🔗 Problem

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

Gemma 2B:

    To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method.

    Step 1: Multiply the first equation by 3 and the second equation by 4 to eliminate the variable s.

    3(40s - 91q = 37) ⇒ 120s - 273q = 111
    4(30s + 12q = 121) ⇒ 120s + 48q = 484

    Step 2: Subtract the second equation from the first equation to eliminate s.

    (120s - 273q = 111) - (120s + 48q = 484)
    -321q = -373

    Step 3: Solve for q.
    q = -373/-321
    q = 1.16 (approximately)

    Step 4: Substitute the value of q into either of the original equations to find s.
    Let's use the first equation: (40s) - (91 × 1.16) = 37
    40s - 105.56 = 37
    40s = 142.56
    s = 142.56/40
    s = 3.564 (approximately)

    Therefore, the values of the variables are:
    q ≈ 1.16
    s ≈ 3.564

    ### Answer: q ≈ 1.16, s ≈ 3.564

Expected answer: {q: 373/321, s: 2291/642}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given system of equations for the values of b and m, we can use the elimination method or the substitution method.

    Using the elimination method:

    Step 1: Multiply the first equation by 79 and the second equation by 44 to eliminate the variable m.
    4 * b * 79 + 44 * m * 79 = 180 * 79
    316 * b + 3476 * m = 14220

    79 * m * 44 - 6 * b * 44 = 48 * 44
    3476 * m - 264 * b = 2112

    Step 2: Subtract the second equation from the first equation to eliminate m.
    316 * b + 3476 * m - (3476 * m - 264 * b) = 14220 - 2112
    316 * b - 264 * b = 12108
    52 * b = 12108
    b = 12108 / 52
    b = 232.8461538461538

    Step 3: Substitute the value of b into either of the original equations to find the value of m.
    Using the first equation: 4 * 232.8461538461538 + 44 * m = 180
    931.3846153846154 + 44 * m = 180
    44 * m = 180 - 931.3846153846154
    44 * m = -751.3846153846154
    m = -751.3846153846154 / 44
    m = -17.0769666666667

    Therefore, the values of b and m are:
    b = 232.8461538461538
    m = -17.0769666666667

    ### Answer: b = 232.8461538461538, m = -17.0769666666667

Expected answer: {b: 3027/145, m: 318/145}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    42 * o - 15 * g = 28
    47 * o + 29 * g = 141

    Step 1: Multiply the first equation by 29 and the second equation by 15 to eliminate the variable g.
    42 * o * 29 - 15 * g * 29 = 28 * 29
    1218 * o - 435 * g = 812

    47 * o * 15 + 29 * g * 15 = 141 * 15
    705 * o + 435 * g = 2115

    Step 2: Subtract the second equation from the first equation to eliminate g.
    (1218 * o - 435 * g) - (705 * o + 435 * g) = 812 - 2115
    513 * o = -1303

    Step 3: Solve for o.
    o = -1303 / 513
    o = -2.54

    Step 4: Substitute the value of o into either of the original equations to find the value of g.
    Let's use the first equation: 42 * o - 15 * g = 28
    42 * (-2.54) - 15 * g = 28
    -106.68 - 15 * g = 28
    -15 * g = 28 + 106.68
    -15 * g = 134.68
    g = -8.98

    Therefore, the values of o and g are:
    o = -2.54
    g = -8.98

    ### Answer: o = -2.54, g = -8.98

Expected answer: {g: 4606/1923, o: 2927/1923}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Step 1: Multiply the first equation by (-5) to eliminate the variable v.
    (-5)((12v - 6o) = 21)
    -60v + 30o = -105

    Step 2: Add the second equation to the modified first equation to eliminate v.
    (76v - 45o) = 50
    -60v + 30o = -105
    -----------
    16o = -55

    Step 3: Solve for o.
    o = -55/16
    o = -3.4375

    Step 4: Substitute the value of o into either of the original equations to find v.
    Let's use the first equation: (12v - 6(-3.4375)) = 21
    12v - (-20.625) = 21
    12v = 41.625
    v = 41.625/12
    v = 3.46875

    Therefore, the values of v and o are:
    v = 3.46875
    o = -3.4375

    ### Answer: v = 3.46875, o = -3.4375

Expected answer: {o: 83/7, v: 215/28}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Step 1: Multiply the first equation by 2 to eliminate the variable f.
    150f + 50g = 276

    Step 2: Multiply the second equation by 75 to eliminate the variable f.
    150f + 6750g = 10725

    Step 3: Subtract the second equation from the first equation to eliminate f.
    50g - 6750g = 276 - 10725
    -6700g = -10449
    g = 1.56

    Step 4: Substitute the value of g into either of the original equations to find f.
    Let's use the first equation: 75f + 25(1.56) = 138
    75f + 39 = 138
    75f = 99
    f = 1.32

    Therefore, the values of f and g in the given system of equations are:
    f = 1.32
    g = 1.56

    ### Answer: f = 1.32, g = 1.56

Expected answer: {f: 1769/1340, g: 10449/6700}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    194 = 4y + 14n   (1)
    7 = 65n - 79y    (2)

    Step 1: Multiply equation (1) by 79 and equation (2) by 4 to eliminate the variable y.
    79(194) = 79(4y + 14n)
    15326 = 316y + 1106n   (3)

    4(7) = 4(65n - 79y)
    28 = 260n - 316y   (4)

    Step 2: Subtract equation (4) from equation (3) to eliminate y.
    15326 - 28 = 1106n - 260n
    15298 = 846n

    n = 15298/846
    n = 18

    Step 3: Substitute the value of n into either equation (1) or (2) to find the value of y.
    Let's use equation (1):
    194 = 4y + 14(18)
    194 = 4y + 252
    4y = 194 - 252
    4y = -58
    y = -58/4
    y = -14.5

    Therefore, the solution to the system of equations is:
    y = -14.5
    n = 18

    ### Answer: y = -14.5, n = 18

Expected answer: {n: 7677/683, y: 6256/683}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we need to use the substitution method or the elimination method.

    Let's use the substitution method.

    Step 1: Isolate one variable in one of the equations.
    From the second equation, we can isolate l:
    6 = 41l - 53g
    41l = 6 + 53g
    l = (6 + 53g) / 41

    Step 2: Substitute the expression for l into the first equation.
    146 = 28g + 26((6 + 53g) / 41)
    146 = 28g + (156 + 1378g) / 41
    146 × 41 = 28g × 41 + 156 + 1378g
    5986 = 1148g + 156 + 1378g
    5830 = 2526g
    g = 5830 / 2526
    g = 2.31 (approximately)

    Step 3: Substitute the value of g into either equation to find l.
    Let's use the second equation:
    6 = 41l - 53(2.31)
    6 = 41l - 122.43
    41l = 128.43
    l = 128.43 / 41
    l = 3.13 (approximately)

    Therefore, the values of l and g that satisfy the given equations are:
    l ≈ 3.13
    g ≈ 2.31

    ### Answer: l ≈ 3.13, g ≈ 2.31

Expected answer: {g: 2915/1263, l: 3953/1263}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 89 and the second equation by -20 to eliminate the variable p.

    89(20p + 56y = 90)
    1780p + 1984y = 8010

    -20(89p + 7y = 98)
    -1780p - 140y = -1960

    Step 2: Add the two equations to eliminate p.
    1984y + (-140y) = 8010 + (-1960)
    1844y = 6050
    y = 6050/1844
    y = 3.28

    Step 3: Substitute the value of y into either of the original equations to find p.
    Let's use the first equation: 20p + 56(3.28) = 90
    20p + 183.68 = 90
    20p = -93.68
    p = -4.68

    Therefore, the values of p and y are:
    p = -4.68
    y = 3.28

    ### Answer: p = -4.68, y = 3.28

Expected answer: {p: 347/346, y: 3025/2422}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we can use the substitution method or the elimination method.

    Using the substitution method:

    Step 1: Isolate one variable in one of the equations.
    From the first equation, we can isolate d:
    72k - 22d = 17
    22d = 72k - 17
    d = (72k - 17) / 22

    Step 2: Substitute the expression for d into the second equation.
    6d + 69k = 96
    6((72k - 17) / 22) + 69k = 96
    (432k - 102) / 22 + 69k = 96
    432k - 102 + 1518k = 2112
    1950k = 2214
    k = 2214 / 1950
    k = 1.135384615 (rounded to 9 decimal places)

    Step 3: Substitute the value of k into either equation to find d.
    Using the first equation:
    72(1.135384615) - 22d = 17
    81.94615 - 22d = 17
    22d = 64.94615
    d = 64.94615 / 22
    d = 2.952098864 (rounded to 9 decimal places)

    Therefore, the values of the variables are:
    k = 1.135384615
    d = 2.952098864

    ### Answer: k = 1.135384615, d = 2.952098864

Expected answer: {d: 1913/650, k: 369/325}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    73h - 33d = 48
    10d + 14h = 91

    Step 1: Multiply the first equation by 10 and the second equation by 73 to eliminate the variable h.
    730h - 330d = 480
    730d + 1022h = 6643

    Step 2: Subtract the second equation from the first equation to eliminate h.
    730h - 330d = 480
    -(730d + 1022h = 6643)
    -----------------
    -1060d = -6163

    Step 3: Divide both sides by -1060 to find the value of d.
    d = 6163/1060
    d = 5.81

    Step 4: Substitute the value of d into either of the original equations to find the value of h.
    Let's use the first equation: 73h - 33(5.81) = 48
    73h - 191.73 = 48
    73h = 239.73
    h = 239.73/73
    h = 3.28

    Therefore, the values of d and h are:
    d = 5.81
    h = 3.28

    ### Answer: d = 5.81, h = 3.28

Expected answer: {d: 5971/1192, h: 3483/1192}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the given system of equations for the values of q and e, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    1) 43q - 44e = 29
    2) 30q + 43e = 130

    Step 1: Multiply the first equation by 30 and the second equation by (-43) to eliminate the variable e.
    30(43q - 44e = 29)
    1290q - 1320e = 870

    -43(30q + 43e = 130)
    -1290q - 1849e = -5590

    Step 2: Add the two equations to eliminate e.
    1290q - 1320e = 870
    -1290q - 1849e = -5590
    ----------
    -529e = -4720

    Step 3: Solve for e by dividing both sides by -529.
    e = 4720 / 529
    e = 8.92 (approximately)

    Step 4: Substitute the value of e into either of the original equations to find q.
    Let's use the first equation: 43q - 44(8.92) = 29
    43q - 393.68 = 29
    43q = 422.68
    q = 422.68 / 43
    q = 9.83 (approximately)

    Therefore, the values of q and e that satisfy the given system of equations are:
    q ≈ 9.83
    e ≈ 8.92

    ### Answer: q ≈ 9.83, e ≈ 8.92

Expected answer: {e: 4720/3169, q: 6967/3169}

Correct? No

🔗 Problem

Problem: Find the values of r and a in the following system of equations: 53 = ((89 * r) - (74 * a)) 109 = ((37 * a) + (6 * r))

Gemma 2B:

    To solve this system of equations, we need to use the substitution method or the elimination method. Let's use the elimination method.

    Step 1: Multiply the first equation by 6 and the second equation by 89 to eliminate the variable r.

    53 × 6 = 318 = (534 × r) - (444 × a)
    109 × 89 = 9701 = (3293 × a) + (534 × r)

    Step 2: Subtract the second equation from the first equation to eliminate r.

    318 - 9701 = -9383 = (534 × r) - (444 × a) - (3293 × a) - (534 × r)
    -9383 = -3737 × a

    Step 3: Solve for a.

    a = -9383 / -3737
    a = 2.51

    Step 4: Substitute the value of a into either of the original equations to find r.

    Let's use the second equation: 109 = (37 × a) + (6 × r)
    109 = (37 × 2.51) + (6 × r)
    109 = 92.87 + 6r
    6r = 109 - 92.87
    6r = 16.13
    r = 16.13 / 6
    r = 2.69

    Therefore, the values of r and a are:
    r = 2.69
    a = 2.51

    ### Answer: r = 2.69, a = 2.51

Expected answer: {a: 9383/3737, r: 271/101}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve for 'g' and 'q', we need to use the given equations and the substitution method.

    Given equations:
    1. 42g + 12q = 169
    2. 12q - 50g = 19

    Step 1: Rearrange the second equation to isolate 'q'.
    12q - 50g = 19
    12q = 50g + 19
    q = (50g + 19) / 12

    Step 2: Substitute the expression for 'q' into the first equation.
    42g + 12((50g + 19) / 12) = 169
    42g + 50g + 19 = 169
    92g = 150
    g = 150 / 92
    g = 25 / 18

    Step 3: Substitute the value of 'g' into the expression for 'q' to find 'q'.
    q = (50(25/18) + 19) / 12
    q = (1250/18 + 19) / 12
    q = (1269/18) / 12
    q = 105.75 / 18
    q = 5.875

    Therefore, the quantities of the two different types of grains in the mix are:
    g = 25/18 kg
    q = 5.875 kg

    ### Answer: g = 25/18 kg, q = 5.875 kg

Expected answer: {g: 75/46, q: 578/69}

Correct? No

🔗 Problem

Problem: Find the values of s and v for the following system of equations: 37 * s - 11 * v = 54 82 * s - 62 * v = 41

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 62 and the second equation by 11 to eliminate the variable v.

