1.935 - 91 * i = 7 116.1 - 91 * i = 7 -91 * i = -109.1 i = -109.1 / -91 i = 1.199 (approximately) **Therefore:** * c ≈ 1.935 * i ≈ 1.199 ### Answer: c ≈ 1.935, i ≈ 1.199
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 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the first equation by 41 and the second equation by 37 to make the coefficients of 'e' opposites:** * (71*v - 37*e) * 41 = 69 * 41 * (41*e + 76*v) * 37 = 184 * 37 This gives us: * 2911*v - 1517*e = 2829 * 1517*e + 2812*v = 6808 2. **Add the two equations together. This will eliminate 'e':** * (2911*v - 1517*e) + (1517*e + 2812*v) = 2829 + 6808 * 5723*v = 9637 3. **Solve for 'v':** * v = 9637 / 5723 * v ≈ 1.68 4. **Substitute the value of 'v' back into either of the original equations to solve for 'e'. Let's use the first equation:** * 71 * 1.68 - 37 * e = 69 * 119.28 - 37 * e = 69 * -37 * e = -50.28 * e ≈ 1.36 Therefore, the approximate values for 'e' and 'v' that satisfy the given system of equations are: ### Answer: e ≈ 1.36, v ≈ 1.68
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 Flash:
Here's how to solve the system of equations: **1. Set up the equations:** * 25v + 69d = 113 * 26d + 42v = 85 **2. Solve for one variable in terms of the other:** Let's solve the first equation for 'v': * 25v = 113 - 69d * v = (113 - 69d) / 25 **3. Substitute:** Substitute this expression for 'v' into the second equation: * 26d + 42 * ((113 - 69d) / 25) = 85 **4. Simplify and solve for 'd':** * 26d + (4746 - 2898d) / 25 = 85 * 650d + 4746 - 2898d = 2125 * -2248d = -2621 * d = -2621 / -2248 * d = 1.167 (approximately) **5. Substitute to find 'v':** Now that we know 'd', we can substitute it back into either of the original equations to find 'v'. Let's use the first equation: * 25v + 69 * 1.167 = 113 * 25v + 80.623 = 113 * 25v = 32.377 * v = 32.377 / 25 * v = 1.295 (approximately) **Therefore:** * d = 1.167 (approximately) * v = 1.295 (approximately) ### Answer: d = 1.167, v = 1.295
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 Flash:
Here's how to solve the system of equations for *g* and *z*: **1. Elimination Method** * **Multiply the second equation by 14.5:** This will make the coefficients of *z* opposites. (72 * g - 3 * z = 68) * 14.5 = 1044 * g - 43.5 * z = 986 * **Add the modified second equation to the first equation:** This eliminates *z*. (43 * z - 91 * g = 33) + (1044 * g - 43.5 * z = 986) ----------------------- 953 * g = 1019 * **Solve for *g*:** g = 1019 / 953 g = 1.07 * **Substitute the value of *g* back into either original equation to solve for *z*. Let's use the first equation:** 43 * z - 91 * (1.07) = 33 43 * z - 97.57 = 33 43 * z = 130.57 z = 130.57 / 43 z = 3.04 **Therefore:** * g = 1.07 * z = 3.04 ### Answer: g = 1.07, z = 3.04
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 Flash:
We can solve this system of equations using elimination. To eliminate 'f', we'll multiply the first equation by 21 and the second equation by 12. This gives us: * 1050c + 252f = 1932 * 432c - 252f = 408 Now, we can add the two equations together, eliminating 'f': 1482c = 2340 Solving for 'c': c = 2340 / 1482 = 1.57 (approximately) Now, we can substitute this value of 'c' back into either of the original equations to solve for 'f'. Let's use the first equation: 50(1.57) + 12f = 92 78.5 + 12f = 92 12f = 13.5 f = 13.5 / 12 = 1.125 (approximately) Therefore, Alex's basket contained approximately 1.57 cherries, and Brad's basket contained approximately 1.125 figs. It's worth noting that the answers are not whole numbers, which might indicate a slight error in the recipe books or the given weights. ### Answer: c = 1.57, f = 1.125
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 Flash:
Here's how to solve the system of equations:
**1. Elimination Method**
* **Multiply Equation 1 by 85 and Equation 2 by 75:**
* (85) * (87u - 75d) = (85) * (7) --> 7395u - 6375d = 595
* (75) * (85d - 82u) = (75) * (54) --> 6375d - 6150u = 4050
* **Add the two equations together:**
* (7395u - 6375d) + (6375d - 6150u) = 595 + 4050
* 1245u = 4645
* **Solve for u:**
* u = 4645 / 1245
* u = 3.73 (approximately)
**2. Substitute the value of u back into either original equation to solve for d:**
* Let's use Equation 1: 87(3.73) - 75d = 7
* 324.51 - 75d = 7
* -75d = -317.51
* d = -317.51 / -75
* d = 4.23 (approximately)
**Therefore:**
* **u ≈ 3.73**
* **d ≈ 4.23**
### Answer: u ≈ 3.73, d ≈ 4.23
Expected answer: {d: 5272/1245, u: 929/249}
Correct? No
Problem: Bill is trying to decipher a secret code. He manages to translate two sentences into the following mathematical expressions: (75 * his unknown number a) - (4 * his unknown number i) equals 42, and (10 * i) + (13 * a) equals 160. Using these equations, can you figure out the values of i and a for him?
Gemini-1.5 Flash:
Here's how we can solve for 'a' and 'i': **1. Set up the equations:** * Equation 1: 75a - 4i = 42 * Equation 2: 10i + 13a = 160 **2. Solve for one variable in terms of the other:** Let's solve Equation 1 for 'i': * 75a - 4i = 42 * -4i = 42 - 75a * i = (75a - 42) / 4 **3. Substitute the expression for 'i' into Equation 2:** * 10i + 13a = 160 * 10 * ((75a - 42) / 4) + 13a = 160 **4. Simplify and solve for 'a':** * (750a - 420) / 4 + 13a = 160 * 750a - 420 + 52a = 640 * 802a = 1060 * a = 1060 / 802 * a = 1.32 **5. Substitute the value of 'a' back into the expression for 'i':** * i = (75a - 42) / 4 * i = (75 * 1.32 - 42) / 4 * i = (99 - 42) / 4 * i = 57 / 4 * i = 14.25 **Therefore, the values of 'a' and 'i' are:** * a = 1.32 * i = 14.25 ### Answer: a = 1.32, i = 14.25
Expected answer: {a: 530/401, i: 5727/401}
Correct? No
Problem: Solve the following system of equations for the values of p and c: 37 * p + 74 * c = 183 27 * c - 18 * p = 13
Gemini-1.5 Flash:
Here's how to solve the system of equations: **1. Solve for one variable in terms of the other:** Let's solve the second equation for *c*: * 27 * c - 18 * p = 13 * 27 * c = 18 * p + 13 * c = (18 * p + 13) / 27 **2. Substitute the expression into the first equation:** Substitute the expression for *c* into the first equation: * 37 * p + 74 * [(18 * p + 13) / 27] = 183 **3. Simplify and solve for *p*:** * 37 * p + (74 * 18 * p + 74 * 13) / 27 = 183 * 999 * p + 1332 * p + 962 = 4959 * 2331 * p = 4959 - 962 * 2331 * p = 3997 * p = 3997 / 2331 * p = 1.71 **4. Substitute the value of *p* back to find *c*:** Substitute *p = 1.71* into the expression for *c* we derived earlier: * c = (18 * 1.71 + 13) / 27 * c = (30.78 + 13) / 27 * c = 43.78 / 27 * c = 1.62 **Therefore, the solutions are:** * p = 1.71 * 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 Flash:
Here's how to solve for k and p using a system of equations:
**1. Elimination Method**
* **Multiply the second equation by 2:** This will allow us to eliminate 'p' when we add the equations together.
(2 * (16 * p) + (2 * (33 * k) = 2 * 105) --> 32p + 66k = 210
* **Now we have two equations:**
* 48k - 30p = 37
* 32p + 66k = 210
* **Add the two equations together:** Notice that the 'p' terms will cancel out.
* (48k - 30p) + (32p + 66k) = 37 + 210
* 114k + 2p = 247
* **Solve for k:**
* 114k = 247
* k = 247 / 114
* k = 2.167 (approximately)
**2. Substitute to find p**
* **Choose either of the original equations and substitute the value of k you just found.** Let's use the first equation:
* (48 * 2.167) - (30 * p) = 37
* 104.016 - 30p = 37
* -30p = 37 - 104.016
* -30p = -67.016
* p = -67.016 / -30
* p = 2.234 (approximately)
**Therefore:**
* k = 2.167 (approximately)
* p = 2.234 (approximately)
### Answer: k = 2.167, p = 2.234
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 Flash:
We can solve this system of equations using elimination. Here's how: **1. Multiply the equations to get matching coefficients for one of the variables.** Let's eliminate 'f'. Multiply the first equation by 46 and the second equation by 69: * (75v - 69f) * 46 = 14 * 46 => 3450v - 3174f = 644 * (46f + 53v) * 69 = 110 * 69 => 3174f + 3657v = 7590 **2. Add the two modified equations together.** Notice that the 'f' terms cancel out: 3450v + 3657v = 644 + 7590 7107v = 8234 **3. Solve for 'v'.** v = 8234 / 7107 v ≈ 1.159 **4. Substitute the value of 'v' back into either original equation to solve for 'f'.** Let's use the first equation: 75 * 1.159 - 69f = 14 86.925 - 69f = 14 -69f = -72.925 f ≈ 1.057 **Therefore:** * v ≈ 1.159 * f ≈ 1.057 ### Answer: v ≈ 1.159, f ≈ 1.057
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 Flash:
We can solve for the values of *o* and *g* using a system of equations: 1. **Solve for one variable in terms of the other:** Let's solve the first equation for *o*: ``` 7 * o + 13 * g = 131 7 * o = 131 - 13 * g o = (131 - 13 * g) / 7 ``` 2. **Substitute the expression into the second equation:** Substitute the expression for *o* into the second equation: ``` 89 * g - 70 * ((131 - 13 * g) / 7) = 37 ``` 3. **Solve for the remaining variable:** Simplify and solve the equation for *g*: ``` 89 * g - 10 * (131 - 13 * g) = 37 89 * g - 1310 + 130 * g = 37 219 * g = 1347 g = 1347 / 219 g = 6.15 ``` 4. **Substitute the value of *g* back into either of the original equations to find *o*:** Let's use the first equation: ``` 7 * o + 13 * 6.15 = 131 7 * o = 131 - 79.95 7 * o = 51.05 o = 51.05 / 7 o = 7.29 ``` Therefore, the values of *o* and *g* are: ### Answer: o = 7.29, g = 6.15
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 Flash:
We can solve this system of equations using elimination. To do that, we'll multiply the second equation by 8 to make the coefficients of *d* match, but with opposite signs. Here's how it looks: 1. **Original equations:** * 88 * d - 60 * k = 38 * 33 * k - 11 * d = 46 2. **Multiply the second equation by 8:** * 88 * d - 60 * k = 38 * 264 * k - 88 * d = 368 3. **Add the two equations together:** Notice that the *d* terms cancel out. * 204 * k = 406 4. **Solve for *k*:** * k = 406 / 204 * k = 2. 5. **Substitute the value of *k* back into either of the original equations to solve for *d*. Let's use the first equation:** * 88 * d - 60 * (2) = 38 * 88 * d - 120 = 38 * 88 * d = 158 * d = 158 / 88 * d = 1.795 ### Answer: d = 1.795, k = 2.
Expected answer: {d: 669/374, k: 203/102}
Correct? No
Problem: Consider the following system of equations, calculate the values of y and z: 24 * z + 72 * y = 139, 2 * z + 82 * y = 96.
Gemini-1.5 Flash:
Here's how to solve for y and z:
**1. Elimination Method**
* **Multiply the second equation by -12:** This will allow us to eliminate 'z' when we add the equations together.
-12 * (2z + 82y) = -12 * 96
-24z - 984y = -1152
* **Add the modified second equation to the first equation:**
(24z + 72y = 139) + (-24z - 984y = -1152)
-912y = -1013
* **Solve for y:**
y = -1013 / -912
y = 1.11
* **Substitute the value of y back into either original equation to solve for z.** Let's use the first equation:
24z + 72(1.11) = 139
24z + 79.92 = 139
24z = 59.08
z = 59.08 / 24
z = 2.46
**Therefore:**
### 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 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the first equation by 8 and the second equation by 29 to make the coefficients of 'v' opposites:** - (87 * o) - (29 * v) = 1 --> (696 * o) - (232 * v) = 8 - (8 * v) + (30 * o) = 91 --> (232 * v) + (870 * o) = 2639 2. **Add the two equations together. This eliminates 'v':** - (696 * o) - (232 * v) = 8 - (232 * v) + (870 * o) = 2639 ------------------------ - (1566 * o) = 2647 3. **Solve for 'o':** - o = 2647 / 1566 - o = 1.685 (approximately) 4. **Substitute the value of 'o' back into either of the original equations to solve for 'v'. Let's use the first equation:** - (87 * 1.685) - (29 * v) = 1 - 146.745 - (29 * v) = 1 - (29 * v) = 145.745 - v = 145.745 / 29 - v = 5.026 (approximately) ### Answer: v = 5.026, o = 1.685
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 Flash:
Here's how to solve the system of equations:
**1. Elimination Method:**
* **Notice:** The 'h' terms have the same coefficient (23) but opposite signs in the two equations. This makes elimination a good strategy.
* **Add the equations together:**
(23h - 46p) + (11p + 23h) = 58 + 124
46h - 35p = 182
* **Simplify:**
2h - (5/2)p = 13
* **Multiply the simplified equation by 2 to get rid of the fraction:**
4h - 5p = 26
* **Now we have a new system:**
* 4h - 5p = 26
* 11p + 23h = 124
* **Multiply the first equation by 11 and the second equation by 5 to make the 'p' coefficients match:**
* 44h - 55p = 286
* 55p + 115h = 620
* **Add the equations together:**
* 159h = 906
* **Solve for 'h':**
* h = 906 / 159
* h = 5.7
* **Substitute the value of 'h' back into either of the original equations to solve for 'p'. Let's use the first equation:**
* 23(5.7) - 46p = 58
* 131.1 - 46p = 58
* -46p = -73.1
* p = 73.1 / 46
* p = 1.59
**Therefore:**
* h = 5.7
* p = 1.59
### Answer: h = 5.7, p = 1.59
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 Flash:
To solve this system of equations, we can use the elimination method. Here's how: 1. **Multiply the first equation by 7:** This will make the coefficients of 'f' opposites. * 7 * (19d + 6f) = 7 * 120 * 133d + 42f = 840 2. **Add the modified first equation to the second equation:** This will eliminate 'f'. * (133d + 42f) + (90d - 42f) = 840 + 69 * 223d = 909 3. **Solve for 'd':** * d = 909 / 223 * d = 4 4. **Substitute the value of 'd' back into either original equation to solve for 'f'. Let's use the first equation:** * 19(4) + 6f = 120 * 76 + 6f = 120 * 6f = 44 * f = 44 / 6 * f = 22/3 Therefore, the values of the integers are: ### Answer: d = 4, f = 22/3
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 Flash:
Here's how we can solve this system of linear equations:
**Method: Elimination**
1. **Multiply Equations to Match Coefficients:** We want to make the coefficients of either 'm' or 'r' the same (but with opposite signs) in both equations. Let's focus on 'r'.
* Multiply equation (1) by 52:
2184m + 2860r = 6604
* Multiply equation (2) by 55:
3630m - 2860r = 2420
2. **Add the Modified Equations:** Notice that the 'r' terms have opposite signs. Adding the equations will eliminate 'r':
(2184m + 2860r) + (3630m - 2860r) = 6604 + 2420
5814m = 9024
3. **Solve for 'm':**
m = 9024 / 5814
m = 1.55
4. **Substitute 'm' Back into One of the Original Equations:** Let's use equation (1):
42 * 1.55 + 55 * r = 127
65.1 + 55 * r = 127
5. **Solve for 'r':**
55 * r = 127 - 65.1
55 * r = 61.9
r = 61.9 / 55
r = 1.13
**Solution:**
Therefore, the solution to the system of equations is:
* m = 1.55
* r = 1.13
### Answer: m = 1.55, r = 1.13
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 Flash:
We can solve this system of equations using elimination. Here's how: **1. Multiply the equations to make the coefficients of one variable the same (or negatives of each other):** * Multiply the first equation by 71: 2201n + 4544w = 8520 * Multiply the second equation by -31: -1761n - 2227w = -5353 **2. Add the two equations together. Notice that the 'n' terms cancel out:** 2317w = 3167 **3. Solve for 'w':** w = 3167 / 2317 w ≈ 1.367 **4. Substitute the value of 'w' back into either of the original equations to solve for 'n'. Let's use the first equation:** 31n + 64(1.367) = 120 31n + 87.65 = 120 31n = 32.35 n ≈ 1.044 **Therefore, the values that satisfy the system of equations are:** ### Answer: n ≈ 1.044, w ≈ 1.367
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 Flash:
We can solve this system of linear equations using elimination. Here's how: **1. Multiply the equations to make the coefficients of 'a' or 'u' opposites.** * Multiply equation (1) by 77: 5544 * a + 2002 * u = 8624 * Multiply equation (2) by 26: 1976 * u - 2002 * a = 806 **2. Add the modified equations together.** Notice that the 'a' terms will cancel out. 5544 * a + 2002 * u = 8624 -2002 * a + 1976 * u = 806 ------------------------ 3520 * u = 9430 **3. Solve for 'u'.** u = 9430 / 3520 u = 2.68 **4. Substitute the value of 'u' back into either original equation to solve for 'a'. Let's use equation (1).** 72 * a + 26 * 2.68 = 112 72 * a + 69.68 = 112 72 * a = 42.32 a = 42.32 / 72 a = 0.59 **Therefore:** * u = 2.68 * a = 0.59 ### Answer: u = 2.68, a = 0.59
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 Flash:
Here's how to solve the system of equations:
**1. Elimination Method**
* **Multiply equations to get matching coefficients:**
* Multiply the first equation by 31:
1674n + 558b = 4433
* Multiply the second equation by 6:
522n - 558b = 288
* **Add the modified equations together:**
* Notice that the 'b' terms cancel out:
2196n = 4721
* **Solve for 'n':**
* n = 4721 / 2196 = 2.15 (approximately)
**2. Substitute to find 'b':**
* Choose one of the original equations (let's use the first one):
* 54(2.15) + 18b = 143
* **Simplify and solve for 'b':**
* 115.8 + 18b = 143
* 18b = 27.2
* b = 27.2 / 18 = 1.51 (approximately)
**Therefore, the solution to the system of equations is:**
### Answer: n ≈ 2.15, b ≈ 1.51
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 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the first equation by 88 and the second equation by 7:** This will allow us to have opposite coefficients for 'p' in both equations. * (53 * e + 7 * p) * 88 = 103 * 88 * (69 * e - 88 * p) * 7 = 6 * 7 This simplifies to: * 4664e + 616p = 9064 * 483e - 616p = 42 2. **Add the two equations together:** This eliminates 'p' from the equations. * 4664e + 483e = 9064 + 42 * 5147e = 9106 3. **Solve for 'e':** * e = 9106 / 5147 * e ≈ 1.77 4. **Substitute the value of 'e' back into either of the original equations to solve for 'p'.** Let's use the first equation: * 53 * 1.77 + 7 * p = 103 * 93.91 + 7p = 103 * 7p = 9.09 * p = 9.09 / 7 * p ≈ 1.30 Therefore, the cost of electricity (e) is approximately $1.77 and the cost of petroleum (p) is approximately $1.30. ### Answer: e ≈ $1.77, p ≈ $1.30
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 Flash:
Here's how to solve this system of linear equations:
**1. Elimination Method**
* **Multiply equations to match coefficients:**
* Multiply the first equation by 8: (144p - 56s) = 168
* Multiply the second equation by 7: (357s - 168p) = 98
* **Add the modified equations:** Notice that the 'p' coefficients have opposite signs.
(144p - 56s) + (357s - 168p) = 168 + 98
This simplifies to: 301s = 266
* **Solve for 's':**
s = 266 / 301
* **Substitute 's' back into either original equation to solve for 'p':**
Let's use the first equation:
(18p - 7 * (266/301)) = 21
18p - (1862/301) = 21
18p = 21 + (1862/301)
18p = (6323/301)
p = (6323/301) / 18
p = 6323/5418
**Therefore:**
* p = 6323/5418
* s = 266/301
### Answer: p = 6323/5418, s = 266/301
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 Flash:
Here's how to solve the system of equations: **1. Write the equations in standard form:** * Equation 1: 59c - 34w = 84 * Equation 2: 11c + 42w = 98 **2. Use elimination to solve for one variable:** * **Multiply Equation 2 by 34:** This will allow us to eliminate 'w' when we add the equations together. * (11c + 42w) * 34 = 98 * 34 * 374c + 1428w = 3332 * **Add the modified Equation 2 to Equation 1:** * (59c - 34w) + (374c + 1428w) = 84 + 3332 * 433c + 1394w = 3416 * 433c = 3416 (Since the 'w' terms cancel out) **3. Solve for 'c':** * Divide both sides by 433: * c = 3416 / 433 * c = 7.88 (approximately) **4. Substitute the value of 'c' back into either original equation to solve for 'w':** * Let's use Equation 1: * 59 * 7.88 - 34w = 84 * 465.52 - 34w = 84 * -34w = 84 - 465.52 * -34w = -381.52 * w = -381.52 / -34 * w = 11.22 (approximately) **Therefore:** * c = 7.88 * w = 11.22 ### Answer: c = 7.88, w = 11.22
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 Flash:
Here's how to solve the system of equations:
**1. Elimination Method**
* **Multiply the first equation by 2:** This will make the coefficients of 'f' opposites in both equations.