    37 * s - 11 * v = 54 (Multiply by 62)
    2294 * s - 682 * v = 3348

    82 * s - 62 * v = 41 (Multiply by 11)
    902 * s - 682 * v = 451

    Step 2: Subtract the second equation from the first equation to eliminate v.

    2294 * s - 682 * v = 3348
    - (902 * s - 682 * v = 451)
    ------------------------------
    1392 * s = 2897

    Step 3: Divide both sides by 1392 to find the value of s.

    s = 2897 / 1392
    s = 2.08

    Step 4: Substitute the value of s into either of the original equations to find the value of v.

    Let's use the first equation: 37 * s - 11 * v = 54
    37 * 2.08 - 11 * v = 54
    77.96 - 11 * v = 54
    -11 * v = -23.96
    v = 2.18

    Therefore, the values of s and v are:
    s = 2.08
    v = 2.18

Expected answer: {s: 2897/1392, v: 2911/1392}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve for the values of c and z, we need to use the system of linear equations given.

    Let's start by rearranging the equations to get them in the standard form:

    41c + 65z = 151
    78z - 48c = 48

    Now, we can use the elimination method to solve for c and z.

    Step 1: Multiply the first equation by 48 and the second equation by 41 to eliminate c.
    41c × 48 + 65z × 48 = 151 × 48
    1968c + 3120z = 7248

    78z × 41 - 48c × 41 = 48 × 41
    3198z - 1968c = 1968

    Step 2: Subtract the second equation from the first equation to eliminate c.
    (1968c + 3120z) - (3198z - 1968c) = 7248 - 1968
    3888c = 5280
    c = 5280/3888
    c = 1.36 (rounded to two decimal places)

    Step 3: Substitute the value of c into either of the original equations to find z.
    Let's use the first equation: 41(1.36) + 65z = 151
    56.16 + 65z = 151
    65z = 151 - 56.16
    65z = 94.84
    z = 94.84/65
    z = 1.46 (rounded to two decimal places)

    Therefore, the values of c and z are:
    c = 1.36
    z = 1.46

    ### Answer: c = 1.36, z = 1.46

Expected answer: {c: 37/27, z: 512/351}

Correct? No

🔗 Problem

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?

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Step 1: Multiply the first equation by 7 and the second equation by 60 to eliminate the variable c.
    420c - 637i = 49
    420i + 4560c = 8880

    Step 2: Subtract the second equation from the first equation to eliminate c.
    -5197i = -8831

    Step 3: Solve for i.
    i = 8831/5197
    i = 1.7

    Step 4: Substitute the value of i into either of the original equations to find c.
    Using the first equation:
    60c - 91(1.7) = 7
    60c - 154.7 = 7
    60c = 161.7
    c = 2.695

    Therefore, the values of c and i are:
    c = 2.695
    i = 1.7

    ### Answer: c = 2.695, i = 1.7

Expected answer: {c: 1931/1048, i: 2087/1834}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 41 and the second equation by 37 to eliminate the variable e.
    41(71v - 37e = 69) ⇒ 2911v - 1517e = 2829
    37(41e + 76v = 184) ⇒ 1517e + 2812v = 6808

    Step 2: Subtract the second equation from the first equation to eliminate e.
    2911v - 1517e = 2829
    - (1517e + 2812v = 6808)
    ---------------------------
    99v - 3034e = -3979

    Since the coefficient of e is -3034, we can set it to 0 by dividing both sides by -3034.
    99v - 3034e = -3979
    99v = -3979 + 3034(0)
    99v = -3979
    v = -3979/99
    v = -40.19

    Step 3: Substitute the value of v into either of the original equations to find e.
    Let's use the first equation: 71v - 37e = 69
    71(-40.19) - 37e = 69
    -2853.49 - 37e = 69
    -37e = 69 + 2853.49
    -37e = 2922.49
    e = -2922.49/-37
    e = 79.01

    Therefore, the values of e and v that satisfy the given simultaneous linear equations are:
    e = 79.01
    v = -40.19

    ### Answer: e = 79.01, v = -40.19

Expected answer: {e: 7820/5723, v: 9637/5723}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Given equations:
    25v + 69d = 113
    26d + 42v = 85

    Step 1: Multiply the first equation by 26 and the second equation by 25 to eliminate the variable 'v'.
    26(25v + 69d) = 2938
    25(26d + 42v) = 2125

    650v + 1794d = 2938
    650d + 1050v = 2125

    Subtracting the second equation from the first equation:
    744d = 813

    d = 813/744
    d = 1.092... ≈ 1.09

    Step 2: Substitute the value of 'd' into either of the original equations to find the value of 'v'.

    Substituting d = 1.09 into the first equation:
    25v + 69(1.09) = 113
    25v + 75.21 = 113
    25v = 37.79
    v = 37.79/25
    v = 1.512 ≈ 1.51

    Therefore, the values of 'd' and 'v' in the given system of equations are:
    d = 1.09
    v = 1.51

    ### Answer: d = 1.09, v = 1.51

Expected answer: {d: 2621/2248, v: 2927/2248}

Correct? No

🔗 Problem

Problem: Solve the following equations for g and z: 43 * z - 91 * g = 33 72 * g - 3 * z = 68

Gemma 2B:

    To solve the system of equations for g and z, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 3 and the second equation by 91 to eliminate the variable z.
    3(43z - 91g = 33) ⇒ 129z - 273g = 99
    91(72g - 3z = 68) ⇒ 6552g - 273z = 6188

    Step 2: Add the two equations to eliminate z.
    129z - 273g = 99
    6552g - 273z = 6188
    -----------------
    6279g = 6287

    Step 3: Solve for g.
    6279g = 6287
    g = 6287/6279
    g ≈ 1

    Step 4: Substitute the value of g into either of the original equations to find z.
    Let's use the second equation: 72g - 3z = 68
    72(1) - 3z = 68
    72 - 3z = 68
    -3z = -4
    z = 4/3

    Therefore, the solutions are:
    g ≈ 1
    z = 4/3

    ### Answer: g ≈ 1, z = 4/3

Expected answer: {g: 3023/2823, z: 8564/2823}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method.

    Step 1: Multiply the first equation by 3 to eliminate the variable c.
    150c + 36f = 276

    Step 2: Multiply the second equation by 2 to eliminate the variable c.
    72c - 42f = 68

    Step 3: Subtract the second equation from the first equation to eliminate c.
    150c + 36f = 276
    - (72c - 42f = 68)
    ------------------
    78f = 208

    Step 4: Solve for f.
    f = 208/78
    f = 2.67 (approximately)

    Step 5: Substitute the value of f into either equation to find c.
    Let's use the first equation: 50c + 12(2.67) = 92
    50c + 32.04 = 92
    50c = 92 - 32.04
    50c = 59.96
    c = 59.96/50
    c = 1.199 (approximately)

    Therefore, the quantity of cherries (c) is approximately 1.199, and the quantity of figs (f) is approximately 2.67.

    ### Answer: c = 1.199, f = 2.67

Expected answer: {c: 30/19, f: 62/57}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve for the values of u and d, we need to use the given system of linear equations:

    1) 87u - 75d = 7
    2) 85d - 82u = 54

    Step 1: Multiply the first equation by 82 and the second equation by 87 to eliminate u.
    (87u - 75d) × 82 = 7 × 82
    7134u - 6150d = 574

    (85d - 82u) × 87 = 54 × 87
    7395d - 7134u = 4698

    Step 2: Add the two equations to eliminate u.
    7134u - 6150d = 574
    7395d - 7134u = 4698
    ----------
    245d = 5272

    Step 3: Solve for d.
    d = 5272/245
    d = 21.5

    Step 4: Substitute the value of d into either of the original equations to find u.
    Using the first equation: 87u - 75(21.5) = 7
    87u - 1612.5 = 7
    87u = 1619.5
    u = 18.6

    Therefore, the values of u and d are:
    u = 18.6
    d = 21.5

    ### Answer: u = 18.6, d = 21.5

Expected answer: {d: 5272/1245, u: 929/249}

Correct? No

🔗 Problem

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?

Gemma 2B:

    To solve this problem, we need to use the system of two linear equations with two unknowns (a and i) and find their values.

    Given equations:
    (75 * a) - (4 * i) = 42
    (10 * i) + (13 * a) = 160

    Step 1: Rearrange the first equation to get i in terms of a.
    (75 * a) - (4 * i) = 42
    -4i = 42 - (75 * a)
    i = (75a - 42) / 4

    Step 2: Substitute the expression for i into the second equation.
    (10 * i) + (13 * a) = 160
    10 * ((75a - 42) / 4) + 13a = 160
    (187.5a - 105) + 13a = 160
    200.5a = 265
    a = 265 / 200.5
    a = 1.32

    Step 3: Substitute the value of a into the expression for i to find the value of i.
    i = (75a - 42) / 4
    i = (75 * 1.32 - 42) / 4
    i = (99 - 42) / 4
    i = 57 / 4
    i = 14.25

    Therefore, the values of a and i are:
    a = 1.32
    i = 14.25

    ### Answer: a = 1.32, i = 14.25

Expected answer: {a: 530/401, i: 5727/401}

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of p and c: 37 * p + 74 * c = 183 27 * c - 18 * p = 13

Gemma 2B:

    To solve the given system of equations for the values of p and c, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    37p + 74c = 183
    27c - 18p = 13

    Step 1: Multiply the first equation by 18 and the second equation by 37 to eliminate p.
    18(37p + 74c = 183)
    666p + 1332c = 3294

    37(27c - 18p = 13)
    999c - 666p = 481

    Step 2: Subtract the second equation from the first equation to eliminate p.
    (1332c - 999c) = (3294 - 481)
    333c = 2813

    Step 3: Solve for c.
    c = 2813 / 333
    c = 8.45 (approximately)

    Step 4: Substitute the value of c into either of the original equations to find p.
    Let's use the second equation: 27c - 18p = 13
    27(8.45) - 18p = 13
    227.15 - 18p = 13
    -18p = -214.15
    p = 11.9 (approximately)

    Therefore, the solution to the system of equations is:
    p ≈ 11.9
    c ≈ 8.45

    ### Answer: p ≈ 11.9, c ≈ 8.45

Expected answer: {c: 3775/2331, p: 3979/2331}

Correct? No

🔗 Problem

Problem: Determine the values of k and p from the following system of equations: (48 * k) - (30 * p) = 37 (16 * p) + (33 * k) = 105.

Gemma 2B:

    To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method.

    Step 1: Multiply the first equation by 2 to eliminate the variable k.
    2(48k - 30p = 37)
    96k - 60p = 74

    Step 2: Multiply the second equation by -3 to eliminate the variable k.
    -3(16p + 33k = 105)
    -48p - 99k = -315

    Step 3: Add the two equations to eliminate k.
    96k - 60p = 74
    -48p - 99k = -315
    --------------
    -108p = -241

    Step 4: Solve for p.
    p = -241/-108
    p = 2.23 (approximately)

    Step 5: Substitute the value of p into either of the original equations to find k.
    Let's use the second equation: 16p + 33k = 105
    16(2.23) + 33k = 105
    35.68 + 33k = 105
    33k = 69.32
    k = 2.1 (approximately)

    Therefore, the values of k and p are:
    k ≈ 2.1
    p ≈ 2.23

    ### Answer: k ≈ 2.1, p ≈ 2.23

Expected answer: {k: 1871/879, p: 1273/586}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this pair of simultaneous equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    1. 75v - 69f = 14
    2. 46f + 53v = 110

    Step 1: Multiply the first equation by 46 and the second equation by 69 to eliminate the variable f.
    46 × (75v - 69f = 14) ⇒ 3450v - 3174f = 644
    69 × (46f + 53v = 110) ⇒ 3174f + 3657v = 7590

    Step 2: Subtract the second equation from the first equation to eliminate f.
    (3450v - 3174f = 644) - (3174f + 3657v = 7590)
    -207v = -6946
    v = 33.5

    Step 3: Substitute the value of v into either of the original equations to find the value of f.
    Let's use the first equation: 75v - 69f = 14
    75(33.5) - 69f = 14
    2512.5 - 69f = 14
    -69f = 14 - 2512.5
    -69f = -2498.5
    f = 36.2

    Therefore, the values that satisfy the given simultaneous equations are:
    v = 33.5
    f = 36.2

    ### Answer: v = 33.5, f = 36.2

Expected answer: {f: 7508/7107, v: 358/309}

Correct? No

🔗 Problem

Problem: Determine the values of o and g from the following system of equations: 7 * o + 13 * g = 131 89 * g - 70 * o = 37.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    7o + 13g = 131
    89g - 70o = 37

    Step 1: Multiply the first equation by 70 and the second equation by 7 to eliminate the variable o.
    490o + 910g = 9,170
    623g - 490o = 259

    Step 2: Subtract the second equation from the first equation to eliminate o.
    490o + 910g = 9,170
    -(623g - 490o = 259)
    --------------
    287g = 8,911

    Step 3: Divide both sides by 287 to find the value of g.
    g = 8,911 / 287
    g = 31

    Step 4: Substitute the value of g into either of the original equations to find the value of o.
    Let's use the first equation: 7o + 13(31) = 131
    7o + 403 = 131
    7o = -272
    o = -272 / 7
    o = -39

    Therefore, the values of o and g are:
    o = -39
    g = 31

    ### Answer: o = -39, g = 31

Expected answer: {g: 449/73, o: 3726/511}

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of d and k: 88 * d - 60 * k = 38 33 * k - 11 * d = 46.

Gemma 2B:

    To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    88 * d - 60 * k = 38
    33 * k - 11 * d = 46

    Step 1: Multiply the first equation by 11 and the second equation by 88 to eliminate the variable d.
    968 * d - 660 * k = 418
    2904 * k - 968 * d = 4048

    Step 2: Subtract the second equation from the first equation to eliminate d.
    968 * d - 660 * k = 418
    - (2904 * k - 968 * d = 4048)
    -----------------------------------
    3564 * k = -3630

    Step 3: Solve for k.
    k = -3630 / 3564
    k = -1.02

    Step 4: Substitute the value of k into either of the original equations to find d.
    Let's use the first equation: 88 * d - 60 * (-1.02) = 38
    88 * d + 61.2 = 38
    88 * d = 38 - 61.2
    88 * d = -23.2
    d = -23.2 / 88
    d = -0.264

    Therefore, the values of d and k are:
    d = -0.264
    k = -1.02

    ### Answer: d = -0.264, k = -1.02

Expected answer: {d: 669/374, k: 203/102}

Correct? No

🔗 Problem

Problem: Consider the following system of equations, calculate the values of y and z: 24 * z + 72 * y = 139, 2 * z + 82 * y = 96.

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Step 1: Multiply the first equation by 2 and the second equation by 24 to eliminate the variable z.
    48z + 144y = 278 (First equation × 2)
    48z + 1968y = 2304 (Second equation × 24)

    Step 2: Subtract the second equation from the first equation to eliminate z.
    48z + 144y = 278
    - (48z + 1968y = 2304)
    --------------
    -1824y = -2026

    Step 3: Solve for y.
    -1824y = -2026
    y = -2026/-1824
    y = 1.11

    Step 4: Substitute the value of y into either of the original equations to find z.
    Let's use the second equation: 2z + 82y = 96
    Substitute y = 1.11
    2z + 82(1.11) = 96
    2z + 91.02 = 96
    2z = 96 - 91.02
    2z = 4.98
    z = 4.98/2
    z = 2.49

    Therefore, the values of y and z are:
    y = 1.11
    z = 2.49

    ### Answer: y = 1.11, z = 2.49

Expected answer: {y: 1013/912, z: 2243/912}

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the variables v and o: (87 * o) - (29 * v) = 1 and (8 * v) + (30 * o) = 91.

Gemma 2B:

    To solve the given system of equations for v and o, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    (87 * o) - (29 * v) = 1
    (8 * v) + (30 * o) = 91

    Step 1: Multiply the first equation by 8 and the second equation by 29 to eliminate the variable v.
    8 × [(87 * o) - (29 * v) = 1]
    696o - 232v = 8

    29 × [(8 * v) + (30 * o) = 91]
    232v + 870o = 2639

    Step 2: Subtract the second equation from the first equation to eliminate v.
    696o - 232v = 8
    -(232v + 870o = 2639)
    --------------
    -174o = -2631

    Step 3: Solve for o.
    -174o = -2631
    o = 2631/174
    o = 15

    Step 4: Substitute the value of o into either of the original equations to find v.
    Let's use the first equation: (87 * o) - (29 * v) = 1
    (87 * 15) - (29 * v) = 1
    1305 - 29v = 1
    29v = 1304
    v = 1304/29
    v = 45

    Therefore, the solution to the system of equations is:
    v = 45
    o = 15

    ### Answer: v = 45, o = 15

Expected answer: {o: 2647/1566, v: 2629/522}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given system of equations for h and p, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    (23 × h) - (46 × p) = 58 ... (1)
    (11 × p) + (23 × h) = 124 ... (2)

    Step 1: Multiply equation (1) by 11 and equation (2) by 46 to eliminate the variable h.
    11 × [(23 × h) - (46 × p) = 58] ⇒ 253h - 506p = 638 ... (3)
    46 × [(11 × p) + (23 × h) = 124] ⇒ 506p + 1058h = 5704 ... (4)

    Step 2: Subtract equation (3) from equation (4) to eliminate h.
    (506p + 1058h = 5704) - (253h - 506p = 638)
    ⇒ 1564p = 6342
    ⇒ p = 4

    Step 3: Substitute the value of p = 4 into either equation (1) or (2) to find the value of h.
    Substituting p = 4 in equation (1):
    (23 × h) - (46 × 4) = 58
    23h - 184 = 58
    23h = 242
    h = 242 / 23
    h = 10.52

    Therefore, the solution to the system of equations is:
    h = 10.52
    p = 4

    ### Answer: h = 10.52, p = 4

Expected answer: {h: 2114/437, p: 22/19}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Step 1: Multiply the first equation by 6 and the second equation by 1 to eliminate the variable f.
    6(19d + 6f = 120) ⇒ 114d + 36f = 720
    1(90d - 42f = 69) ⇒ 90d - 42f = 69

    Step 2: Subtract the second equation from the first equation to eliminate f.
    (114d + 36f = 720) - (90d - 42f = 69)
    24d + 78f = 651

    Step 3: Divide both sides of the equation by 78 to find the value of f.
    f = 651/78
    f = 8.346... ≈ 8 (since we are dealing with integers)

    Step 4: Substitute the value of f into either of the original equations to find the value of d.
    Let's use the first equation: 19d + 6(8) = 120
    19d + 48 = 120
    19d = 72
    d = 72/19
    d = 3.789... ≈ 4 (since we are dealing with integers)

    Therefore, the values of d and f that satisfy the given system of equations are:
    d = 4
    f = 8

    ### Answer: d = 4, f = 8

Expected answer: {d: 909/223, f: 3163/446}

Correct? No

🔗 Problem

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'?

Gemma 2B:

    To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by (-66) and the second equation by 42 to eliminate the variable 'm'.

    (-66)(42m + 55r = 127) ⇒ -2772m - 3630r = -8382
    (42)(66m - 52r = 44) ⇒ 2772m - 2184r = 1848

    Step 2: Add the two equations to eliminate 'm'.

    -3630r = -8382
    -2184r = 1848
    -----------
    -5814r = -6534

    Step 3: Divide both sides by -5814 to find the value of 'r'.

    r = -6534/-5814
    r = 1.124 (approximately)

    Step 4: Substitute the value of 'r' into either of the original equations to find the value of 'm'.

    Let's use the first equation: 42m + 55(1.124) = 127
    42m + 61.82 = 127
    42m = 65.18
    m = 65.18/42
    m = 1.552 (approximately)

    Therefore, the solution to the system of linear equations is:
    r = 1.124
    m = 1.552

    ### Answer: r = 1.124, m = 1.552

Expected answer: {m: 1504/969, r: 363/323}

Correct? No

🔗 Problem

Problem: Determine the values of n and w that satisfy the following system of linear equations simultaneously: 31n + 64w = 120 57w + 71n = 173

Gemma 2B:

    To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by -71 and the second equation by 31 to eliminate the variable n.

    -71(31n + 64w = 120) ⇒ -2191n - 4464w = -8520
    31(57w + 71n = 173) ⇒ 1767w + 2201n = 5353

    Step 2: Add the two equations to eliminate n.

    (-2191n - 4464w = -8520)
    (+1767w + 2201n = 5353)
    --------------
    -697w = -3167

    Step 3: Solve for w.
    -697w = -3167
    w = -3167/-697
    w = 4.54

    Step 4: Substitute the value of w into either of the original equations to find n.
    Let's use the first equation: 31n + 64(4.54) = 120
    31n + 290.56 = 120
    31n = -170.56
    n = -5.5

    Therefore, the values of n and w that satisfy the system of linear equations simultaneously are:
    n = -5.5
    w = 4.54

    ### Answer: n = -5.5, w = 4.54

Expected answer: {n: 4232/2777, w: 3157/2777}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 76 and the second equation by 26 to eliminate the variable u.

    76 × (72a + 26u = 112)
    5472a + 1976u = 8512

    26 × (76u - 77a = 31)
    1976u - 2002a = 806

    Step 2: Subtract the second equation from the first equation to eliminate u.

    (5472a + 1976u) - (1976u - 2002a) = 8512 - 806
    7474a = 7706

    a = 7706 / 7474
    a = 1.03

    Step 3: Substitute the value of a into either of the original equations to find u.

    Let's use the first equation: 72(1.03) + 26u = 112
    74.16 + 26u = 112
    26u = 112 - 74.16
    26u = 37.84
    u = 37.84 / 26
    u = 1.455

    Therefore, the values of u and a are:
    u = 1.455
    a = 1.03

    ### Answer: u = 1.455, a = 1.03

Expected answer: {a: 3853/3737, u: 5428/3737}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    54n + 18b = 143 (1)
    87n - 93b = 48 (2)

    Step 1: Multiply the first equation by (-87) and the second equation by 54 to eliminate the variable n.
    (-87)(54n + 18b = 143)
    -4698n - 1566b = -12441

    54(87n - 93b = 48)
    4698n - 5022b = 2592

    Step 2: Add the two equations to eliminate n.
    -1566b = -12441
    -5022b = 2592
    -----------
    -6588b = -9849

    Step 3: Divide both sides by -6588 to find the value of b.
    b = -9849 / -6588
    b = 1.5

    Step 4: Substitute the value of b into either of the original equations to find the value of n.
    Let's use the first equation: 54n + 18(1.5) = 143
    54n + 27 = 143
    54n = 116
    n = 116/54
    n = 2.148148148...

    Therefore, the values of b and n are:
    b = 1.5
    n = 2.148148148...

    ### Answer: b = 1.5, n = 2.148148148...

Expected answer: {b: 3283/2196, n: 4721/2196}

Correct? No

🔗 Problem

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)?

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    53e + 7p = 103
    69e - 88p = 6

    Step 1: Multiply the first equation by (-88) and the second equation by 7 to eliminate the variable p.
    (-88)(53e + 7p = 103)
    (-4664e - 616p = -9064)

    7(69e - 88p = 6)
    483e - 616p = 42

    Step 2: Add the two equations to eliminate p.
    (-4664e - 616p = -9064)
    (483e - 616p = 42)
    -----------------
    -4181e = -9022

    Step 3: Solve for e.
    -4181e = -9022
    e = 9022/4181
    e = 2.16

    Step 4: Substitute the value of e into either of the original equations to find p.
    Let's use the first equation: 53e + 7p = 103
    53(2.16) + 7p = 103
    114.48 + 7p = 103
    7p = -11.48
    p = -1.64

    Therefore, the cost of electricity (e) is $2.16, and the cost of petroleum (p) is -$1.64.

    ### Answer: e = 2.16, p = -1.64

Expected answer: {e: 9106/5147, p: 6789/5147}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of linear equations, we can use the substitution method or the elimination method. Let's use the elimination method.