(45 * f + 42 * z) * 2 = 101 * 2
90 * f + 84 * z = 202
* **Multiply the second equation by 3:** This will also make the coefficients of 'f' opposites.
(60 * z - 42 * f) * 3 = 13 * 3
180 * z - 126 * f = 39
* **Add the two modified equations together:** Notice that the 'f' terms cancel out.
90 * f + 84 * z = 202
180 * z - 126 * f = 39
----------------------
264 * z = 241
* **Solve for 'z':**
z = 241 / 264
z = 0.9136...
* **Substitute the value of 'z' back into either of the original equations to solve for 'f'. Let's use the first equation:**
45 * f + 42 * (0.9136...) = 101
45 * f + 38.36... = 101
45 * f = 101 - 38.36...
45 * f = 62.64...
f = 62.64... / 45
f = 1.392...
**Therefore:**
### Answer: f = 1.392..., z = 0.9136...
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 Flash:
Here's how to solve for 'p' and 'u' using the system of equations:
**1. Elimination Method**
* **Multiply equations to make coefficients match:**
* Multiply the first equation by 15: (26 * p - 21 * u = 6) * 15 => 390p - 315u = 90
* Multiply the second equation by -26: (15 * p + 58 * u = 83) * -26 => -390p - 1508u = -2158
* **Add the modified equations:**
* (390p - 315u = 90) + (-390p - 1508u = -2158) => -1823u = -2068
* **Solve for 'u':**
* u = -2068 / -1823
* u = 1.134 (approximately)
* **Substitute the value of 'u' back into either original equation to solve for 'p'. Let's use the first equation:**
* 26p - 21 * 1.134 = 6
* 26p - 23.814 = 6
* 26p = 29.814
* p = 29.814 / 26
* p = 1.147 (approximately)
**Therefore:**
* p ≈ 1.147
* u ≈ 1.134
### Answer: p ≈ 1.147, u ≈ 1.134
Expected answer: {p: 2091/1823, u: 2068/1823}
Correct? No
Problem: Solve for the values of n and s using these two simultaneous equations: 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75
Gemini-1.5 Flash:
Here's how to solve for the values of n and s using the given simultaneous equations:
**1. Elimination Method**
* **Multiply Equation 2 by 38:** This will allow us to eliminate 's' when we add the equations together.
* (59 * n + 8 * s) * 38 = 75 * 38
* 2242 * n + 304 * s = 2850
* **Add the modified Equation 2 to Equation 1:**
* (92 * n - 38 * s) + (2242 * n + 304 * s) = 62 + 2850
* 2334 * n + 266 * s = 2912
* **Simplify the equation:**
* 1167 * n + 133 * s = 1456
* **Solve for n:**
* n = (1456 - 133 * s) / 1167
**2. Substitution Method**
* **Substitute the value of n from the simplified equation into either Equation 1 or Equation 2.** Let's use Equation 1:
* 92 * [(1456 - 133 * s) / 1167] - 38 * s = 62
* **Simplify and solve for s:**
* (133872 - 12296 * s) / 1167 - 38 * s = 62
* 133872 - 12296 * s - 44506 * s = 72214
* -56802 * s = -61658
* s = -61658 / -56802
* **s = 1.086** (approximately)
**3. Substitute the value of s back into the equation for n:**
* n = (1456 - 133 * 1.086) / 1167
* n = (1456 - 144.58) / 1167
* **n = 1.115** (approximately)
### Answer: n = 1.115, s = 1.086
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 Flash:
Here's how to solve the simultaneous equations:
**1. Elimination Method**
* **Multiply equations to make coefficients match:**
* Multiply the first equation by 58: (58)(87z - 35e) = (58)(63) => 5046z - 2030e = 3654
* Multiply the second equation by 35: (35)(81z + 58e) = (35)(154) => 2835z + 2030e = 5390
* **Add the modified equations:** Notice the 'e' terms cancel out:
* 5046z - 2030e = 3654
* 2835z + 2030e = 5390
* ------------------
* 7881z = 9044
* **Solve for 'z':**
* z = 9044 / 7881
* z = 1.147 (approximately)
* **Substitute 'z' back into either original equation to solve for 'e':**
* Let's use the first equation: 87(1.147) - 35e = 63
* 99.999 - 35e = 63
* -35e = -36.999
* e = 1.057 (approximately)
**Therefore:**
* z ≈ 1.147
* e ≈ 1.057
### Answer: z ≈ 1.147, e ≈ 1.057
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 Flash:
Here's how to solve the system of equations: **1. Elimination Method** * **Multiply equation (1) by 9:** This will allow us to eliminate 't' when we subtract the equations. * (99 * t) + (396 * h) = 990 * **Subtract the modified equation (1) from equation (2):** * (98 * t) - (36 * h) = 18 * -(99 * t) - (396 * h) = -990 * -t - 432 * h = -972 * **Solve for 'h':** * -432 * h = -972 + t * h = (972 - t) / 432 **2. Substitution Method** * **Substitute the expression for 'h' from step 1 into equation (1):** * (11 * t) + (44 * [(972 - t) / 432]) = 110 * **Simplify and solve for 't':** * (11 * t) + [(972 - t) / 9] = 110 * (99 * t) + (972 - t) = 990 * 98 * t = 18 * t = 18 / 98 * **t = 9/49** * **Substitute the value of 't' back into the expression for 'h':** * h = (972 - (9/49)) / 432 * h = (47493 / 49) / 432 * **h = 103 / 49** **Therefore:** * **t = 9/49** * **h = 103/49** ### Answer: t = 9/49, h = 103/49
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 Flash:
We can solve this system of equations using elimination. Here's how:
1. **Multiply equations to match coefficients:**
- Multiply the first equation by 41:
(41 * (69 * b) - (41 * 50 * o)) = (41 * 55)
-> 2829b - 2050o = 2255
- Multiply the second equation by 50:
(50 * (41 * o) + (50 * 18 * b)) = (50 * 85)
-> 2050o + 900b = 4250
2. **Add the modified equations:**
Notice that the 'o' terms have opposite signs and equal magnitudes. Adding the equations eliminates 'o':
(2829b - 2050o) + (2050o + 900b) = 2255 + 4250
-> 3729b = 6505
3. **Solve for 'b':**
Divide both sides by 3729:
b = 6505 / 3729
-> b = 1.745 (approximately)
4. **Substitute 'b' into one of the original equations to solve for 'o':**
Let's use the first equation:
(69 * 1.745) - (50 * o) = 55
-> 120.405 - 50o = 55
-> 50o = 120.405 - 55
-> 50o = 65.405
5. **Solve for 'o':**
Divide both sides by 50:
o = 65.405 / 50
-> o = 1.308 (approximately)
### Answer: b = 1.745, o = 1.308
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 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the second equation by 16 to make the coefficients of 'p' opposites:** (6 * q + 5 * p = 177) * 16 => 96q + 80p = 2832 2. **Add the modified second equation to the first equation:** (67q - 80p = 65) + (96q + 80p = 2832) => 163q = 2897 3. **Solve for 'q':** 163q = 2897 => q = 2897 / 163 => q = 17.8 4. **Substitute the value of 'q' back into either original equation to solve for 'p'. Let's use the second equation:** 6 * (17.8) + 5 * p = 177 => 106.8 + 5p = 177 5. **Solve for 'p':** 5p = 177 - 106.8 => 5p = 70.2 => p = 70.2 / 5 => p = 14.04 ### Answer: p = 14.04, q = 17.8
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 Flash:
To solve this system of equations, we can use elimination. We can multiply the second equation by 48/41, then add the two equations together. This will eliminate 'f' and allow us to solve for 'k'. Here's how it works: 1. **Multiply the second equation by 48/41:** (33 * k - 41 * f) * (48/41) = 31 * (48/41) This simplifies to: 38.4 * k - 48 * f = 36.24 2. **Add the two equations together:** (10 * k + 48 * f) + (38.4 * k - 48 * f) = 161 + 36.24 This simplifies to: 48.4 * k = 197.24 3. **Solve for k:** k = 197.24 / 48.4 k = 4.08 4. **Substitute the value of k back into either of the original equations to solve for f. Let's use the first equation:** 10 * 4.08 + 48 * f = 161 40.8 + 48 * f = 161 48 * f = 120.2 f = 120.2 / 48 f = 2.5 Therefore, the values of k and f that satisfy the system of equations are: ### Answer: k = 4.08, f = 2.5
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 Flash:
Here's how to solve the system of equations: **1. Elimination Method** * **Align the equations:** ``` 31 = 58l - n 103 = 73l + n ``` * **Add the equations together:** Notice that the 'n' terms have opposite signs. Adding the equations will eliminate 'n'. ``` 134 = 131l ``` * **Solve for 'l':** ``` l = 134 / 131 l = 1.023 ``` * **Substitute the value of 'l' back into either original equation to solve for 'n'. Let's use the first equation:** ``` 31 = 58(1.023) - n 31 = 59.334 - n n = 59.334 - 31 n = 28.334 ``` **Therefore, the solution to the system of equations is:** * **l = 1.023** * **n = 28.334** ### Answer: l = 1.023, n = 28.334
Expected answer: {l: 134/131, n: 3711/131}
Correct? No
Problem: Find the values of y and j that satisfy the following system of equations: 29y - 28j = 39 25y + 37j = 116
Gemini-1.5 Flash:
We can solve this system of equations using elimination. 1. **Multiply the first equation by 25 and the second equation by 29:** This will allow us to have the same coefficient for 'y' in both equations. * (29y - 28j) * 25 = 725y - 700j = 975 * (25y + 37j) * 29 = 725y + 1073j = 3364 2. **Subtract the first modified equation from the second modified equation:** This eliminates 'y' and allows us to solve for 'j'. * (725y + 1073j) - (725y - 700j) = 3364 - 975 * 1773j = 2389 * j = 2389/1773 = 1.35 3. **Substitute the value of 'j' back into either of the original equations to solve for 'y':** We'll use the first equation. * 29y - 28(1.35) = 39 * 29y - 37.8 = 39 * 29y = 76.8 * y = 76.8 / 29 = 2.65 Therefore, the values of y and j that satisfy the system of equations are: ### 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 Flash:
Here's how to solve the system of equations: **1. Elimination Method** * **Multiply the first equation by 17 and the second equation by 85:** This will make the coefficients of 'a' opposites. * (93u - 85a) * 17 = 74 * 17 => 1581u - 1445a = 1258 * (96u + 17a) * 85 = 186 * 85 => 8160u + 1445a = 15810 * **Add the two equations together:** This eliminates 'a'. * 1581u - 1445a + 8160u + 1445a = 1258 + 15810 * 9741u = 17068 * **Solve for 'u':** * u = 17068 / 9741 * u = 1.75 (approximately) * **Substitute the value of 'u' back into either of the original equations to solve for 'a'. Let's use the first equation:** * 93 * 1.75 - 85a = 74 * 162.75 - 85a = 74 * -85a = 74 - 162.75 * -85a = -88.75 * a = -88.75 / -85 * a = 1.04 (approximately) **Solution:** * u = 1.75 (approximately) * a = 1.04 (approximately) ### Answer: u ≈ 1.75, a ≈ 1.04
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 Flash:
Here's how to solve for the values of 'u' and 'g' using the given system of equations: **1. Elimination Method:** * **Multiply the first equation by 23 and the second equation by 61:** This will make the coefficients of 'g' opposites: (23) * (97u - 61g) = (23) * (15) --> 2231u - 1403g = 345 (61) * (85u + 23g) = (61) * (186) --> 5185u + 1403g = 11346 * **Add the two equations together:** The 'g' terms will cancel out: 2231u - 1403g + 5185u + 1403g = 345 + 11346 7416u = 11691 * **Solve for 'u':** u = 11691 / 7416 u = 1.575 (approximately) * **Substitute the value of 'u' back into either original equation to solve for 'g'.** Let's use the first equation: 97 * (1.575) - 61g = 15 153.075 - 61g = 15 -61g = -138.075 g = -138.075 / -61 g = 2.264 (approximately) **Therefore:** * **u ≈ 1.575** * **g ≈ 2.264** ### Answer: u ≈ 1.575, g ≈ 2.264
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 Flash:
Here's how to solve the system of equations:
**1. Elimination Method**
* **Notice:** Both equations have the same coefficient for 'b' (67). This makes it easy to eliminate 'b'.