    Given equations:
    (18p - 7s) = 21
    (51s - 24p) = 14

    Step 1: Multiply the first equation by 24 and the second equation by 18 to eliminate the variable p.
    24(18p - 7s) = 24(21)
    432p - 168s = 504

    18(51s - 24p) = 18(14)
    918s - 432p = 252

    Step 2: Add the two equations to eliminate p.
    432p - 168s = 504
    918s - 432p = 252
    --------------
    750s = 756

    Step 3: Solve for s.
    s = 756 / 750
    s = 1

    Step 4: Substitute the value of s into either of the original equations to find p.
    Let's use the first equation: (18p - 7(1)) = 21
    18p - 7 = 21
    18p = 28
    p = 28/18
    p = 7/4

    Therefore, the values of p and s that satisfy both equations are:
    p = 7/4
    s = 1

    ### Answer: p = 7/4, s = 1

Expected answer: {p: 1169/750, s: 126/125}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    59c - 34w = 84 (Equation 1)
    11c + 42w = 98 (Equation 2)

    Step 1: Multiply Equation 1 by 11 and Equation 2 by 59 to eliminate the variable c.
    649c - 374w = 924 (11 × Equation 1)
    649c + 2478w = 5782 (59 × Equation 2)

    Step 2: Subtract the second equation from the first equation to eliminate c.
    -2852w = -4858
    w = 4858 / 2852
    w = 1.7

    Step 3: Substitute the value of w into either of the original equations to find c.
    Let's use Equation 1: 59c - 34(1.7) = 84
    59c - 57.8 = 84
    59c = 141.8
    c = 141.8 / 59
    c = 2.4

    Therefore, the solution to the system of equations is:
    c = 2.4
    w = 1.7

    ### Answer: c = 2.4, w = 1.7

Expected answer: {c: 1715/713, w: 2429/1426}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    45f + 42z = 101 (1)
    60z - 42f = 13 (2)

    Step 1: Multiply the first equation by (-42) and the second equation by 45 to eliminate the variable f.
    (-42)(45f + 42z = 101)
    -1890f - 1764z = -4242

    45(60z - 42f = 13)
    2700z - 1890f = 585

    Step 2: Add the two equations to eliminate f.
    -1764z = -4242
    +2700z = 585
    -------
    936z = -3657

    z = -3657/936
    z = -3.91 (approximately)

    Step 3: Substitute the value of z into either of the original equations to find f.
    Let's use the first equation: 45f + 42(-3.91) = 101
    45f - 164.22 = 101
    45f = 265.22
    f = 265.22/45
    f = 5.89 (approximately)

    Therefore, the solution to the system of equations is:
    f ≈ 5.89
    z ≈ -3.91

    ### Answer: f ≈ 5.89, z ≈ -3.91

Expected answer: {f: 919/744, z: 1609/1488}

Correct? No

🔗 Problem

Problem: Calculate the values of p and u from the following system of equations: 26 * p - 21 * u = 6 15 * p + 58 * u = 83.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    26p - 21u = 6
    15p + 58u = 83

    Step 1: Multiply the first equation by 15 and the second equation by 26 to eliminate the variable p.
    15(26p - 21u = 6)
    390p - 315u = 90

    26(15p + 58u = 83)
    390p + 1508u = 2158

    Step 2: Subtract the second equation from the first equation to eliminate p.
    390p - 315u = 90
    -(390p + 1508u = 2158)
    --------------
    -1823u = -2068

    Step 3: Divide both sides by -1823 to find the value of u.
    u = 2068/1823
    u ≈ 1.134

    Step 4: Substitute the value of u into either of the original equations to find the value of p.
    Let's use the first equation: 26p - 21(1.134) = 6
    26p - 23.814 = 6
    26p = 29.814
    p = 29.814/26
    p ≈ 1.147

    Therefore, the values of p and u are:
    p ≈ 1.147
    u ≈ 1.134

    ### Answer: p ≈ 1.147, u ≈ 1.134

Expected answer: {p: 2091/1823, u: 2068/1823}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve for the values of n and s using the given simultaneous equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    1) 92n - 38s = 62
    2) 59n + 8s = 75

    Step 1: Multiply the first equation by 8 and the second equation by 38 to eliminate the variable s.
    8 × (92n - 38s = 62)
    736n - 304s = 496

    38 × (59n + 8s = 75)
    2242n + 304s = 2850

    Step 2: Subtract the second equation from the first equation to eliminate s.
    (736n - 304s = 496) - (2242n + 304s = 2850)
    -1506n = -2354
    n = 2354/1506
    n = 1.56 (approximately)

    Step 3: Substitute the value of n into either of the original equations to find the value of s.
    Let's use the first equation: 92n - 38s = 62
    92(1.56) - 38s = 62
    143.52 - 38s = 62
    -38s = -81.52
    s = 2.14 (approximately)

    Therefore, the values of n and s are:
    n ≈ 1.56
    s ≈ 2.14

    ### Answer: n ≈ 1.56, s ≈ 2.14

Expected answer: {n: 1673/1489, s: 1621/1489}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given pair of simultaneous equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    87z - 35e = 63
    81z + 58e = 154

    Step 1: Multiply the first equation by 81 and the second equation by -87 to eliminate the variable z.
    81(87z - 35e) = 7107z - 2835e = 5103
    -87(81z + 58e) = -7047z - 5066e = -12113

    Step 2: Add the two equations to eliminate z.
    5103 + (-12113) = -7010

    Step 3: Simplify the resulting equation to find the value of e.
    -7010 = 0
    e = 200

    Step 4: Substitute the value of e into either of the original equations to find the value of z.
    Let's use the first equation: 87z - 35(200) = 63
    87z - 7000 = 63
    87z = 7063
    z = 81

    Therefore, the values of e and z are:
    e = 200
    z = 81

    ### Answer: e = 200, z = 81

Expected answer: {e: 2765/2627, z: 9044/7881}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given system of simultaneous linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    (11 * t) + (44 * h) = 110
    (98 * t) - (36 * h) = 18

    Step 1: Multiply the first equation by -9 to eliminate the variable t.
    -9(11t + 44h = 110)
    -99t - 396h = -990

    Step 2: Multiply the second equation by 1 to keep it unchanged.
    98t - 36h = 18

    Step 3: Add the two equations to eliminate the variable t.
    (-99t - 396h = -990)
    (98t - 36h = 18)
    -----------
    -432h = -972

    Step 4: Divide both sides by -432 to find the value of h.
    h = -972 / -432
    h = 2.25

    Step 5: Substitute the value of h into either of the original equations to find the value of t.
    Let's use the first equation: (11t) + (44 * 2.25) = 110
    11t + 99 = 110
    11t = 11
    t = 1

    Therefore, the values of h and t are:
    h = 2.25
    t = 1

    ### Answer: h = 2.25, t = 1

Expected answer: {h: 481/214, t: 108/107}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    (69 * b) - (50 * o) = 55
    (41 * o) + (18 * b) = 85

    Step 1: Multiply the first equation by 41 and the second equation by 50 to eliminate the variable 'o'.
    41(69b - 50o = 55)
    2839b - 2050o = 2255

    50(41o + 18b = 85)
    2050o + 900b = 4250

    Step 2: Subtract the second equation from the first equation to eliminate 'o'.
    2839b - 2050o = 2255
    - (2050o + 900b = 4250)
    -----------------------
    789b = -1995

    b = -1995/789
    b = -2.525 (approximately)

    Step 3: Substitute the value of 'b' in either of the original equations to find the value of 'o'.
    Let's use the first equation: (69 * b) - (50 * o) = 55
    (69 * (-2.525)) - (50 * o) = 55
    -174.225 - 50o = 55
    -50o = 55 + 174.225
    -50o = 229.225
    o = -229.225/(-50)
    o = 4.585 (approximately)

    Therefore, the values of 'b' and 'o' are:
    b = -2.525
    o = 4.585

    ### Answer: b = -2.525, o = 4.585

Expected answer: {b: 6505/3729, o: 1625/1243}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    67q - 80p = 65
    6q + 5p = 177

    Step 1: Multiply the first equation by 6 and the second equation by 67 to eliminate the variable q.
    6(67q - 80p = 65)
    402q - 480p = 390

    67(6q + 5p = 177)
    402q + 335p = 11889

    Step 2: Subtract the second equation from the first equation to eliminate q.
    402q - 480p = 390
    -(402q + 335p = 11889)
    --------------
    -815p = -11499

    Step 3: Divide both sides by -815 to find the value of p.
    p = 11499/815
    p = 14.1

    Step 4: Substitute the value of p into either of the original equations to find the value of q.
    Let's use the second equation: 6q + 5(14.1) = 177
    6q + 70.5 = 177
    6q = 106.5
    q = 106.5/6
    q = 17.75

    Therefore, the values of p and q are:
    p = 14.1
    q = 17.75

    ### Answer: p = 14.1, q = 17.75

Expected answer: {p: 11469/815, q: 2897/163}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    10k + 48f = 161
    33k - 41f = 31

    Step 1: Multiply the first equation by 33 and the second equation by 10 to eliminate the variable k.
    330k + 1584f = 5313
    330k - 410f = 310

    Step 2: Subtract the second equation from the first equation to eliminate k.
    1994f = 5003

    Step 3: Solve for f.
    f = 5003/1994
    f = 2.51 (approximately)

    Step 4: Substitute the value of f into either of the original equations to find k.
    Let's use the first equation: 10k + 48(2.51) = 161
    10k + 120.48 = 161
    10k = 40.52
    k = 4.052

    Therefore, the values of k and f that satisfy the given system of equations are:
    k ≈ 4.052
    f ≈ 2.51

    ### Answer: k ≈ 4.052, f ≈ 2.51

Expected answer: {f: 5003/1994, k: 8089/1994}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the given system of equations, we can use the substitution method or the elimination method. Let's use the substitution method.

    Given equations:
    31 = 58l - n
    103 = n + 73l

    Step 1: Express one variable in terms of the other variable from one of the equations.
    From the first equation, we can express n in terms of l.
    n = 58l - 31

    Step 2: Substitute the expression for n into the second equation.
    103 = n + 73l
    103 = (58l - 31) + 73l
    103 = 131l - 31

    Step 3: Solve for l by rearranging the terms.
    131l - 31 = 103
    131l = 134
    l = 134/131
    l ≈ 1.023

    Step 4: Substitute the value of l into either of the original equations to find n.
    Let's use the first equation: 31 = 58l - n
    31 = 58(1.023) - n
    31 = 59.334 - n
    n = 59.334 - 31
    n = 28.334

    Therefore, the solution to the system of equations is:
    l ≈ 1.023
    n ≈ 28.334

    ### Answer: l ≈ 1.023, n ≈ 28.334

Expected answer: {l: 134/131, n: 3711/131}

Correct? No

🔗 Problem

Problem: Find the values of y and j that satisfy the following system of equations: 29y - 28j = 39 25y + 37j = 116

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 25 and the second equation by 29 to eliminate the variable y.
    25(29y - 28j = 39) ⇒ 725y - 700j = 975
    29(25y + 37j = 116) ⇒ 725y + 1073j = 3364

    Step 2: Subtract the second equation from the first equation to eliminate y.
    725y - 700j = 975
    -(725y + 1073j = 3364)
    -----------------
    -1773j = -2389

    Step 3: Divide both sides by -1773 to find the value of j.
    j = 2389/1773
    j = 1.347 (rounded to 3 decimal places)

    Step 4: Substitute the value of j into either of the original equations to find the value of y.
    Let's use the first equation: 29y - 28(1.347) = 39
    29y - 37.716 = 39
    29y = 76.716
    y = 76.716/29
    y = 2.645 (rounded to 3 decimal places)

    Therefore, the values of y and j that satisfy the given system of equations are:
    y = 2.645
    j = 1.347

Expected answer: {j: 2389/1773, y: 4691/1773}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 17 and the second equation by -85 to eliminate the variable 'a'.
    17(93u - 85a = 74) ⇒ 1581u - 1445a = 1258
    -85(96u + 17a = 186) ⇒ -8160u - 1445a = -15810

    Step 2: Add the two equations to eliminate 'a'.
    (1581u - 1445a = 1258) + (-8160u - 1445a = -15810)
    -6579u = -14552

    Step 3: Solve for 'u'.
    -6579u = -14552
    u = -14552/-6579
    u = 2.21

    Step 4: Substitute the value of 'u' into either of the original equations to find 'a'.
    Let's use the first equation: 93u - 85a = 74
    93(2.21) - 85a = 74
    205.53 - 85a = 74
    -85a = -131.53
    a = 1.55

    Therefore, the solutions are:
    u = 2.21
    a = 1.55

    ### Answer: u = 2.21, a = 1.55

Expected answer: {a: 3398/3247, u: 1004/573}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this pair of simultaneous equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 23 and the second equation by 61 to eliminate the variable g.

    97 * u - 61 * g = 15 (Multiply by 23)
    2231u - 1403g = 345

    85 * u + 23 * g = 186 (Multiply by 61)
    5185u + 1403g = 11346

    Step 2: Subtract the second equation from the first equation to eliminate g.

    2231u - 1403g = 345
    -5185u - 1403g = -11346
    -----------------
    -2954u = -10001

    Step 3: Solve for u.
    -2954u = -10001
    u = 10001/2954
    u = 3.386 (rounded to 3 decimal places)

    Step 4: Substitute the value of u into either of the original equations to find g.
    Let's use the first equation: 97 * u - 61 * g = 15
    97 * 3.386 - 61g = 15
    328.042 - 61g = 15
    -61g = -313.042
    g = 5.131 (rounded to 3 decimal places)

    Therefore, the values of u and g are:
    u = 3.386
    g = 5.131

    ### Answer: u = 3.386, g = 5.131

Expected answer: {g: 1863/824, u: 1299/824}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    (17 * r) + (67 * b) = 103 (1)
    (85 * r) + (67 * b) = 187 (2)

    Step 1: Multiply the first equation by (-5) to eliminate the variable b.
    (-5) × (17 * r + 67 * b = 103)
    -85 * r - 335 * b = -515

    Step 2: Add the resulting equation from Step 1 to the second equation to eliminate the variable b.
    (85 * r + 67 * b = 187) + (-85 * r - 335 * b = -515)
    0 * b = -328

    Step 3: Solve for r using the second equation.
    85 * r + 67 * b = 187
    85 * r = 187 - 67 * b
    85 * r = 187 (since b = 0)
    r = 187 / 85
    r = 2.2

    Step 4: Substitute the value of r into either equation to find the value of b.
    (17 * 2.2) + (67 * b) = 103
    37.4 + 67 * b = 103
    67 * b = 103 - 37.4
    67 * b = 65.6
    b = 65.6 / 67
    b = 0.98

    Therefore, the values of r and b that satisfy the given system of equations are:
    r = 2.2
    b = 0.98

    ### Answer: r = 2.2, b = 0.98

Expected answer: {b: 82/67, r: 21/17}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method.

    Step 1: Multiply the first equation by 38 and the second equation by 54 to eliminate the variable t.

    38(85n - 54t = 41)
    3230n - 2052t = 1558

    54(38t + 30n = 126)
    2052t + 1620n = 6804

    Step 2: Subtract the second equation from the first equation to eliminate t.

    3230n - 2052t = 1558
    - (2052t + 1620n = 6804)
    ----------------------------
    1610n = -5246

    Step 3: Solve for n.
    n = -5246/1610
    n = -3.26

    Step 4: Substitute the value of n into either of the original equations to find t.
    Let's use the first equation: (85n) - (54t) = 41
    (85(-3.26)) - (54t) = 41
    -277.1 - 54t = 41
    -54t = 41 + 277.1
    -54t = 318.1
    t = -5.89

    Therefore, the values of t and n are:
    t = -5.89
    n = -3.26

    ### Answer: t = -5.89, n = -3.26

Expected answer: {n: 4181/2425, t: 948/485}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we need to use the substitution method or the elimination method.

    Let's use the substitution method:

    Step 1: Rearrange the first equation to express one variable in terms of the other.
    75u - 69q = 5
    75u = 5 + 69q
    u = (5 + 69q) / 75

    Step 2: Substitute the expression for u into the second equation.
    48u + 25q = 140
    48((5 + 69q) / 75) + 25q = 140
    (240 + 3312q) / 75 + 25q = 140
    3312q + 1875q = 10500
    5187q = 10500
    q = 10500 / 5187
    q = 2

    Step 3: Substitute the value of q into the expression for u to find the value of u.
    u = (5 + 69(2)) / 75
    u = (5 + 138) / 75
    u = 143 / 75
    u = 1.9

    Therefore, the values of u and q are:
    u = 1.9
    q = 2

    ### Answer: u = 1.9, q = 2

Expected answer: {q: 180/91, u: 515/273}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Step 1: Multiply the first equation by 49 and the second equation by 20 to eliminate the variable x.

    49((20v) + (17x)) = 7,231
    20((24x) - (49v)) = 440

    Step 2: Subtract the second equation from the first equation to eliminate x.

    7,231 - 440 = 6,791
    980v + 833x = 7,231
    -980v - 98x = -440
    --------------
    1,666x = 6,791

    Step 3: Solve for x.
    x = 6,791 / 1,666
    x = 4.08

    Step 4: Substitute the value of x into either of the original equations to find v.
    Let's use the first equation: (20v) + (17(4.08)) = 149
    20v + 69.36 = 149
    20v = 79.64
    v = 3.98

    Therefore, the values of v and x are:
    v = 3.98
    x = 4.08

    ### Answer: v = 3.98, x = 4.08

Expected answer: {v: 3202/1313, x: 7741/1313}

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of t and g: 39 * t + 41 * g = 147 94 * t - 91 * g = 42.

Gemma 2B:

    To solve the given system of equations for the values of t and g, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    39t + 41g = 147
    94t - 91g = 42

    Step 1: Multiply the first equation by 94 and the second equation by 39 to eliminate one of the variables (let's eliminate t).
    39t × 94 + 41g × 94 = 147 × 94
    3666t + 3854g = 13818

    94t × 39 - 91g × 39 = 42 × 39
    3666t - 3549g = 1638

    Step 2: Subtract the second equation from the first equation to eliminate t.
    (3854g + 3666t) - (-3549g + 3666t) = 13818 - 1638
    7403g = 12180

    Step 3: Solve for g by dividing both sides by 7403.
    g = 12180 / 7403
    g = 1.645 (approximately)

    Step 4: Substitute the value of g into either of the original equations to find the value of t.
    Let's use the first equation: 39t + 41(1.645) = 147
    39t + 67.645 = 147
    39t = 147 - 67.645
    39t = 79.355
    t = 79.355 / 39
    t = 2.035 (approximately)

    Therefore, the solution to the system of equations is:
    t ≈ 2.035
    g ≈ 1.645

    ### Answer: t ≈ 2.035, g ≈ 1.645

Expected answer: {g: 12180/7403, t: 15099/7403}

Correct? No

🔗 Problem

Problem: Determine the values of c and o that satisfy the following system of equations: 6c + 55o = 112, and 55c - 48o = 36.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    6c + 55o = 112
    55c - 48o = 36

    Step 1: Multiply the first equation by -48 and the second equation by 6 to eliminate the variable c.
    -288c - 2640o = -5376
    330c - 288o = 216

    Step 2: Add the two equations to eliminate c.
    42o = -5160

    Step 3: Solve for o.
    o = -5160/42
    o = -122.857142857...
    o ≈ -123

    Step 4: Substitute the value of o into either of the original equations to find c.
    Let's use the first equation: 6c + 55(-123) = 112
    6c - 6765 = 112
    6c = 6877
    c = 1146

    Therefore, the values of c and o that satisfy the given system of equations are:
    c = 1146
    o = -123

    ### Answer: c = 1146, o = -123

Expected answer: {c: 7356/3313, o: 5944/3313}

Correct? No

🔗 Problem

Problem: Determine the values of u and w that satisfy the following system of equations: 8w + 68u = 170, 71w - 94u = 8.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 71 and the second equation by 8 to eliminate the variable w.
    8(71)w + 68(71)u = 170(71)
    568w + 4828u = 12070

    71(8)w - 94(8)u = 8(8)
    568w - 752u = 64

    Step 2: Subtract the second equation from the first equation to eliminate w.
    4828u - (-752u) = 12070 - 64
    5580u = 12006

    Step 3: Solve for u.
    u = 12006 / 5580
    u = 2.15

    Step 4: Substitute the value of u into either of the original equations to find w.
    Let's use the first equation: 8w + 68(2.15) = 170
    8w + 146.2 = 170
    8w = 23.8
    w = 2.975

    Therefore, the values that satisfy the system of equations are:
    u = 2.15
    w = 2.975

    ### Answer: u = 2.15, w = 2.975

Expected answer: {u: 667/310, w: 459/155}

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of r and f: 68*r + 8*f = 92 97*f + 38*r = 182

Gemma 2B:

    To solve the given system of equations for the values of r and f, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    68r + 8f = 92 ... (1)
    97f + 38r = 182 ... (2)

    Step 1: Multiply equation (1) by 97 and equation (2) by 68 to eliminate the variable f.
    97 × (68r + 8f = 92)
    6556r + 776f = 8884

    68 × (97f + 38r = 182)
    6596f + 2584r = 12376

    Step 2: Subtract the second equation from the first equation to eliminate f.
    (6556r + 776f = 8884) - (6596f + 2584r = 12376)
    -5820f + 3972r = -3492

    Step 3: Divide both sides of the equation by 3972 to find the value of r.
    -5820f + 3972r = -3492
    r = -3492 / 3972
    r = -0.879 (rounded to 3 decimal places)

    Step 4: Substitute the value of r into either of the original equations to find the value of f.
    Let's use equation (1): 68r + 8f = 92
    68(-0.879) + 8f = 92
    -59.772 + 8f = 92
    8f = 92 + 59.772
    8f = 151.772
    f = 151.772 / 8
    f = 18.972 (rounded to 3 decimal places)

    Therefore, the solution to the system of equations is:
    r = -0.879
    f = 18.972

    ### Answer: r = -0.879, f = 18.972

Expected answer: {f: 2220/1573, r: 1867/1573}

Correct? No

🔗 Problem

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).

Gemma 2B:

    To solve this problem, we need to use the system of linear equations and find the values of r and d that satisfy both equations.

    Given equations:
    67d - 37r = 50
    39r - 21d = 16

    Step 1: Rearrange the equations to get the standard form (Ax + By = C).
    67d - 37r = 50 (Multiply both sides by -1)
    -67d + 37r = -50

    39r - 21d = 16

    Step 2: Use the elimination method to solve for one variable in terms of the other.
    Multiply the first equation by 39 and the second equation by 67 to eliminate d.
    (-67d + 37r) × 39 = -50 × 39
    -2613d + 1443r = -1950

    39r - 21d = 16 (Multiply by 67)
    2613r - 1407d = 1072

    Step 3: Add the two equations to eliminate d.
    1443r - 1950 + 2613r - 1072 = 0
    4056r = 3022
    r = 3022/4056
    r = 0.75 (Cost of each ragdoll)

    Step 4: Substitute the value of r into either equation to find d.
    Let's use the first equation: 67d - 37(0.75) = 50
    67d - 27.75 = 50
    67d = 77.75
    d = 77.75/67
    d = 1.16 (Cost of each apple)

    Therefore, the cost of each ragdoll (r) is $0.75, and the cost of each apple (d) is $1.16.

    ### Answer: r = 0.75, d = 1.16

Expected answer: {d: 1271/918, r: 1061/918}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given system of simultaneous linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    163 = 50x + a (First equation)
    24 = 98x - 84a (Second equation)

    Step 1: Multiply the first equation by 98 and the second equation by 50 to eliminate the variable x.
    98(163) = 98(50x + a)
    15,974 = 4,900x + 98a

    50(24) = 50(98x - 84a)
    1,200 = 4,900x - 4,200a

    Step 2: Subtract the second equation from the first equation to eliminate x.
    15,974 - 1,200 = 4,900x + 98a - 4,900x + 4,200a
    14,774 = 4,298a

    Step 3: Solve for a.
    a = 14,774 / 4,298
    a = 3.44 (approximately)

    Step 4: Substitute the value of a into either of the original equations to find x.
    Let's use the first equation: 163 = 50x + a
    163 = 50x + 3.44
    50x = 163 - 3.44
    50x = 159.56
    x = 159.56 / 50
    x = 3.19 (approximately)

    Therefore, the values of x and a are:
    x ≈ 3.19
    a ≈ 3.44

    ### Answer: x ≈ 3.19, a ≈ 3.44

Expected answer: {a: 7387/2149, x: 6858/2149}

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of p and b: 67 * p + 18 * b = 110 82 * b - 44 * p = 58.

Gemma 2B:

    To solve the given system of equations for the values of p and b, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    67p + 18b = 110
    82b - 44p = 58

    Step 1: Multiply the first equation by 44 and the second equation by 67 to eliminate p.
    67(67p + 18b) = 4490 + 1206b
    82(82b - 44p) = 6724b - 3608p

    Step 2: Subtract the second equation from the first equation to eliminate p.
    4490 + 1206b
    -(-3608p + 6724b)
    -----------------------
    4490 - 6724b + 1206b = -2518b

    Step 3: Solve for b.
    -2518b = -2518
    b = 1

    Step 4: Substitute the value of b into either of the original equations to find p.
    Let's use the first equation: 67p + 18(1) = 110
    67p + 18 = 110
    67p = 92
    p = 92/67
    p = 1.373134328 (approximately)

    Therefore, the values of p and b that satisfy the given system of equations are:
    p ≈ 1.373134328
    b = 1

    ### Answer: p ≈ 1.373134328, b = 1

Expected answer: {b: 4363/3143, p: 3988/3143}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this problem, we need to use the given simultaneous equations and find the values of c and e.

    Given equations:
    63c - 36e = 27
    8c + 32e = 69

    Step 1: Rearrange the equations to get one variable in terms of the other.
    From the first equation, we can express e in terms of c.
    63c - 36e = 27
    36e = 63c - 27
    e = (63c - 27) / 36

    Step 2: Substitute the expression for e into the second equation.
    8c + 32((63c - 27) / 36) = 69
    8c + (2016c - 864) / 36 = 69
    8c + 56c - 24 = 69
    64c = 93
    c = 93/64

    Step 3: Substitute the value of c into the expression for e to find the value of e.
    e = (63c - 27) / 36
    e = (63 × (93/64) - 27) / 36
    e = (5859/64 - 27) / 36
    e = (5832/64) / 36
    e = 162/64
    e = 2.53125

    Therefore, the values of c and e are:
    c = 93/64 ≈ 1.453125
    e = 2.53125

    ### Answer: c = 93/64 ≈ 1.453125, e = 2.53125

Expected answer: {c: 93/64, e: 459/256}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the given system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    26x + 29p = 107
    61x - 51p = 4

    Step 1: Multiply the first equation by a suitable constant to make the coefficients of one variable (say, x) the same in both equations.
    Multiplying the first equation by 61/26, we get:
    61x + (29 × 61/26)p = 107 × 61/26
    61x + 67.5p = 2497

    Step 2: Subtract the second equation from the modified first equation to eliminate the variable x.
    (61x + 67.5p = 2497) - (61x - 51p = 4)
    118.5p = 2493

    Step 3: Solve for p.
    p = 2493 / 118.5
    p = 21

    Step 4: Substitute the value of p into either of the original equations to find the value of x.
    Let's use the first equation: 26x + 29(21) = 107
    26x + 609 = 107
    26x = -502
    x = -19.3076923077

    Therefore, the values of x and p are:
    x = -19.3076923077
    p = 21

    ### Answer: x = -19.3076923077, p = 21

Expected answer: {p: 6423/3095, x: 5573/3095}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method.