* **Subtract the first equation from the second equation:**
(85r + 67b) - (17r + 67b) = 187 - 103
* **Simplify:**
68r = 84
* **Solve for 'r':**
r = 84 / 68 = 1.235
**2. Substitute to find 'b'**
* **Choose either of the original equations and substitute the value of 'r' you just found.** Let's use the first equation:
17 * 1.235 + 67b = 103
* **Simplify and solve for 'b':**
21 + 67b = 103
67b = 82
b = 82 / 67 = 1.224
**Therefore:**
* r = 1.235
* b = 1.224
### Answer: r = 1.235, b = 1.224
Expected answer: {b: 82/67, r: 21/17}
Correct? No
Problem: Given the following system of equations, solve for the values of t and n: 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126.
Gemini-1.5 Flash:
Here's how to solve the system of equations using elimination: **1. Multiply equations to make coefficients match:** * Multiply the first equation by 30 and the second equation by 54: * (1) 2550n - 1620t = 1230 * (2) 2052t + 1620n = 6804 **2. Add the modified equations:** * Notice that the 'n' terms have opposite signs. Adding the equations will eliminate 'n': * 4102t = 8034 **3. Solve for 't':** * Divide both sides by 4102: * t = 8034 / 4102 * t = 2 **4. Substitute the value of 't' into either original equation to solve for 'n':** * Let's use the first equation: * (85 * n) - (54 * 2) = 41 * 85n - 108 = 41 * 85n = 149 * n = 149 / 85 * n = 1.75 **Therefore:** * t = 2 * n = 1.75 ### Answer: t = 2, n = 1.75
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 Flash:
Here's how to solve the system of equations: **1. Translate the word problem into equations:** * Equation 1: 75u - 69q = 5 * Equation 2: 48u + 25q = 140 **2. Solve for one variable in terms of the other:** Let's solve Equation 1 for 'u': * 75u = 5 + 69q * u = (5 + 69q) / 75 **3. Substitute the expression for 'u' into Equation 2:** * 48 * ((5 + 69q) / 75) + 25q = 140 **4. Simplify and solve for 'q':** * (16/25) * (5 + 69q) + 25q = 140 * 3.2 + 41.28q + 25q = 140 * 66.28q = 136.8 * q = 136.8 / 66.28 * q = 2 **5. Substitute the value of 'q' back into the expression for 'u':** * u = (5 + 69 * 2) / 75 * u = (5 + 138) / 75 * u = 143 / 75 * u = 1.91 **Therefore:** * u = 1.91 * q = 2 ### Answer: u = 1.91, q = 2
Expected answer: {q: 180/91, u: 515/273}
Correct? No
Problem: Determine the values of v and x from the following system of equations: ((20 * v) + (17 * x)) = 149 and ((24 * x) - (49 * v)) = 22.
Gemini-1.5 Flash:
Here's how to solve the system of equations: **1. Elimination Method** * **Multiply the first equation by 24 and the second equation by 17:** This will make the coefficients of 'x' in both equations the same (but with opposite signs). * (24 * (20 * v) + (24 * 17 * x)) = (24 * 149) => 480v + 408x = 3576 * (17 * (24 * x) - (17 * 49 * v)) = (17 * 22) => 408x - 833v = 374 * **Subtract the second equation from the first equation:** This eliminates 'x'. * (480v + 408x) - (408x - 833v) = 3576 - 374 * 1313v = 3202 * **Solve for 'v':** * v = 3202 / 1313 = 2.44 (approximately) **2. Substitute 'v' back into either original equation to solve for 'x':** * Let's use the first equation: (20 * v) + (17 * x) = 149 * Substitute v = 2.44: (20 * 2.44) + (17 * x) = 149 * Simplify: 48.8 + 17x = 149 * Subtract 48.8 from both sides: 17x = 100.2 * Solve for 'x': x = 100.2 / 17 = 5.9 (approximately) **Therefore:** * v ≈ 2.44 * x ≈ 5.9 ### Answer: v ≈ 2.44, x ≈ 5.9
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 Flash:
Here's how to solve the system of equations:
**1. Elimination Method**
* **Multiply equations to match coefficients:** Our goal is to make the coefficients of either 't' or 'g' the same (but opposite signs) in both equations. Let's eliminate 'g':
* Multiply the first equation by 91: (39 * 91)t + (41 * 91)g = 147 * 91
* Multiply the second equation by 41: (94 * 41)t - (91 * 41)g = 42 * 41
* **Add the equations:** Notice that the 'g' terms will cancel out:
(39 * 91)t + (41 * 91)g = 147 * 91
(94 * 41)t - (91 * 41)g = 42 * 41
-----------------------------
(39 * 91 + 94 * 41)t = (147 * 91 + 42 * 41)
* **Solve for 't':**
3549t + 3834t = 13383 + 1722
7383t = 15105
t = 15105 / 7383
t = 2.045 (approximately)
**2. Substitute to find 'g':**
* Choose either of the original equations and substitute the value of 't' we just found. Let's use the first equation:
39 * 2.045 + 41 * g = 147
79.755 + 41 * g = 147
* **Solve for 'g':**
41 * g = 147 - 79.755
41 * g = 67.245
g = 67.245 / 41
g = 1.64 (approximately)
**Therefore:**
* t ≈ 2.045
* g ≈ 1.64
### Answer: t ≈ 2.045, g ≈ 1.64
Expected answer: {g: 12180/7403, t: 15099/7403}
Correct? No
Problem: Determine the values of c and o that satisfy the following system of equations: 6c + 55o = 112, and 55c - 48o = 36.
Gemini-1.5 Flash:
We can solve this system of equations using elimination. 1. **Multiply the first equation by 48 and the second equation by 55:** This will allow us to eliminate 'o' when we add the equations together. - (6c + 55o) * 48 = 288c + 2640o = 5376 - (55c - 48o) * 55 = 3025c - 2640o = 1980 2. **Add the two equations together:** Notice that the 'o' terms cancel out. - 288c + 3025c = 5376 + 1980 - 3313c = 7356 3. **Solve for 'c':** - c = 7356 / 3313 - c = 2.22 (approximately) 4. **Substitute the value of 'c' back into either original equation 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 - o = 1.80 (approximately) Therefore, the values of c and o that satisfy the system of equations are: ### Answer: c ≈ 2.22 and 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 Flash:
We can solve this system of equations using elimination. 1. **Multiply the first equation by 71 and the second equation by 8.** This will allow us to have opposite coefficients for 'w' in both equations. * (8w + 68u) * 71 = 170 * 71 --> 568w + 4818u = 12070 * (71w - 94u) * 8 = 8 * 8 --> 568w - 752u = 64 2. **Subtract the second equation from the first equation.** This eliminates 'w'. * (568w + 4818u) - (568w - 752u) = 12070 - 64 * 5570u = 12006 3. **Solve for 'u'.** * u = 12006 / 5570 * 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 equation. * 8w + 68 * 2.15 = 170 * 8w + 146.2 = 170 * 8w = 23.8 * w = 23.8 / 8 * w = 2.975 Therefore, the values of u and w that satisfy the system of equations are: ### 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 Flash:
Here's how to solve the system of equations:
**1. Elimination Method**
* **Multiply equations to match coefficients:**
* Multiply the first equation by 97:
6596r + 784f = 8924
* Multiply the second equation by 8:
776f + 304r = 1456
* **Subtract the second equation from the first:**
(6596r + 784f) - (304r + 776f) = 8924 - 1456
6292r = 7468
* **Solve for r:**
r = 7468 / 6292
r = 1.186
* **Substitute the value of r back into either original equation to solve for f.** Let's use the first equation:
68 * 1.186 + 8f = 92
80.73 + 8f = 92
8f = 11.27
f = 11.27 / 8
f = 1.409
**Therefore:**
* r = 1.186
* f = 1.409
### Answer: r = 1.186, f = 1.409
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 Flash:
We have two equations: 1. 67d - 37r = 50 2. 39r - 21d = 16 To solve for 'd' and 'r', we can use a method called elimination. Here's how: **1. Multiply the equations to make the coefficients of 'd' (or 'r') the same:** * Multiply equation 1 by 21: 1407d - 777r = 1050 * Multiply equation 2 by 67: 2613r - 1407d = 1072 **2. Add the modified equations together:** * Notice that the 'd' terms cancel out: (1407d - 777r) + (2613r - 1407d) = 1050 + 1072 * This simplifies to: 1836r = 2122 **3. Solve for 'r':** * Divide both sides by 1836: r = 2122/1836 * Therefore, r ≈ 1.15 **4. Substitute the value of 'r' back into either original equation to solve for 'd':** * Let's use equation 1: 67d - 37(1.15) = 50 * Simplify: 67d - 42.55 = 50 * Add 42.55 to both sides: 67d = 92.55 * Divide both sides by 67: d ≈ 1.38 **Therefore:** * The cost of each apple (d) is approximately $1.38. * The cost of each ragdoll (r) is approximately $1.15. ### Answer: d ≈ $1.38, r ≈ $1.15.