    Given equations:
    (29 × l) + (17 × g) = 68
    (31 × g) - (10 × l) = 17

    Step 1: Multiply the first equation by 31 and the second equation by 29 to eliminate the variable g.
    31 × [(29 × l) + (17 × g) = 68]
    31 × (29 × l) + 31 × (17 × g) = 68 × 31
    899l + 527g = 2108

    29 × [(31 × g) - (10 × l) = 17]
    29 × (31 × g) - 29 × (10 × l) = 17 × 29
    899g - 290l = 493

    Step 2: Subtract the second equation from the first equation to eliminate g.
    (899l + 527g = 2108) - (899g - 290l = 493)
    817l = 1615

    Step 3: Solve for l.
    l = 1615 / 817
    l = 1.975 (approximately)

    Step 4: Substitute the value of l into either of the original equations to find g.
    Let's use the first equation: (29 × l) + (17 × g) = 68
    (29 × 1.975) + (17 × g) = 68
    57.075 + 17g = 68
    17g = 68 - 57.075
    17g = 10.925
    g = 10.925 / 17
    g = 0.643 (approximately)

    Therefore, the values of l and g are:
    l ≈ 1.975
    g ≈ 0.643

    ### Answer: l ≈ 1.975, g ≈ 0.643

Expected answer: {g: 1173/1069, l: 1819/1069}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 2 to eliminate the variable a.
    66a - 108c = 40

    Step 2: Subtract the second equation from the modified first equation.
    66a - 108c = 40
    -(24a - 26c = 29)
    -----------------
    42a - 82c = 11

    Step 3: Divide both sides of the equation by -82 to find the value of c.
    c = 11/82

    Step 4: Substitute the value of c into either of the original equations to find the value of a.
    Let's use the first equation: 33a - 54(11/82) = 20
    33a - 594/82 = 20
    33a = 20 + 594/82
    a = (20 + 594/82) / 33
    a = 614/82 / 33
    a = 614 / 2706
    a = 0.227 (approximately)

    Therefore, the values are:
    a ≈ 0.227
    c ≈ -0.134

    ### Answer: a = 0.227, c = -0.134

Expected answer: {a: 523/219, c: 159/146}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve a system of two linear equations with two variables, we can use various methods, such as the substitution method, elimination method, or matrix method. Let's use the elimination method here.

    Given equations:
    71d + 9j = 132
    43j - 84d = 91

    Step 1: Multiply the first equation by 84 and the second equation by 71 to eliminate the variable d.
    71d × 84 + 9j × 84 = 132 × 84
    5964d + 756j = 11088

    43j × 71 - 84d × 71 = 91 × 71
    3053j - 5964d = 6461

    Step 2: Subtract the second equation from the first equation to eliminate d.
    (5964d + 756j) - (3053j - 5964d) = 11088 - 6461
    9817j = 4627

    Step 3: Solve for j by dividing both sides by 9817.
    j = 4627 / 9817
    j = 0.472 (approximately)

    Step 4: Substitute the value of j into either of the original equations to find d.
    Let's use the first equation: 71d + 9(0.472) = 132
    71d + 4.248 = 132
    71d = 127.752
    d = 1.8 (approximately)

    Therefore, the solution to the system of equations is:
    d ≈ 1.8
    j ≈ 0.472

    ### Answer: d ≈ 1.8, j ≈ 0.472

Expected answer: {d: 4857/3809, j: 17549/3809}

Correct? No

🔗 Problem

Problem: Determine the values of o and n by solving this system of equations: (81 * o) + (26 * n) = 163 (87 * n) + (23 * o) = 143.

Gemma 2B:

    To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method.

    Step 1: Multiply the first equation by 87 and the second equation by 81 to eliminate the variable n.

    (81 × 87 × o) + (26 × 87 × n) = 163 × 87
    (87 × 23 × o) + (81 × 87 × n) = 143 × 81

    7047o + 2262n = 14191
    2001o + 7047n = 11583

    Step 2: Subtract the second equation from the first equation to eliminate n.

    (7047o + 2262n) - (2001o + 7047n) = 14191 - 11583
    5046o - 4785n = 2608

    Step 3: Set the coefficient of n to zero by dividing both sides by -4785.

    5046o - 4785n = 2608
    -4785n = -2608
    n = 2608 / 4785
    n = 0.545 (rounded to 3 decimal places)

    Step 4: Substitute the value of n into either of the original equations to find the value of o.

    (81 × o) + (26 × 0.545) = 163
    81o + 14.17 = 163
    81o = 148.83
    o = 148.83 / 81
    o = 1.838 (rounded to 3 decimal places)

    Therefore, the values of o and n are:
    o = 1.838
    n = 0.545

    ### Answer: o = 1.838, n = 0.545

Expected answer: {n: 7834/6449, o: 10463/6449}

Correct? No

🔗 Problem

Problem: Solve the following system of equations for w and s: 46 * s - 61 * w = 2 61 * s + 67 * w = 178.

Gemma 2B:

    To solve the given system of equations for w and s, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    46s - 61w = 2
    61s + 67w = 178

    Step 1: Multiply the first equation by 61 and the second equation by 46 to eliminate the variable s.
    61(46s - 61w = 2)
    2806s - 3721w = 122

    46(61s + 67w = 178)
    2806s + 3082w = 8168

    Step 2: Subtract the second equation from the first equation to eliminate s.
    2806s - 3721w = 122
    - (2806s + 3082w = 8168)
    ----------------------------
    -6803w = -8046

    Step 3: Solve for w.
    -6803w = -8046
    w = 8046/6803
    w = 1.18 (approximately)

    Step 4: Substitute the value of w into either of the original equations to find s.
    Let's use the first equation: 46s - 61(1.18) = 2
    46s - 71.98 = 2
    46s = 73.98
    s = 73.98/46
    s = 1.61 (approximately)

    Therefore, the solution to the system of equations is:
    w = 1.18
    s = 1.61

    ### Answer: w = 1.18, s = 1.61

Expected answer: {s: 10992/6803, w: 8066/6803}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 8 and the second equation by 51 to eliminate the variable i.
    408k - 64i = 688
    1989i + 408k = 9843

    Step 2: Subtract the second equation from the first equation to eliminate the variable k.
    -2053i = -9155
    i = 9155/2053
    i = 4.46 (approximately)

    Step 3: Substitute the value of i into either of the original equations to find the value of k.
    Let's use the first equation: 51k - 8(4.46) = 86
    51k - 35.68 = 86
    51k = 121.68
    k = 121.68/51
    k = 2.39 (approximately)

    Therefore, the values of i and k in the given system of equations are:
    i = 4.46
    k = 2.39

    ### Answer: i = 4.46, k = 2.39

Expected answer: {i: 9155/2053, k: 4898/2053}

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of l and m: 26m + 17l = 146 93m + l = 126

Gemma 2B:

    To solve the given system of equations for the values of l and m, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    26m + 17l = 146
    93m + l = 126

    Step 1: Multiply the first equation by (-1) to make the coefficients of l opposite in sign.
    -26m - 17l = -146

    Step 2: Add the two equations to eliminate the variable l.
    (-26m - 17l) + (93m + l) = (-146) + 126
    67m = -20

    Step 3: Divide both sides by 67 to find the value of m.
    m = -20/67

    Step 4: Substitute the value of m into either of the original equations to find the value of l.
    Let's use the second equation: 93m + l = 126
    Substituting m = -20/67, we get:
    93(-20/67) + l = 126
    (-1860/67) + l = 126
    l = 126 + 1860/67
    l = 1986/67

    Therefore, the values of l and m are:
    m = -20/67
    l = 1986/67

    ### Answer: m = -20/67, l = 1986/67

Expected answer: {l: 10302/1555, m: 1996/1555}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by -29 and the second equation by 8 to eliminate the variable 'f'.

    -29(8f + 39h = 66)
    -232f - 1131h = -1914

    8(29f + 46h = 119)
    232f + 368h = 952

    Step 2: Add the two equations together to eliminate 'f'.

    -1131h + 368h = -1914 + 952
    -763h = -962
    h = 962/763
    h = 1.26 (approximately)

    Step 3: Substitute the value of 'h' into either of the original equations to find the value of 'f'.

    Let's use the first equation: (8f) + (39 × 1.26) = 66
    8f + 49.14 = 66
    8f = 16.86
    f = 2.11 (approximately)

    Therefore, the values of 'h' and 'f' are:
    h = 1.26
    f = 2.11

    ### Answer: h = 1.26, f = 2.11

Expected answer: {f: 1605/763, h: 962/763}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    22b + 23k = 109
    62k - 69b = 14

    Step 1: Multiply the first equation by 62 and the second equation by 23 to eliminate k.
    22b(62) + 23k(62) = 109(62)
    1364b + 1426k = 6758

    62k(23) - 69b(23) = 14(23)
    1426k - 1587b = 322

    Step 2: Subtract the second equation from the first equation to eliminate k.
    (1364b + 1426k) - (1426k - 1587b) = 6758 - 322
    2951b = 6436

    Step 3: Solve for b.
    b = 6436 / 2951
    b = 2.18

    Step 4: Substitute the value of b into either of the original equations to find k.
    Let's use the first equation: 22b + 23k = 109
    22(2.18) + 23k = 109
    47.96 + 23k = 109
    23k = 109 - 47.96
    23k = 61.04
    k = 61.04 / 23
    k = 2.65

    Therefore, the values of k and b that satisfy the given system of equations are:
    k = 2.65
    b = 2.18

    ### Answer: k = 2.65, b = 2.18

Expected answer: {b: 6436/2951, k: 7829/2951}

Correct? No

🔗 Problem

Problem: Solve the following system of linear equations for the values of m and s: 1. 84m - 55s = 1 2. 63m - 2s = 98.

Gemma 2B:

    To solve the given system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 2 and the second equation by 3 to eliminate the variable m.
    2(84m - 55s = 1) ⇒ 168m - 110s = 2
    3(63m - 2s = 98) ⇒ 189m - 6s = 294

    Step 2: Subtract the second equation from the first equation to eliminate m.
    (168m - 110s = 2) - (189m - 6s = 294)
    -21m - 104s = -292

    Step 3: Divide both sides of the equation by -104 to find the value of s.
    -104s = -292
    s = 292/104
    s = 2.8076923076923075

    Step 4: Substitute the value of s into either of the original equations to find the value of m.
    Let's use the first equation: 84m - 55(2.8076923076923075) = 1
    84m - 153.42307692307692 = 1
    84m = 154.42307692307692
    m = 154.42307692307692 / 84
    m = 1.8383451085702472

    Therefore, the solution to the system of linear equations is:
    m = 1.8383451085702472
    s = 2.8076923076923075

    ### Answer: m = 1.8383451085702472, s = 2.8076923076923075

Expected answer: {m: 1796/1099, s: 389/157}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Step 1: Multiply the first equation by 51 and the second equation by 6 to eliminate the variable b.

    1. (306b) + (255k) = 5355
    2. (570k) - (306b) = 402

    Step 2: Subtract the second equation from the first equation to eliminate b.

    (306b) + (255k) = 5355
    -(570k) + (306b) = -402
    ----------------------------
    -315k = 4953

    Step 3: Solve for k by dividing both sides by -315.

    k = -4953/-315
    k = 15.75

    Step 4: Substitute the value of k into either of the original equations to find the value of b.

    Let's use the first equation: (6b) + (5 * 15.75) = 105
    6b + 78.75 = 105
    6b = 26.25
    b = 4.375

    Therefore, the values of b and k are:
    b = 4.375
    k = 15.75

    ### Answer: b = 4.375, k = 15.75

Expected answer: {b: 1928/165, k: 1919/275}

Correct? No

🔗 Problem

Problem: Determine the values of d and u from the following pair of simultaneous linear equations: 33d + 62u = 114 and 32u + 40d = 83.

Gemma 2B:

    To solve this pair of simultaneous linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    33d + 62u = 114
    32u + 40d = 83

    Step 1: Multiply the first equation by 32 and the second equation by 33 to eliminate one of the variables (let's eliminate d).
    32(33d + 62u) = 3648
    33(32u + 40d) = 2739

    Step 2: Subtract the second equation from the first equation.
    3648 - 2739 = 909
    1056u + 1056d - 1320d = 909
    1056u - 264d = 909

    Step 3: Divide both sides of the equation by the coefficient of u to find the value of u.
    1056u/1056 - 264d/1056 = 909/1056
    u - 0.25d = 0.86
    u = 0.86 + 0.25d

    Step 4: Substitute the value of u into one of the original equations to find the value of d.
    Let's use the first equation: 33d + 62u = 114
    33d + 62(0.86 + 0.25d) = 114
    33d + 53.32 + 15.5d = 114
    48.5d = 60.68
    d = 60.68/48.5
    d = 1.25

    Step 5: Substitute the value of d into the expression for u to find the value of u.
    u = 0.86 + 0.25(1.25)
    u = 0.86 + 0.3125
    u = 1.1725

    Therefore, the values of d and u are:
    d = 1.25
    u = 1.1725

    ### Answer: d = 1.25, u = 1.1725

Expected answer: {d: 749/712, u: 1821/1424}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of equations with two variables (c and p), we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    5c + 28p = 102
    20c - 67p = 13

    Step 1: Multiply the first equation by 4 and the second equation by 1 to eliminate the variable c.
    20c + 112p = 408
    20c - 67p = 13

    Step 2: Subtract the second equation from the first equation to eliminate c.
    20c + 112p = 408
    - (20c - 67p = 13)
    ---------------
    179p = 395

    Step 3: Solve for p by dividing both sides by 179.
    p = 395 / 179
    p = 2.21

    Step 4: Substitute the value of p into either of the original equations to find c.
    Let's use the first equation: 5c + 28(2.21) = 102
    5c + 61.88 = 102
    5c = 40.12
    c = 8.024

    Therefore, the values of the variables are:
    c = 8.024
    p = 2.21

    ### Answer: c = 8.024, p = 2.21

Expected answer: {c: 7198/895, p: 395/179}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 76 and the second equation by 58 to eliminate the variable r.
    (58r + 12q) × 76 = 4408r + 912q = 11,172
    (97q - 76r) × 58 = 5,626q - 4,408r = 4,408

    Step 2: Subtract the second equation from the first equation to eliminate r.
    4,408r + 912q = 11,172
    -(5,626q - 4,408r = 4,408)
    --------------
    -4,714q = 6,764

    Step 3: Solve for q.
    -4,714q = 6,764
    q = -6,764 / -4,714
    q = 1.435 (approximately)

    Step 4: Substitute the value of q into either of the original equations to find r.
    Let's use the first equation: 58r + 12(1.435) = 147
    58r + 17.22 = 147
    58r = 147 - 17.22
    58r = 129.78
    r = 129.78 / 58
    r = 2.238 (approximately)

    Therefore, the values of r and q are:
    r ≈ 2.238
    q ≈ 1.435

    ### Answer: r ≈ 2.238, q ≈ 1.435

Expected answer: {q: 7790/3269, r: 13347/6538}

Correct? No

🔗 Problem

Problem: Resolve the provided system of simultaneous linear equations for v and k: 47 * v - 3 * k = 10 73 * v + 2 * k = 128.

Gemma 2B:

    To solve the given system of simultaneous linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    47v - 3k = 10
    73v + 2k = 128

    Step 1: Multiply the first equation by 2 to eliminate the variable k.
    94v - 6k = 20

    Step 2: Subtract the second equation from the modified first equation.
    94v - 6k - (73v + 2k) = 20 - 128
    21v - 8k = -108

    Step 3: Divide both sides of the equation by -8 to find the value of k.
    -21v/8 + k = 108/8
    k = 13.5

    Step 4: Substitute the value of k into either of the original equations to find the value of v.
    Let's use the first equation: 47v - 3(13.5) = 10
    47v - 40.5 = 10
    47v = 50.5
    v = 50.5/47
    v ≈ 1.074468085106383

    Therefore, the solution to the system of simultaneous linear equations is:
    v ≈ 1.074468085106383
    k = 13.5

    ### Answer: v ≈ 1.074468085106383, k = 13.5

Expected answer: {k: 5286/313, v: 404/313}

Correct? No

🔗 Problem

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?

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Given equations:
    89p - 41q = 60
    27p + 91q = 124

    Step 1: Multiply the first equation by 27 and the second equation by 89 to eliminate p.
    27(89p - 41q) = 27(60)
    2403p - 1107q = 1620

    89(27p + 91q) = 89(124)
    2403p + 8099q = 11036

    Step 2: Subtract the second equation from the first equation to eliminate p.
    2403p - 1107q = 1620
    -(2403p + 8099q = 11036)
    -----------------
    -9206q = -9416

    Step 3: Solve for q.
    q = -9416/-9206
    q = 1.023 (approximately)

    Step 4: Substitute the value of q into either of the original equations to find p.
    Let's use the first equation: 89p - 41(1.023) = 60
    89p - 41.943 = 60
    89p = 101.943
    p = 101.943/89
    p = 1.145 (approximately)

    Therefore, the values of p and q are:
    p ≈ 1.145
    q ≈ 1.023

    ### Answer: p ≈ 1.145, q ≈ 1.023

Expected answer: {p: 5272/4603, q: 4708/4603}

Correct? No

🔗 Problem

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

Gemma 2B:

    To find the values of u and d that make the given equations true, we need to solve the system of two linear equations with two unknowns.

    Given equations:
    (88 * u) - (96 * d) = 12
    (27 * u) + (36 * d) = 71

    Step 1: Solve for one of the variables, say d, in terms of u from the first equation.
    (88 * u) - (96 * d) = 12
    -96d = 12 - 88u
    d = (12 - 88u) / (-96)
    d = (88u - 12) / 96

    Step 2: Substitute the expression for d in the second equation.
    (27 * u) + (36 * ((88u - 12) / 96)) = 71
    27u + (3312u - 432) / 96 = 71
    27u + 34.5u - 4.5 = 71
    61.5u = 75.5
    u = 75.5 / 61.5
    u = 1.227272727...

    Step 3: Substitute the value of u in the expression for d to find the value of d.
    d = (88u - 12) / 96
    d = (88 * 1.227272727... - 12) / 96
    d = (108 - 12) / 96
    d = 96 / 96
    d = 1

    Therefore, the values of u and d that make the given equations true are:
    u = 1.227272727... (approximately 1.23)
    d = 1

    ### Answer: u = 1.23, d = 1

Expected answer: {d: 1481/1440, u: 151/120}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    20n - 18f = 9
    3n - 2f = 70

    Step 1: Multiply the second equation by 10 to eliminate the variable n.
    30n - 20f = 700

    Step 2: Subtract the first equation from the modified second equation.
    10n - 2f = 691
    -18f = -9
    f = 0.5

    Step 3: Substitute the value of f into either of the original equations to find the value of n.
    Let's use the second equation: 3n - 2(0.5) = 70
    3n - 1 = 70
    3n = 71
    n = 23.67

    Therefore, the values of f and n are:
    f = 0.5
    n = 23.67

    ### Answer: f = 0.5, n = 23.67

Expected answer: {f: 1373/14, n: 621/7}

Correct? No

🔗 Problem

Problem: Determine the values of j and l from the following system of equations: 7 * l + 97 * j = 133 22 * j + 37 * l = 76.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    7l + 97j = 133
    22j + 37l = 76

    Step 1: Multiply the first equation by -22 and the second equation by 7 to eliminate the variable j.
    (-22)(7l + 97j = 133)
    -154l - 2134j = -2926

    7(22j + 37l = 76)
    154j + 259l = 532

    Step 2: Add the two equations to eliminate j.
    (-154l - 2134j = -2926)
    (154j + 259l = 532)
    -----------------
    105l = -2394

    Step 3: Solve for l.
    l = -2394/105
    l = -22.8

    Step 4: Substitute the value of l into either of the original equations to find j.
    Let's use the first equation: 7l + 97j = 133
    7(-22.8) + 97j = 133
    -159.6 + 97j = 133
    97j = 292.6
    j = 292.6/97
    j = 3.02

    Therefore, the values of j and l are:
    j = 3.02
    l = -22.8

    ### Answer: j = 3.02, l = -22.8

Expected answer: {j: 1463/1145, l: 1482/1145}

Correct? No

🔗 Problem

Problem: Determine the values of d and f from the following system of equations: 21d - 46f = 53 8d + 84f = 133.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 8 and the second equation by 21 to eliminate the variable d.

    21d - 46f = 53 (Multiply by 8)
    168d - 368f = 424

    8d + 84f = 133 (Multiply by 21)
    168d + 1764f = 2793

    Step 2: Subtract the second equation from the first equation to eliminate d.

    168d - 368f = 424
    -(168d + 1764f = 2793)
    -----------------
    -2132f = -2369

    Step 3: Divide both sides by -2132 to find the value of f.

    f = -2369/-2132
    f = 1.11

    Step 4: Substitute the value of f into either of the original equations to find the value of d.

    Let's use the first equation: 21d - 46(1.11) = 53
    21d - 51.06 = 53
    21d = 104.06
    d = 4.96

    Therefore, the values of d and f are:
    d = 4.96
    f = 1.11

    ### Answer: d = 4.96, f = 1.11

Expected answer: {d: 5285/1066, f: 2369/2132}

Correct? No

🔗 Problem

Problem: Determine the values of u and w that satisfy the following system of equations: 74u + 11w = 116 66u - 68w = 21.

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Step 1: Multiply the first equation by 66 and the second equation by 74 to eliminate the variable u.
    74u × 66 + 11w × 66 = 116 × 66
    4884u + 726w = 7656

    66u × 74 - 68w × 74 = 21 × 74
    4884u - 5032w = 1554

    Step 2: Subtract the second equation from the first equation to eliminate u.
    (4884u + 726w = 7656) - (4884u - 5032w = 1554)
    5758w = 6102

    Step 3: Solve for w.
    w = 6102 / 5758
    w = 1.06 (approximately)

    Step 4: Substitute the value of w into either of the original equations to find u.
    Let's use the first equation: 74u + 11(1.06) = 116
    74u + 11.66 = 116
    74u = 104.34
    u = 1.41 (approximately)

    Therefore, the values of u and w that satisfy the given system of equations are:
    u ≈ 1.41
    w ≈ 1.06

    ### Answer: u ≈ 1.41, w ≈ 1.06

Expected answer: {u: 8119/5758, w: 3051/2879}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Step 1: Multiply the first equation by 47 and the second equation by 76 to eliminate the variable j.
    76 * f - 42 * j = 96 (Multiply by 47)
    3572 * f - 1974 * j = 4512

    47 * j + 34 * f = 136 (Multiply by 76)
    3572 * f + 2572 * j = 10336

    Step 2: Subtract the second equation from the first equation to eliminate j.
    3572 * f - 1974 * j = 4512
    - (3572 * f + 2572 * j = 10336)
    -----------------------------------
    -4546 * j = -5824

    Step 3: Divide both sides by -4546 to find the value of j.
    j = 5824 / 4546
    j = 1.28

    Step 4: Substitute the value of j into either of the original equations to find the value of f.
    Let's use the first equation: 76 * f - 42 * (1.28) = 96
    76 * f - 53.76 = 96
    76 * f = 149.76
    f = 149.76 / 76
    f = 1.97

    Therefore, the values of f and j that satisfy the given system of equations are:
    f = 1.97
    j = 1.28

    ### Answer: f = 1.97, j = 1.28

Expected answer: {f: 1278/625, j: 884/625}

Correct? No

🔗 Problem

Problem: Consider the following system of equations. Find the values of u and e: 68u + 32e = 174 63e - 69u = 1.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 63 and the second equation by 68 to eliminate the variable u.
    (68u + 32e) × 63 = 4284u + 2016e
    (63e - 69u) × 68 = 4284e - 4692u

    Step 2: Subtract the second equation from the first equation to eliminate u.
    4284u + 2016e
    - (4284e - 4692u)
    --------------
    6708e = 10976

    Step 3: Solve for e.
    e = 10976 / 6708
    e = 1.636 (rounded to 3 decimal places)

    Step 4: Substitute the value of e into either of the original equations to find u.
    Let's use the second equation: 63e - 69u = 1
    63(1.636) - 69u = 1
    102.668 - 69u = 1
    -69u = -101.668
    u = 1.473 (rounded to 3 decimal places)

    Therefore, the values of u and e are:
    u = 1.473
    e = 1.636

    ### Answer: u = 1.473, e = 1.636

Expected answer: {e: 6037/3246, u: 5465/3246}

Correct? No

🔗 Problem

Problem: Find the values of p and n that satisfy the following system of linear equations: 57p + 4n = 129 and 24p + 39n = 154.

Gemma 2B:

    To solve this system of linear equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 24 and the second equation by 57 to eliminate the variable p.
    24(57p + 4n = 129) = 1368p + 96n = 3096
    57(24p + 39n = 154) = 1368p + 2223n = 8778

    Step 2: Subtract the second equation from the first equation to eliminate p.
    (1368p + 96n = 3096) - (1368p + 2223n = 8778)
    -2127n = -5682
    n = 5682/2127
    n = 2.67 (rounded to two decimal places)

    Step 3: Substitute the value of n into either of the original equations to find the value of p.
    Let's use the first equation: 57p + 4(2.67) = 129
    57p + 10.68 = 129
    57p = 118.32
    p = 118.32/57
    p = 2.08 (rounded to two decimal places)

    Therefore, the values of p and n that satisfy the given system of linear equations are:
    p = 2.08
    n = 2.67

    ### Answer: p = 2.08, n = 2.67

Expected answer: {n: 1894/709, p: 4415/2127}

Correct? No

🔗 Problem

Problem: Determine the values of y and z from the given system of equations: 66 * z - 44 * y = 32 60 * y + 14 * z = 116.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 14 and the second equation by 44 to eliminate the variable z.
    66 * z - 44 * y = 32 (Multiply by 14)
    924 * z - 616 * y = 448

    60 * y + 14 * z = 116 (Multiply by 44)
    2640 * y + 616 * z = 5104

    Step 2: Subtract the second equation from the first equation to eliminate z.
    (924 * z - 616 * y = 448)
    - (2640 * y + 616 * z = 5104)
    -1716 * y = -4656

    Step 3: Solve for y.
    -1716 * y = -4656
    y = 4656 / 1716
    y = 2.71

    Step 4: Substitute the value of y into either of the original equations to find z.
    Let's use the first equation: 66 * z - 44 * y = 32
    66 * z - 44 * (2.71) = 32
    66 * z - 119.24 = 32
    66 * z = 151.24
    z = 151.24 / 66
    z = 2.29

    Therefore, the values of y and z are:
    y = 2.71
    z = 2.29

    ### Answer: y = 2.71, z = 2.29

Expected answer: {y: 901/572, z: 439/286}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    83t - 58h = 45 (1)
    69h + 11t = 124 (2)

    Step 1: Multiply the first equation by 69 and the second equation by -83 to eliminate the variable h.
    (83t - 58h) × 69 = 5727t - 4002h = 3105
    (69h + 11t) × (-83) = -5727h - 913t = -10292

    Step 2: Add the resulting equations to eliminate h.
    5727t - 4002h = 3105
    -5727h - 913t = -10292
    --------------
    4814t = -7187

    Step 3: Solve for t.
    4814t = -7187
    t = -7187 / 4814
    t = -1.49 (approximately)

    Step 4: Substitute the value of t into either of the original equations to find h.
    Let's use the first equation: 83t - 58h = 45
    83(-1.49) - 58h = 45
    -123.67 - 58h = 45
    -58h = 168.67
    h = -2.91 (approximately)

    Therefore, the solution to the system of equations is:
    t = -1.49
    h = -2.91

    ### Answer: t = -1.49, h = -2.91

Expected answer: {h: 9797/6365, t: 10297/6365}

Correct? No

🔗 Problem

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?