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 Flash:
Here's how to solve the system of equations: **1. Express one variable in terms of the other.** Let's solve the first equation for *a*: * 163 = 50x + a * a = 163 - 50x **2. Substitute the expression for the variable into the second equation.** Substitute the expression for *a* (163 - 50x) into the second equation: * 24 = 98x - 84(163 - 50x) **3. Solve for *x*.** * 24 = 98x - 13752 + 4200x * 13776 = 4298x * x = 13776 / 4298 * x = 3.2 **4. Substitute the value of *x* back into the expression for *a*.** * a = 163 - 50(3.2) * a = 163 - 160 * a = 3 **Solution:** Therefore, the solution to the system of equations is: * **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 Flash:
We can solve this system of equations using elimination. To do that, we need to make the coefficients of either 'p' or 'b' the same in both equations. Let's multiply the first equation by 22 and the second equation by 9. This will give us the same coefficient for 'p' in both equations: * Equation 1 * 22: 1474 * p + 396 * b = 2420 * Equation 2 * 9: 738 * b - 396 * p = 522 Now, we can add the two equations together: * (1474 * p + 396 * b) + (738 * b - 396 * p) = 2420 + 522 * 1474 * p + 396 * b + 738 * b - 396 * p = 2942 * 1078 * p + 1134 * b = 2942 Now we have a new equation with only 'p' and 'b'. Let's rearrange the first original equation to solve for 'p': * 67 * p + 18 * b = 110 * 67 * p = 110 - 18 * b * p = (110 - 18 * b) / 67 Now we can substitute this value of 'p' into the new equation we derived: * 1078 * ((110 - 18 * b) / 67) + 1134 * b = 2942 * 1798 * (110 - 18 * b) + 76122 * b = 197214 * 197780 - 32364 * b + 76122 * b = 197214 * 43758 * b = -556 * b = -556 / 43758 Now we can substitute this value of 'b' back into the equation for 'p': * p = (110 - 18 * (-556 / 43758)) / 67 * p = (110 + 9996/43758) / 67 * p = (4814440/43758) / 67 * p = 4814440 / 2931826 So, the solutions for the system of equations are: * p = 4814440 / 2931826 * b = -556 / 43758 ### Answer: p = 4814440 / 2931826, b = -556 / 43758
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 Flash:
Here's how to solve the simultaneous equations: **1. Set up the equations:** * 63c - 36e = 27 * 8c + 32e = 69 **2. Simplify the equations (optional):** We can simplify the equations by dividing both sides of the first equation by 9 and the second equation by 8: * 7c - 4e = 3 * c + 4e = 69/8 **3. Solve for one variable:** We can use elimination to solve for 'c'. Notice that the 'e' terms have opposite signs in the two equations. Add the two equations together: 7c - 4e = 3 + c + 4e = 69/8 ------------------ 8c = 87/8 Now, solve for 'c': c = (87/8) / 8 c = 87/64 **4. Substitute to find the other variable:** Substitute the value of 'c' (87/64) into either of the original equations. Let's use the second equation: 8(87/64) + 32e = 69 Simplify and solve for 'e': 87/8 + 32e = 69 32e = 69 - 87/8 32e = 465/8 e = (465/8) / 32 e = 465/256 **Therefore:** * c = 87/64 * e = 465/256 ### Answer: c = 87/64, e = 465/256
Expected answer: {c: 93/64, e: 459/256}
Correct? No
Problem: Solve the following system of linear equations to find the values of x and p: 26 * x + 29 * p = 107 61 * x - 51 * p = 4
Gemini-1.5 Flash:
Here's how to solve the system of equations using elimination:
1. **Multiply the equations to make the coefficients of one variable opposites:**
* Multiply the first equation by 51:
(26 * x + 29 * p = 107) * 51 => 1326x + 1479p = 5467
* Multiply the second equation by 29:
(61 * x - 51 * p = 4) * 29 => 1769x - 1479p = 116
2. **Add the two equations together:**
Notice that the 'p' terms cancel out:
1326x + 1479p = 5467
1769x - 1479p = 116
------------------
3095x = 5583
3. **Solve for x:**
3095x = 5583
x = 5583 / 3095
x = 1.8
4. **Substitute the value of x back into either original equation to solve for p.** Let's use the first equation:
26 * (1.8) + 29 * p = 107
46.8 + 29p = 107
29p = 60.2
p = 60.2 / 29
p = 2.076
**Therefore, the solutions are:**
* x = 1.8
* p = 2.076
### Answer: x = 1.8, p = 2.076
Expected answer: {p: 6423/3095, x: 5573/3095}
Correct? No
Problem: Find the values for g and l from the system of equations below. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17.
Gemini-1.5 Flash:
Here's how to solve this system of equations using elimination: **1. Multiply the equations to create opposite coefficients for one variable.** * Multiply the first equation by 10: (290 * l) + (170 * g) = 680 * Multiply the second equation by 17: (527 * g) - (170 * l) = 289 **2. Add the two equations together.** This will eliminate 'l'. (290 * l) + (170 * g) = 680 (527 * g) - (170 * l) = 289 ---------------------- (697 * g) = 969 **3. Solve for 'g'.** g = 969 / 697 g = 1.39 (approximately) **4. Substitute the value of 'g' back into either original equation to solve for 'l'.** Let's use the first equation: (29 * l) + (17 * 1.39) = 68 (29 * l) + 23.63 = 68 (29 * l) = 44.37 l = 44.37 / 29 l = 1.53 (approximately) **Therefore:** ### Answer: g = 1.39, l = 1.53
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 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the first equation by 26 and the second equation by 54:** This will allow us to have coefficients of 'c' that are opposites. * 858*a - 1404*c = 520 * 1296*a - 1404*c = 1566 2. **Subtract the second equation from the first equation:** This eliminates 'c'. * -438*a = -1046 3. **Solve for 'a':** Divide both sides by -438. * a = -1046 / -438 = 2.39 4. **Substitute the value of 'a' back into either original equation to solve for 'c'.** Let's use the first equation: * 33 * 2.39 - 54*c = 20 * 78.87 - 54*c = 20 * -54*c = 20 - 78.87 = -58.87 * c = -58.87 / -54 = 1.09 **Therefore:** * a = 2.39 * c = 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 Flash:
We can solve this system of equations using the elimination method. Here's how:
1. **Multiply equations to make coefficients match:**
* Multiply the first equation by 43:
3053d + 387j = 5676
* Multiply the second equation by 9:
387j - 756d = 819
2. **Subtract the second equation from the first equation:**
* Notice the 'j' terms have the same coefficient, so they cancel out when subtracting.
* (3053d + 387j) - (387j - 756d) = 5676 - 819
* This simplifies to 3809d = 4857
3. **Solve for 'd':**
* Divide both sides by 3809:
d = 4857 / 3809
d ≈ 1.275
4. **Substitute the value of 'd' back into either original equation to solve for 'j'.** Let's use the first equation:
* 71(1.275) + 9j = 132
* 90.525 + 9j = 132
* 9j = 41.475
* j ≈ 4.61
Therefore, the solution to the system of equations is:
* d ≈ 1.275
* j ≈ 4.61
### Answer: d ≈ 1.275, j ≈ 4.61
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 Flash:
Here's how to solve this system of equations: **1. Elimination Method:** * **Multiply equations to match coefficients:** - Multiply the first equation by 87: (81 * o * 87) + (26 * n * 87) = 163 * 87 - Multiply the second equation by 26: (87 * n * 26) + (23 * o * 26) = 143 * 26 * **Subtract the second equation from the first:** - This will eliminate 'n' since the coefficients are now the same but with opposite signs. - (81 * o * 87) - (23 * o * 26) = (163 * 87) - (143 * 26) - Simplify: (7047 * o) - (598 * o) = 14181 - 3718 - Further simplification: 6449 * o = 10463 * **Solve for 'o':** - o = 10463 / 6449 - o = 1.62 (rounded to two decimal places) * **Substitute the value of 'o' back into either original equation to solve for 'n'.** Let's use the first equation: - (81 * 1.62) + (26 * n) = 163 - 131.22 + 26 * n = 163 - 26 * n = 163 - 131.22 - 26 * n = 31.78 - n = 31.78 / 26 - n = 1.22 (rounded to two decimal places) **Therefore:** * **o = 1.62** * **n = 1.22** ### Answer: o = 1.62, n = 1.22
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 Flash:
Here's how to solve the system of equations:
**1. Elimination Method**
* **Multiply the first equation by 67 and the second equation by 61.** This will allow us to eliminate 'w' when we add the equations together.
* (46 * s - 61 * w) * 67 = 2 * 67
* (61 * s + 67 * w) * 61 = 178 * 61
* This simplifies to:
* 3082 * s - 4087 * w = 134
* 3721 * s + 4087 * w = 10858
* **Add the two equations together:** Notice that the 'w' terms cancel out.
* 3082 * s + 3721 * s = 134 + 10858
* 6803 * s = 10992
* **Solve for 's':**
* s = 10992 / 6803
* s = 1.615 (rounded to three decimal places)
* **Substitute the value of 's' back into either of the original equations to solve for 'w'. Let's use the first equation:**
* 46 * 1.615 - 61 * w = 2
* 74.29 - 61 * w = 2
* -61 * w = 2 - 74.29
* -61 * w = -72.29
* w = -72.29 / -61
* w = 1.185 (rounded to three decimal places)
**Therefore:**
* **s = 1.615**
* **w = 1.185**
### Answer: s = 1.615, w = 1.185
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 Flash:
Here's how to solve the system of equations:
**1. Elimination Method**
* **Multiply the first equation by 39 and the second equation by 8:** This will allow us to have opposite coefficients for the 'i' term.
* (51*k - 8*i) * 39 = 1989*k - 312*i = 3354
* (39*i + 8*k) * 8 = 312*i + 64*k = 1544
* **Add the two modified equations together:** Notice that the 'i' terms cancel out.
* 1989*k + 64*k = 3354 + 1544
* 2053*k = 4898
* **Solve for 'k':**
* k = 4898 / 2053 = 2.385 (approximately)
**2. Substitution Method**
* **Solve the first equation for 'i':**
* -8*i = 86 - 51*k
* i = (86 - 51*k) / -8
* **Substitute this expression for 'i' into the second equation:**
* 39 * [(86 - 51*k) / -8] + 8*k = 193
* **Simplify and solve for 'k':**
* -498.75 + 248.625*k + 8*k = 193
* 256.625*k = 691.75
* k = 691.75 / 256.625 = 2.694 (approximately)
**3. Find 'i'**
* **Substitute the value of 'k' back into either of the original equations.** Let's use the first equation:
* 51 * 2.694 - 8*i = 86
* 137.394 - 8*i = 86
* -8*i = -51.394
* i = 6.424 (approximately)
### Answer: i = 6.424, k = 2.694
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 Flash:
We can solve this system of equations using elimination. 1. **Multiply the second equation by -17:** This will allow us to eliminate 'l' when we add the equations together. -17(93m + l) = -17(126) -1581m - 17l = -2142 2. **Add the modified second equation to the first equation:** 26m + 17l = 146 -1581m - 17l = -2142 ------------------ -1555m = -1996 3. **Solve for 'm':** m = -1996 / -1555 m = 1.28 4. **Substitute the value of 'm' back into either original equation to solve for 'l'. Let's 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 Flash:
Here's how to solve the system of equations to find the values of 'h' and 'f':
**1. Elimination Method:**
* **Multiply Equation 1 by -29:**
-232f - 1131h = -1914
* **Multiply Equation 2 by 8:**
232f + 368h = 952
* **Add the two modified equations together:**
-763h = -962
* **Solve for 'h':**
h = -962 / -763
h = 1.26
**2. Substitute 'h' back into either original equation to solve for 'f':**
Let's use Equation 1:
(8 * f) + (39 * 1.26) = 66
8f + 49.14 = 66
8f = 16.86
f = 16.86 / 8
f = 2.11
**Therefore:**
* **h = 1.26**
* **f = 2.11**
### Answer: h = 1.26, f = 2.11
Expected answer: {f: 1605/763, h: 962/763}
Correct? No
Problem: Find the values of k and b that satisfy the following system of equations: 22 * b + 23 * k = 109 and 62 * k - 69 * b = 14.
Gemini-1.5 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the first equation by 69 and the second equation by 23:** This will make the coefficients of 'b' in both equations have the same absolute value but opposite signs. * (69) * (22 * b + 23 * k) = (69) * (109) => 1518 * b + 1587 * k = 7521 * (23) * (62 * k - 69 * b) = (23) * (14) => 1426 * k - 1587 * b = 322 2. **Add the two modified equations together:** This eliminates 'b'. * (1518 * b + 1587 * k) + (1426 * k - 1587 * b) = 7521 + 322 * 3013 * k = 7843 3. **Solve for 'k':** * k = 7843 / 3013 = 2.60 4. **Substitute the value of 'k' back into either of the original equations to solve for 'b'. Let's use the first equation:** * 22 * b + 23 * (2.60) = 109 * 22 * b + 59.8 = 109 * 22 * b = 49.2 * b = 49.2 / 22 = 2.24 Therefore, the values that satisfy the system of equations are: ### Answer: k = 2.60, b = 2.24
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 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the second equation by 27.5:** This will make the coefficients of 's' in both equations opposites. (27.5) * (63m - 2s) = (27.5) * 98 This gives us: 1732.5m - 55s = 2695 2. **Subtract the first equation from the modified second equation:** (1732.5m - 55s) - (84m - 55s) = 2695 - 1 This simplifies to: 1648.5m = 2694 3. **Solve for 'm':** m = 2694 / 1648.5 m = 1.63 (approximately) 4. **Substitute the value of 'm' back into either of the original equations to solve for 's'.** Let's use the first equation: 84 * 1.63 - 55s = 1 137.52 - 55s = 1 -55s = -136.52 s = 2.48 (approximately) ### Answer: m = 1.63, s = 2.48
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 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the first equation by 51 and the second equation by 6 to make the coefficients of 'b' the same but opposite in sign:** - Equation 1 * 51: (306 * b) + (255 * k) = 5355 - Equation 2 * 6: (570 * k) - (306 * b) = 402 2. **Add the two modified equations together:** (306 * b) + (255 * k) = 5355 (570 * k) - (306 * b) = 402 ----------------------- (825 * k) = 5757 3. **Solve for k:** k = 5757 / 825 k = 7 4. **Substitute the value of k (7) back into either of the original equations to solve for b.** Let's use the first equation: (6 * b) + (5 * 7) = 105 (6 * b) + 35 = 105 (6 * b) = 70 b = 70 / 6 b = 11.67 (approximately) ### Answer: b = 11.67, k = 7
Expected answer: {b: 1928/165, k: 1919/275}
Correct? No
Problem: Determine the values of d and u from the following pair of simultaneous linear equations: 33d + 62u = 114 and 32u + 40d = 83.
Gemini-1.5 Flash:
We can solve this system of equations using elimination. Here's how:
1. **Multiply the first equation by 40 and the second equation by -33.** This will allow us to eliminate 'd' when we add the equations together.
* (33d + 62u) * 40 = 114 * 40 => 1320d + 2480u = 4560
* (32u + 40d) * -33 = 83 * -33 => -1056u - 1320d = -2739
2. **Add the two modified equations together.** The 'd' terms will cancel out:
* 1320d + 2480u = 4560
* -1056u - 1320d = -2739
* ------------------------
* 1424u = 1821
3. **Solve for 'u':**
* u = 1821 / 1424
* u = 1.28
4. **Substitute the value of 'u' back into either of the original equations to solve for 'd'.** Let's use the first equation:
* 33d + 62(1.28) = 114
* 33d + 79.36 = 114
* 33d = 34.64
* d = 34.64 / 33
* d = 1.05
Therefore:
### Answer: d = 1.05 and u = 1.28
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 Flash:
Here's how to solve this system of equations using the elimination method: 1. **Multiply the first equation by 4:** This will make the coefficients of 'c' in both equations multiples of each other. * (5 * c + 28 * p = 102) * 4 => 20 * c + 112 * p = 408 2. **Now we have the following system of equations:** * 20 * c + 112 * p = 408 * 20 * c - 67 * p = 13 3. **Subtract the second equation from the first equation:** This eliminates 'c'. * (20 * c + 112 * p = 408) - (20 * c - 67 * p = 13) => 179 * p = 395 4. **Solve for 'p':** * 179 * p = 395 => p = 395 / 179 => p ≈ 2.207 5. **Substitute the value of 'p' back into either original equation to solve for 'c'. Let's use the first equation:** * 5 * c + 28 * (2.207) = 102 * 5 * c + 61.80 = 102 * 5 * c = 40.20 * c = 40.20 / 5 => c = 8.04 ### Answer: c ≈ 8.04, p ≈ 2.207
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 Flash:
We can solve this system of linear equations using a method called elimination. Here's how:
1. **Multiply the first equation by 76 and the second equation by 58:**
This will allow us to have the coefficients of 'r' in both equations be multiples of each other.
- Equation 1: (58 * r + 12 * q = 147) * 76 => 4408 * r + 912 * q = 11172
- Equation 2: (97 * q - 76 * r = 76) * 58 => 5626 * q - 4408 * r = 4408
2. **Add the two modified equations together:**
Notice that the 'r' terms cancel out:
4408 * r + 912 * q = 11172
-4408 * r + 5626 * q = 4408
-------------------------
6538 * q = 15580
3. **Solve for 'q':**
Divide both sides by 6538:
q = 15580 / 6538
q = 2.38 (approximately)
4. **Substitute the value of 'q' back into either of the original equations to solve for 'r'.** Let's use the first equation:
58 * r + 12 * 2.38 = 147
58 * r + 28.56 = 147
5. **Solve for 'r':**
Subtract 28.56 from both sides:
58 * r = 118.44
Divide both sides by 58:
r = 2.04 (approximately)
Therefore, the values of r and q are:
### 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 Flash:
Here's how to solve the system of equations: **1. Elimination Method:** * **Multiply the second equation by 3/2:** This will allow us to eliminate 'k' when we add the equations together. (3/2) * (73v + 2k) = (3/2) * 128 109.5v + 3k = 192 * **Add the modified second equation to the first equation:** (47v - 3k) + (109.5v + 3k) = 10 + 192 156.5v = 202 * **Solve for 'v':** v = 202 / 156.5 v ≈ 1.29 **2. Substitute to find 'k':** * **Choose one of the original equations and substitute the value of 'v':** Let's use the first equation. 47 * 1.29 - 3k = 10 60.63 - 3k = 10 * **Solve for 'k':** -3k = 10 - 60.63 -3k = -50.63 k ≈ 16.88 **Therefore, the solution to the system of equations is:** * v ≈ 1.29 * 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 Flash:
Here's how to solve the system of equations: **1. Set up the equations:** * 89p - 41q = 60 * 27p + 91q = 124 **2. Solve for one variable in terms of the other:** Let's solve the first equation for *p*: * 89p = 60 + 41q * p = (60 + 41q) / 89 **3. Substitute the expression for *p* into the second equation:** * 27 * [(60 + 41q) / 89] + 91q = 124 **4. Simplify and solve for *q*:** * (1620 + 1107q) / 89 + 91q = 124 * 1620 + 1107q + 8119q = 11016 * 9226q = 9396 * q = 9396 / 9226 * q ≈ 1.02 **5. Substitute the value of *q* back into either of the original equations to solve for *p*. Let's use the first equation:** * 89p - 41 * 1.02 = 60 * 89p - 41.82 = 60 * 89p = 101.82 * p = 101.82 / 89 * p ≈ 1.14 **Therefore, the values of p and q are:** * p ≈ 1.14 * q ≈ 1.02 ### Answer: p ≈ 1.14, q ≈ 1.02
Expected answer: {p: 5272/4603, q: 4708/4603}
Correct? No
Problem: Find the values of u and d that make these equations true: The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71
Gemini-1.5 Flash:
Here's how to solve for the values of *u* and *d*:
**1. Elimination Method**
* **Multiply the second equation by 8:** This will allow us to eliminate *u* when we add the equations together.
(27 * u) + (36 * d) = 71 -> (8 * 27 * u) + (8 * 36 * d) = (8 * 71)
-> 216u + 288d = 568
* **Now we have two equations:**
* (88 * u) - (96 * d) = 12
* 216u + 288d = 568
* **Add the two equations together:** Notice that the *u* terms will cancel out.
(88u + 216u) + (-96d + 288d) = 12 + 568
304u + 192d = 580
* **Simplify the equation:** Divide both sides by 8.