Gemma 2B:

    To solve this system of equations, we need to use the elimination method or the substitution method.

    Let's use the elimination method:

    Step 1: Multiply the first equation by 2 to eliminate the variable h.
    2 × ((79 * h) - (61 * o)) = 9 × 2
    158h - 122o = 18

    Step 2: Multiply the second equation by -5 to eliminate the variable h.
    -5 × ((16 * h) - (12 * o)) = -5 × 9
    -80h + 60o = -45

    Step 3: Add the two equations to eliminate h.
    158h - 122o = 18
    -80h + 60o = -45
    --------------
    78o = -27

    Step 4: Solve for o.
    o = -27/78
    o = -0.346... ≈ -0.35

    Step 5: Substitute the value of o into either of the original equations to find h.
    Let's use the second equation: (16 * h) - (12 * (-0.35)) = 9
    16h + 4.2 = 9
    16h = 4.8
    h = 0.3

    Therefore, the values of o and h are:
    o = -0.35
    h = 0.3

    ### Answer: o = -0.35, h = 0.3

Expected answer: {h: 63/4, o: 81/4}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve for the variables s and y in the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    4y + 56s = 139
    65y - 90s = 16

    Step 1: Multiply the first equation by 65 and the second equation by 4 to eliminate the variable y.
    260y + 3640s = 9035
    260y - 360s = 64

    Step 2: Subtract the second equation from the first equation to eliminate y.
    4000s = 8971

    Step 3: Solve for s.
    s = 8971 / 4000
    s = 2.24275

    Step 4: Substitute the value of s into either of the original equations to find y.
    Let's use the first equation: 4y + 56(2.24275) = 139
    4y + 125.594 = 139
    4y = 13.406
    y = 3.3515

    Therefore, the solution to the system of equations is:
    s = 2.24275
    y = 3.3515

    ### Answer: s = 2.24275, y = 3.3515

Expected answer: {s: 8971/4000, y: 6703/2000}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve the pair of simultaneous equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    42d - 4z = 75
    40d + 45z = 162

    Step 1: Multiply the first equation by 5 and the second equation by 6 to eliminate the variable z.
    210d - 20z = 375
    240d + 270z = 972

    Step 2: Subtract the second equation from the first equation to eliminate z.
    (210d - 20z) - (240d + 270z) = 375 - 972
    -30d - 290z = -597

    Step 3: Divide both sides of the equation by -30 to find the value of d.
    d = 19.9

    Step 4: Substitute the value of d into either of the original equations to find the value of z.
    Let's use the first equation: 42d - 4z = 75
    42(19.9) - 4z = 75
    835.8 - 4z = 75
    -4z = -760.8
    z = 190.2

    Therefore, the values of d and z are:
    d = 19.9
    z = 190.2

    ### Answer: d = 19.9, z = 190.2

Expected answer: {d: 4023/2050, z: 1902/1025}

Correct? No

🔗 Problem

Problem: Determine the values of 'a' and 'o' from the following system of equations: (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75.

Gemma 2B:

    To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method.

    Given equations:
    (42 * o) + (24 * a) = 142
    (42 * a) + (6 * o) = 75

    Step 1: Multiply the first equation by 6 and the second equation by 24 to eliminate the variable 'o'.
    6 × [(42 * o) + (24 * a) = 142]
    252o + 144a = 852

    24 × [(42 * a) + (6 * o) = 75]
    1008a + 144o = 1800

    Step 2: Subtract the second equation from the first equation to eliminate 'o'.
    (252o + 144a = 852) - (1008a + 144o = 1800)
    -864a = -948
    a = 1.1

    Step 3: Substitute the value of 'a' into either of the original equations to find the value of 'o'.
    Let's use the second equation: (42 * a) + (6 * o) = 75
    (42 * 1.1) + (6 * o) = 75
    46.2 + 6o = 75
    6o = 28.8
    o = 4.8

    Therefore, the values of 'a' and 'o' are:
    a = 1.1
    o = 4.8

    ### Answer: a = 1.1, o = 4.8

Expected answer: {a: 383/270, o: 347/135}

Correct? No

🔗 Problem

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.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 16 and the second equation by 2 to eliminate the variable p.
    39*r - 2*p = 45 (Multiply by 16)
    624*r - 32*p = 720

    16*p - 84*r = 36 (Multiply by 2)
    32*p - 168*r = 72

    Step 2: Add the two equations to eliminate p.
    624*r - 32*p = 720
    32*p - 168*r = 72
    -----------------
    456*r = 792

    Step 3: Solve for r.
    456*r = 792
    r = 792/456
    r = 24/16
    r = 1.5

    Step 4: Substitute the value of r into either of the original equations to find p.
    Let's use the first equation: 39*r - 2*p = 45
    39(1.5) - 2*p = 45
    58.5 - 2*p = 45
    -2*p = -13.5
    p = 6.75

    Therefore, the values of p and r are:
    p = 6.75
    r = 1.5

    ### Answer: p = 6.75, r = 1.5

Expected answer: {p: 216/19, r: 33/19}

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method.

    Step 1: Multiply the first equation by 8 and the second equation by 23 to eliminate the variable 'd'.
    8 × (23d - 52w = 7)
    184d - 416w = 56

    23 × (64w - 8d = 58)
    1472w - 184d = 1334

    Step 2: Add the two equations to eliminate 'd'.
    184d - 416w = 56
    1472w - 184d = 1334
    ----------
    1056w = 1390

    Step 3: Solve for 'w'.
    w = 1390 / 1056
    w = 1.316037736 (rounded to 9 decimal places)

    Step 4: Substitute the value of 'w' into either of the original equations to find 'd'.
    Let's use the first equation: (23d) - (52 × 1.316037736) = 7
    23d - 68.632 = 7
    23d = 75.632
    d = 3.288347826 (rounded to 9 decimal places)

    Therefore, the values of 'd' and 'w' are:
    d = 3.288347826
    w = 1.316037736

    ### Answer: d = 3.288347826, w = 1.316037736

Expected answer: {d: 433/132, w: 695/528}

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of n and z: (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19.

Gemma 2B:

    To solve the given system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    (71 * z) + (60 * n) = 184 ... (1)
    (52 * z) - (47 * n) = 19 ... (2)

    Step 1: Multiply equation (1) by 52 and equation (2) by 71 to eliminate the variable z.
    52 × (71 * z + 60 * n = 184)
    3692z + 3120n = 9568 ... (3)

    71 × (52 * z - 47 * n = 19)
    3692z - 3337n = 1349 ... (4)

    Step 2: Subtract equation (4) from equation (3) to eliminate z.
    (3120n + 3692z) - (-3337n + 3692z) = (9568 - 1349)
    6457n = 8219
    n = 8219 / 6457
    n = 1.273 (approximately)

    Step 3: Substitute the value of n into either equation (1) or (2) to find the value of z.
    Substituting n = 1.273 in equation (1):
    (71 * z) + (60 * 1.273) = 184
    71z + 76.38 = 184
    71z = 107.62
    z = 107.62 / 71
    z = 1.515 (approximately)

    Therefore, the solution to the system of equations is:
    n ≈ 1.273
    z ≈ 1.515

    ### Answer: n ≈ 1.273, z ≈ 1.515

Expected answer: {n: 8219/6457, z: 9788/6457}

Correct? No

🔗 Problem

Problem: Given the following system of equations, find the values of e and z: 70 * e + 90 * z = 189 99 * z - 84 * e = 20

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by 84 and the second equation by 70 to eliminate the variable e.

    70 * e + 90 * z = 189 (Multiply by 84)
    5880 * e + 7560 * z = 15876

    99 * z - 84 * e = 20 (Multiply by 70)
    6930 * z - 5880 * e = 1400

    Step 2: Subtract the second equation from the first equation to eliminate e.

    5880 * e + 7560 * z = 15876
    - (6930 * z - 5880 * e = 1400)
    ----------------------------------
    630 * z = 14476

    Step 3: Solve for z.
    z = 14476 / 630
    z = 23

    Step 4: Substitute the value of z into either of the original equations to find e.
    Let's use the first equation: 70 * e + 90 * 23 = 189
    70 * e + 2070 = 189
    70 * e = 189 - 2070
    70 * e = -1881
    e = -1881 / 70
    e = -26.87

    Therefore, the values of e and z are:
    e = -26.87
    z = 23

    ### Answer: e = -26.87, z = 23

Expected answer: {e: 1879/1610, z: 1234/1035}

Correct? No

🔗 Problem

Problem: Find the values of o and f that satisfy the following system of equations: 1) 26o - 44f = 95 2) 87f + 11o = 169.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Step 1: Multiply the first equation by a suitable number to eliminate one of the variables.
    Multiplying the first equation by 11, we get:
    286o - 484f = 1045

    Step 2: Multiply the second equation by a suitable number to eliminate the other variable.
    Multiplying the second equation by 26, we get:
    2262f + 286o = 4394

    Step 3: Subtract the second equation from the first equation to eliminate the variable 'o'.
    286o - 484f = 1045
    - (2262f + 286o = 4394)
    --------------------------
    -2746f = -3349

    Step 4: Solve for 'f'.
    -2746f = -3349
    f = 3349/2746
    f = 1.22 (approximately)

    Step 5: Substitute the value of 'f' into either of the original equations to find the value of 'o'.
    Let's use the first equation: 26o - 44(1.22) = 95
    26o - 53.68 = 95
    26o = 148.68
    o = 148.68/26
    o = 5.72 (approximately)

    Therefore, the values of 'o' and 'f' that satisfy the given system of equations are:
    o ≈ 5.72
    f ≈ 1.22

Expected answer: {f: 3349/2746, o: 15701/2746}

Correct? No

🔗 Problem

Problem: Calculate the values of z and p using the following system of equations: 40*z - 20*p = 51 6*z + 58*p = 165.

Gemma 2B:

    To solve this system of equations, we can use the elimination method or the substitution method. Let's use the elimination method.

    Given equations:
    40z - 20p = 51
    6z + 58p = 165

    Step 1: Multiply the first equation by 6 and the second equation by -40 to eliminate the variable z.
    240z - 120p = 306
    -240z - 2320p = -6600

    Step 2: Add the two equations to eliminate z.
    -2440p = -6294

    Step 3: Divide both sides by -2440 to find the value of p.
    p = 6294/2440
    p = 2.58

    Step 4: Substitute the value of p into either of the original equations to find the value of z.
    Let's use the first equation: 40z - 20(2.58) = 51
    40z - 51.6 = 51
    40z = 102.6
    z = 102.6/40
    z = 2.565

    Therefore, the values of z and p are:
    z = 2.565
    p = 2.58

    ### Answer: z = 2.565, p = 2.58

Expected answer: {p: 3147/1220, z: 3129/1220}

Correct? No

🔗 Problem

Problem: Find the values of s and f from the given system of equations: (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93.

Gemma 2B:

    To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method.

    Step 1: Multiply the first equation by 3 and the second equation by 71 to eliminate the variable s.

    3(71s - 89f = 5)
    213s - 267f = 15

    71(3f + 10s = 93)
    213f + 710s = 6603

    Step 2: Subtract the second equation from the first equation to eliminate s.

    213s - 267f = 15
    -(213f + 710s = 6603)
    --------------
    -480f = -6588

    Step 3: Divide both sides by -480 to find the value of f.

    f = 6588/480
    f = 13.725

    Step 4: Substitute the value of f into either of the original equations to find the value of s.

    Let's use the first equation: 71s - 89(13.725) = 5
    71s - 1222.325 = 5
    71s = 1227.325
    s = 1227.325/71
    s = 17.285

    Therefore, the values of s and f are:
    s = 17.285
    f = 13.725

Expected answer: {f: 6553/1103, s: 8292/1103}

Correct? No