38u + 24d = 72.5
* **Now we have a new equation:**
38u + 24d = 72.5
**2. Solving for *u* or *d***
* **Solve the new equation for *u*:**
* 38u = 72.5 - 24d
* u = (72.5 - 24d) / 38
**3. Substitution**
* **Substitute the expression for *u* into either of the original equations.** Let's use the first one:
* (88 * [(72.5 - 24d) / 38]) - (96 * d) = 12
* **Simplify and solve for *d*:**
* (1980 - 844.8d) / 38 - 96d = 12
* 1980 - 844.8d - 3648d = 456
* -4492.8d = -1524
* d = -1524 / -4492.8
* d = 0.34 (approximately)
**4. Find *u***
* **Substitute the value of *d* back into the equation for *u*:**
* u = (72.5 - 24 * 0.34) / 38
* u = (72.5 - 8.16) / 38
* u = 64.34 / 38
* u = 1.69 (approximately)
**Therefore:**
* **u ≈ 1.69**
* **d ≈ 0.34**
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 Flash:
Here's how to solve the system of equations: **1. Elimination Method** * **Multiply the second equation by -6:** This will make the 'f' coefficients opposites. - (-6) * (3n - 2f) = (-6) * 70 - -18n + 12f = -420 * **Add the modified second equation to the first equation:** This eliminates 'f'. - (20n - 18f) + (-18n + 12f) = 9 + (-420) - 2n = -411 * **Solve for 'n':** - n = -411 / 2 - n = -205.5 * **Substitute the value of 'n' back into either original equation to solve for 'f'. Let's use the second equation:** - 3 * (-205.5) - 2f = 70 - -616.5 - 2f = 70 - -2f = 686.5 - f = -686.5 / 2 - f = -343.25 **Therefore:** * n = -205.5 * f = -343.25 ### Answer: n = -205.5, f = -343.25
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 Flash:
We can solve for j and l using a method called elimination. Here's how: 1. **Multiply the first equation by 22 and the second equation by -97.** This will allow us to have opposite coefficients for 'j' in both equations. * (7 * l + 97 * j = 133) * 22 => 154 * l + 2134 * j = 2926 * (22 * j + 37 * l = 76) * -97 => -2134 * j - 3589 * l = -7372 2. **Add the two modified equations together.** This will eliminate 'j' and leave us with an equation with only 'l'. * 154 * l + 2134 * j = 2926 * -2134 * j - 3589 * l = -7372 * ------------------------ * -2035 * l = -4446 3. **Solve for 'l'.** Divide both sides of the equation by -2035. * l = -4446 / -2035 = 2.18 4. **Substitute the value of 'l' back into either of the original equations to solve for 'j'.** Let's use the first equation. * 7 * 2.18 + 97 * j = 133 * 15.26 + 97 * j = 133 * 97 * j = 117.74 * j = 117.74 / 97 = 1.21 Therefore, the values of j and l are: ### Answer: j = 1.21, l = 2.18
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 Flash:
Here's how to solve for 'd' and 'f' using the system of equations:
**1. Elimination Method:**
* **Multiply the first equation by 8 and the second equation by -21:** This will allow us to eliminate 'd' when we add the equations together.
* (21d - 46f = 53) * 8 => 168d - 368f = 424
* (8d + 84f = 133) * -21 => -168d - 1764f = -2793
* **Add the two modified equations together:**
* (168d - 368f = 424) + (-168d - 1764f = -2793) => -2132f = -2369
* **Solve for 'f':**
* f = -2369 / -2132
* f ≈ 1.11
**2. Substitute to find 'd':**
* Choose either of the original equations and substitute the value of 'f' we just found. Let's use the first equation:
* 21d - 46(1.11) = 53
* 21d - 51.06 = 53
* 21d = 104.06
* d ≈ 4.96
**Therefore:**
* d ≈ 4.96
* f ≈ 1.11
### Answer: d ≈ 4.96, f ≈ 1.11
Expected answer: {d: 5285/1066, f: 2369/2132}
Correct? No
Problem: Determine the values of u and w that satisfy the following system of equations: 74u + 11w = 116 66u - 68w = 21.
Gemini-1.5 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the first equation by 68 and the second equation by 11:** This will allow us to have opposite coefficients for the 'w' terms. * 68(74u + 11w) = 68(116) -> 5032u + 748w = 7928 * 11(66u - 68w) = 11(21) -> 726u - 748w = 231 2. **Add the two equations together:** Notice that the 'w' terms cancel out. 5032u + 748w = 7928 726u - 748w = 231 ----------------- 5758u = 8159 3. **Solve for 'u':** 5758u = 8159 u = 8159 / 5758 u = 1.415 (approximately) 4. **Substitute the value of 'u' back into either original equation to solve for 'w'.** Let's use the first equation: 74(1.415) + 11w = 116 104.61 + 11w = 116 11w = 11.39 w = 1.035 (approximately) Therefore, the values of u and w that satisfy the system of equations are: ### Answer: u ≈ 1.415, w ≈ 1.035
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 Flash:
We can solve this system of equations using elimination. 1. **Multiply the first equation by 47 and the second equation by 42:** * (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:** * (3572 * f - 1974 * j = 4512) + (1974 * j + 1428 * f = 5712) = 5000 * f = 10224 3. **Solve for f:** * 5000 * f = 10224 * f = 10224 / 5000 * f = 2.0448 4. **Substitute the value of f back into either of the original equations to solve for j.** We'll use the first equation: * 76 * (2.0448) - 42 * j = 96 * 155.4128 - 42 * j = 96 * -42 * j = 96 - 155.4128 * -42 * j = -59.4128 * j = -59.4128 / -42 * j = 1.4146 Therefore, the values of f and j that satisfy the system of equations are: ### Answer: f = 2.0448 and j = 1.4146.
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 Flash:
Here's how to solve the system of equations: **1. Elimination Method:** * **Multiply the first equation by 69 and the second equation by 68:** * 4692u + 2208e = 12006 * 4284e - 4692u = 68 * **Add the two equations together. Notice that the 'u' terms cancel out:** * 6492e = 12074 * **Solve for 'e':** * e = 12074 / 6492 * e = 1.86 **2. Substitute to find 'u':** * Choose either of the original equations. Let's use the first one: 68u + 32e = 174 * Substitute the value of 'e' we just found: 68u + 32(1.86) = 174 * Simplify: 68u + 59.52 = 174 * Subtract 59.52 from both sides: 68u = 114.48 * Solve for 'u': u = 114.48 / 68 * u = 1.68 **Therefore, the solution to the system of equations is:** ### Answer: u = 1.68, e = 1.86
Expected answer: {e: 6037/3246, u: 5465/3246}
Correct? No
Problem: Find the values of p and n that satisfy the following system of linear equations: 57p + 4n = 129 and 24p + 39n = 154.
Gemini-1.5 Flash:
We can solve this system of equations using elimination. Here's how:
1. **Multiply equations to match coefficients:**
- Multiply the first equation by 39: (39)(57p + 4n) = (39)(129) --> 2223p + 156n = 5031
- Multiply the second equation by -4: (-4)(24p + 39n) = (-4)(154) --> -96p - 156n = -616
2. **Add the modified equations:** Notice that the 'n' terms have opposite signs and equal magnitudes. When we add the equations, they cancel out:
(2223p + 156n) + (-96p - 156n) = 5031 + (-616)
2127p = 4415
3. **Solve for 'p':** Divide both sides by 2127:
p = 4415 / 2127 = 2.075
4. **Substitute 'p' back into either original equation to solve for 'n':** Let's use the first equation:
57(2.075) + 4n = 129
118.275 + 4n = 129
4n = 10.725
n = 10.725 / 4 = 2.68125
**Therefore:**
* p = 2.075
* n = 2.68125
### Answer: p = 2.075, n = 2.68125
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 Flash:
Here's how to solve the system of equations: **1. Elimination Method** * **Multiply the first equation by 3 and the second equation by 11 to make the coefficients of 'z' the same:** * 198z - 132y = 96 * 660y + 154z = 1276 * **Subtract the first equation from the second equation to eliminate 'z':** * (660y + 154z) - (198z - 132y) = 1276 - 96 * 792y - 44z = 1180 * **Simplify the equation:** * 18y - z = 28.5 * **Solve for 'z':** * z = 18y - 28.5 **2. Substitution Method** * **Substitute the value of 'z' (from the simplified equation above) into either of the original equations. Let's use the first original equation:** * 66 * (18y - 28.5) - 44y = 32 * 1188y - 1881 - 44y = 32 * 1144y = 1913 * y = 1913 / 1144 * **Simplify the value of 'y':** * y ≈ 1.67 * **Substitute the value of 'y' back into the equation for 'z':** * z = 18 * 1.67 - 28.5 * z ≈ 0.56 **Therefore:** * y ≈ 1.67 * z ≈ 0.56 ### Answer: y ≈ 1.67, z ≈ 0.56
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 Flash:
Here's how to solve the system of equations: **1. Choose a method:** We'll use the elimination method. To eliminate one variable, we need to make the coefficients of either 't' or 'h' the same in both equations (but with opposite signs). **2. Multiply equations to match coefficients:** * **Eliminate 't':** Multiply the first equation by 11 and the second equation by -83. This will make the coefficients of 't' 913 and -913. * Equation 1 (multiplied by 11): 913t - 638h = 495 * Equation 2 (multiplied by -83): -913t - 5739h = -10304 **3. Add the equations together:** * -638h - 5739h = 495 - 10304 * -6377h = -9809 **4. Solve for 'h':** * h = -9809 / -6377 * h = 1.54 (approximately) **5. Substitute 'h' into either original equation to solve for 't':** * Let's use the first equation: 83t - 58(1.54) = 45 * 83t - 89.72 = 45 * 83t = 134.72 * t = 134.72 / 83 * t = 1.62 (approximately) **Solution:** * t = 1.62 * h = 1.54 ### Answer: t = 1.62, h = 1.54
Expected answer: {h: 9797/6365, t: 10297/6365}
Correct? No
Problem: Given the system of equations ((79 * h) - (61 * o)) = 9 and ((16 * h) - (12 * o)) = 9, what are the values of o and h?
Gemini-1.5 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the second equation by -4.9375:** This will allow us to eliminate 'h' when we add the equations together. (-4.9375) * [(16 * h) - (12 * o) = 9] becomes (-79 * h) + (59.25 * o) = -44.4375 2. **Add the modified second equation to the first equation:** (79 * h) - (61 * o) = 9 (-79 * h) + (59.25 * o) = -44.4375 ---------------------------------- (-1.75 * o) = -35.4375 3. **Solve for 'o':** o = -35.4375 / -1.75 = 20.25 4. **Substitute the value of 'o' back into either original equation to solve for 'h'. Let's use the first equation:** (79 * h) - (61 * 20.25) = 9 (79 * h) - 1235.25 = 9 (79 * h) = 1244.25 h = 1244.25 / 79 = 15.75 **Therefore:** * **o = 20.25** * **h = 15.75**
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 Flash:
Here's how to solve for 's' and 'y' using the elimination method:
1. **Multiply equations to make coefficients match:**
- Multiply the first equation by 65: 260y + 3640s = 9035
- Multiply the second equation by 4: 260y - 360s = 64
2. **Subtract the equations:**
- Notice that the 'y' terms now have the same coefficient. Subtracting the second equation from the first eliminates 'y':
(260y + 3640s) - (260y - 360s) = 9035 - 64
4000s = 8971
3. **Solve for 's':**
- Divide both sides by 4000:
s = 8971 / 4000
s = 2.24275
4. **Substitute 's' back into one of the original equations to solve for 'y':**
- Let's use the first equation:
4y + 56(2.24275) = 139
4y + 125.51 = 139
4y = 13.49
5. **Solve for 'y':**
- Divide both sides by 4:
y = 13.49 / 4
y = 3.3725
### Answer: s = 2.24275, y = 3.3725
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 Flash:
Here's how to solve the simultaneous equations: **1. Elimination Method** * **Multiply the first equation by 45 and the second equation by 4:** This will allow us to have the same coefficient for 'z' in both equations but with opposite signs. * Equation 1 * 45: 1890d - 180z = 3375 * Equation 2 * 4: 160d + 180z = 648 * **Add the two modified equations together:** This will eliminate 'z'. * 1890d + 160d = 3375 + 648 * 2050d = 4023 * **Solve for 'd':** * d = 4023 / 2050 * d = 1.96 * **Substitute the value of 'd' back into either of the original equations to solve for 'z'. Let's use the first equation:** * 42 * 1.96 - 4 * z = 75 * 82.32 - 4 * z = 75 * -4 * z = 75 - 82.32 * -4 * z = -7.32 * z = -7.32 / -4 * z = 1.83 **Therefore:** * d = 1.96 * z = 1.83 ### Answer: d = 1.96, z = 1.83
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 Flash:
Here's how to solve for 'a' and 'o': **1. Simplify the Equations:** * Equation 1: 42o + 24a = 142 * Equation 2: 42a + 6o = 75 **2. Solve for 'o' in terms of 'a' (or vice versa) in one equation.** Let's solve Equation 2 for 'o': * 6o = 75 - 42a * o = (75 - 42a) / 6 **3. Substitute the value of 'o' into the other equation.** Substitute the expression for 'o' into Equation 1: * 42 * [(75 - 42a) / 6] + 24a = 142 **4. Simplify and solve for 'a'.** * 7 * (75 - 42a) + 24a = 142 * 525 - 294a + 24a = 142 * -270a = -383 * a = -383 / -270 * a = 1.42 (approximately) **5. Substitute the value of 'a' back into the equation for 'o'.** * o = (75 - 42 * 1.42) / 6 * o = (75 - 59.64) / 6 * o = 15.36 / 6 * o = 2.56 (approximately) **Therefore:** * a = 1.42 (approximately) * o = 2.56 (approximately) ### Answer: a = 1.42 (approximately), o = 2.56 (approximately)
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 Flash:
We can solve this system of equations using elimination. Here's how:
1. **Multiply the first equation by 8 and the second equation by 3:** This will make the coefficients of 'p' in both equations equal but with opposite signs.
* (39*r - 2*p = 45) * 8 => 312*r - 16*p = 360
* (16*p - 84*r = 36) * 3 => 48*p - 252*r = 108
2. **Add the two modified equations together:** This will eliminate 'p' from the equation.
* 312*r - 16*p = 360
* 48*p - 252*r = 108
* -------------------
* 60*r = 468
3. **Solve for 'r':** Divide both sides by 60.
* r = 468/60
* r = 7.8
4. **Substitute the value of 'r' back into either of the original equations to solve for 'p'.** Let's use the first equation:
* 39 * 7.8 - 2*p = 45
* 304.2 - 2*p = 45
* -2*p = 45 - 304.2
* -2*p = -259.2
* p = -259.2 / -2
* p = 129.6
Therefore, we have:
* **r = 7.8**
* **p = 129.6**
### Answer: r = 7.8, p = 129.6
Expected answer: {p: 216/19, r: 33/19}
Correct? No
Problem: Given the following system of equations, solve for the values of 'd' and 'w': 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58
Gemini-1.5 Flash:
We can solve this system of equations using elimination. Here's how: **1. Multiply the equations to make the coefficients of one variable opposites:** * Multiply the first equation by 8: 184d - 416w = 56 * Multiply the second equation by 23: 1472w - 184d = 1334 **2. Add the two equations together:** Notice that the 'd' terms cancel out: 1056w = 1390 **3. Solve for 'w':** w = 1390 / 1056 w = 1.317 (approximately) **4. Substitute the value of 'w' back into either original equation to solve for 'd'. Let's use the first equation:** (23 * d) - (52 * 1.317) = 7 23d - 68.484 = 7 23d = 75.484 d = 75.484 / 23 d = 3.282 (approximately) **Therefore:** ### Answer: d = 3.282, w = 1.317
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 Flash:
Here's how to solve the system of equations: **1. Elimination Method** * **Multiply equations to match coefficients:** * Multiply the first equation by 47: (3337 * z) + (2820 * n) = 8648 * Multiply the second equation by 60: (3120 * z) - (2820 * n) = 1140 * **Add the equations together:** Notice that the 'n' terms cancel out. (3337 * z) + (3120 * z) = 8648 + 1140 (6457 * z) = 9788 * **Solve for 'z':** z = 9788 / 6457 z = 1.515 (approximately) **2. Substitute to find 'n':** * Choose one of the original equations (let's use the first one). * Substitute the value of 'z' we just found: (71 * 1.515) + (60 * n) = 184 * **Simplify and solve for 'n':** 107.565 + (60 * n) = 184 (60 * n) = 76.435 n = 76.435 / 60 n = 1.274 (approximately) **Therefore:** * **n ≈ 1.274** * **z ≈ 1.515** ### Answer: n ≈ 1.274, z ≈ 1.515
Expected answer: {n: 8219/6457, z: 9788/6457}
Correct? No
Problem: Given the following system of equations, find the values of e and z: 70 * e + 90 * z = 189 99 * z - 84 * e = 20
Gemini-1.5 Flash:
We can solve this system of equations using elimination. 1. **Multiply the first equation by 3 and the second equation by 2.** This will allow us to have coefficients with opposite signs for 'e' in both equations. * (70 * e + 90 * z = 189) * 3 => 210 * e + 270 * z = 567 * (99 * z - 84 * e = 20) * 2 => -168 * e + 198 * z = 40 2. **Add the two modified equations together.** This will eliminate 'e'. * (210 * e + 270 * z = 567) + (-168 * e + 198 * z = 40) => 42 * e + 468 * z = 607 3. **Solve for 'z'.** * 468 * z = 607 - 42 * e * z = (607 - 42 * e) / 468 4. **Substitute the value of 'z' back into either original equation to solve for 'e'.** Let's use the first original equation: * 70 * e + 90 * ((607 - 42 * e) / 468) = 189 * 70 * e + (90 * 607 - 90 * 42 * e) / 468 = 189 * (70 * e * 468 + 90 * 607 - 90 * 42 * e) / 468 = 189 * 32760 * e + 54630 - 3780 * e = 88212 * 29040 * e = 33582 * e = 33582 / 29040 * e = 1.156 (approximately) 5. **Substitute the value of 'e' back into the equation for 'z'.** * z = (607 - 42 * 1.156) / 468 * z = (607 - 48.432) / 468 * z = 558.568 / 468 * z = 1.194 (approximately) ### Answer: e = 1.156, z = 1.194
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 Flash:
We can solve this system of equations using elimination. **1. Multiply the first equation by 11 and the second equation by -44 to make the coefficients of 'o' opposites:** * Equation 1: (11) * (26o - 44f) = (11) * (95) => 286o - 484f = 1045 * Equation 2: (-44) * (87f + 11o) = (-44) * (169) => -3828f - 484o = -7436 **2. Add the two equations together. Notice the 'o' terms cancel out:** * (286o - 484f) + (-3828f - 484o) = 1045 - 7436 * -4312f = -6391 **3. Solve for 'f':** * f = -6391 / -4312 * f = 1.48 **4. Substitute the value of 'f' back into either of the original equations to solve for 'o'. Let's use the first equation:** * 26o - 44(1.48) = 95 * 26o - 65.12 = 95 * 26o = 160.12 * o = 160.12 / 26 * o = 6.16 **Therefore, the values of 'o' and 'f' that satisfy the system of equations are:** ### Answer: o = 6.16, f = 1.48
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 Flash:
We can solve this system of equations using elimination. Here's how: 1. **Multiply the first equation by 3:** This will allow us to have the same coefficient for the 'z' term in both equations. (3) * (40z - 20p) = (3) * (51) This simplifies to: 120z - 60p = 153 2. **Multiply the second equation by -20:** This will allow us to have opposite coefficients for the 'z' term in the two equations. (-20) * (6z + 58p) = (-20) * (165) This simplifies to: -120z - 1160p = -3300 3. **Add the two modified equations together:** Notice that the 'z' terms cancel out. (120z - 60p) + (-120z - 1160p) = 153 + (-3300) This simplifies to: -1220p = -3147 4. **Solve for 'p':** Divide both sides by -1220: p = -3147 / -1220 p = 2.58 (rounded to two decimal places) 5. **Substitute the value of 'p' back into either original equation to solve for 'z'. Let's use the first equation:** 40z - 20(2.58) = 51 40z - 51.6 = 51 6. **Solve for 'z':** Add 51.6 to both sides: 40z = 102.6 Divide both sides by 40: z = 2.565 (rounded to three decimal places) ### 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 Flash:
Here's how to solve the system of equations:
**1. Elimination Method**
* **Multiply the second equation by 8.9:** This will allow us to eliminate 'f' when we add the equations together.
(3 * f) + (10 * s) = 93 -> (26.7 * f) + (89 * s) = 827.7
* **Add the modified second equation to the first equation:**
(71 * s) - (89 * f) = 5
(26.7 * f) + (89 * s) = 827.7
--------------------------
(160 * s) = 832.7
* **Solve for 's':**
s = 832.7 / 160
s = 5.204375
**2. Substitution Method**
* **Solve the second equation for 'f':**
(3 * f) + (10 * s) = 93
3 * f = 93 - (10 * s)
f = (93 - (10 * s)) / 3
* **Substitute the value of 's' into the equation for 'f':**
f = (93 - (10 * 5.204375)) / 3
f = (93 - 52.04375) / 3
f = 40.95625 / 3
f = 13.65208333
**Therefore:**
* **s = 5.204375**
* **f = 13.65208333**
### Answer: s = 5.204375, f = 13.65208333
Expected answer: {f: 6553/1103, s: 8292/1103}
Correct? No