🔗 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:

     Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we'll use the elimination method.

    First, we'll multiply the first equation by 12 and the second equation by 91 to make the coefficients of q the same:

    1. (480 * s) - (1092 * q) = 444
    2. (2730 * s) + (1092 * q) = 10991

    Next, we'll add the two equations to eliminate q:

    (480 * s) + (2730 * s) = 444 + 10991
    (3210 * s) = 11435

    Now, we'll solve for s:

    s = 11435 / 3210
    s = 3.562305295950156

    Now that we have the value of s, we can substitute it back into one of the original equations to find q. Let's use the first equation:

    (40 * 3.562305295950156) - (91 * q) = 37
    142.49221183800624 - (91 * q) = 37

    Now, we'll solve for q:

    - (91 * q) = 37 - 142.49221183800624
    - (91 * q) = -105.49221183800624

    q = -105.49221183800624 / -91
    q = 1.1592550751429257

    ### Answer: q = 1.1592550751429257, s = 3.562305295950156

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:

     Reasoning: Let's solve the system of equations using substitution. First, we can solve the first equation for b: 4 * b = 180 - 44 * m
    b = (180 - 44 * m) / 4
    Now, we can substitute this expression for b into the second equation: 79 * m - 6 * ((180 - 44 * m) / 4) = 48
    Simplifying the equation: 79 * m - (6 * 180) / 4 + (6 * 44 * m) / 4 = 48
    79 * m - 270 + 66 * m = 48
    Combining like terms: 145 * m = 318
    m = 318 / 145
    m = 2.19310344828
    Now, we can substitute the value of m back into the expression for b: b = (180 - 44 * 2.19310344828) / 4
    b = (180 - 96.4965476244) / 4
    b = 83.5034523756 / 4
    b = 20.8758630939
    ### Answer: m = 2.19310344828, b = 20.8758630939

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 29 and the second equation by 15 to make the coefficients of g in both equations equal:

    29 * (42 * o - 15 * g) = 29 * 28

    15 * (47 * o + 29 * g) = 15 * 141

    Simplifying these equations, we get:

    1218 * o - 435 * g = 812

    705 * o + 435 * g = 2115

    Now, we can add these two equations to eliminate the g term:

    1218 * o - 435 * g + 705 * o + 435 * g = 812 + 2115

    Simplifying this equation, we get:

    1923 * o = 2927

    Dividing both sides by 1923, we get:

    o = 2927 / 1923

    o = 1.522

    Now, we can substitute the value of o back into one of the original equations to solve for g. Let's use the first equation:

    42 * o - 15 * g = 28

    42 * 1.522 - 15 * g = 28

    63.924 - 15 * g = 28

    Subtracting 63.924 from both sides, we get:

    -15 * g = -35.924

    Dividing both sides by -15, we get:

    g = 2.395

    ### Answer: o = 1.522, g = 2.395

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:

     Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.
    First, we can multiply the first equation by 45 and the second equation by 6 to make the coefficients of o the same:
    45((12 * v) - (6 * o)) = 45 * 21
    6((76 * v) - (45 * o)) = 6 * 50
    This gives us:
    540v - 270o = 945
    456v - 270o = 300
    Now, we can subtract the second equation from the first to eliminate o:
    (540v - 270o) - (456v - 270o) = 945 - 300
    This simplifies to:
    84v = 645
    Now, we can solve for v:
    v = 645 / 84
    v = 7.678571428571429
    Now that we have the value of v, we can substitute it back into one of the original equations to solve for o. Let's use the first equation:
    ((12 * 7.678571428571429) - (6 * o)) = 21
    This simplifies to:
    92.14285714285714 - 6o = 21
    Now, we can solve for o:
    6o = 92.14285714285714 - 21
    6o = 71.14285714285714
    o = 71.14285714285714 / 6
    o = 11.857142857142858
    ### Answer: v = 7.678571428571429, o = 11.857142857142858

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.
    First, we can multiply the first equation by 2 and the second equation by 75 to make the coefficients of f the same:
    150 * f + 50 * g = 276
    150 * f + 6750 * g = 10725
    Next, we can subtract the first equation from the second equation to eliminate f:
    6700 * g = 10449
    Now, we can solve for g:
    g = 10449 / 6700 = <<10449/6700=1.56>>1.56
    Finally, we can substitute the value of g back into one of the original equations to solve for f:
    75 * f + 25 * 1.56 = 138
    75 * f + 39 = 138
    75 * f = 99
    f = 99 / 75 = <<99/75=1.32>>1.32
    ### 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:

     Reasoning: First, we can solve the first equation for y: y = (194 - 14 * n) / 4.
    Substituting this expression for y into the second equation, we get: 7 = 65 * n - 79 * ((194 - 14 * n) / 4).
    Simplifying this equation, we get: 7 = 65 * n - 194 * 19.75 + 14 * n * 19.75.
    Combining like terms, we get: 7 = 79 * n - 3825.5.
    Solving for n, we get: n = (7 + 3825.5) / 79 = <<(7+3825.5)/79=48.5>>48.5.
    Substituting this value of n back into the expression for y, we get: y = (194 - 14 * 48.5) / 4 = <<(194-14*48.5)/4=1.5>>1.5.
    ### Answer: y = 1.5, n = 48.5

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:

     Reasoning: Let's solve the system of equations using substitution or elimination method. We can use the elimination method by multiplying the first equation by 41 and the second equation by 26 to eliminate l.
    First equation multiplied by 41: 6026 = 1148 * g + 1066 * l
    Second equation multiplied by 26: 156 = 1066 * l - 1378 * g
    Subtracting the second equation from the first equation: 5870 = 2526 * g
    Solving for g: g = 5870 / 2526 = <<5870/2526=2.32>>2.32
    Substituting g = 2.32 into the first equation: 146 = 28 * 2.32 + 26 * l
    Solving for l: l = (146 - 28 * 2.32) / 26 = <<(146 - 28 * 2.32) / 26=3.2>>3.2
    ### Answer: l = 3.2, g = 2.32

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:

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

    First, we can multiply equation 1 by 7 and equation 2 by 56 to make the coefficients of y the same:

    1) 140 * p + 392 * y = 630
    2) 4984 * p + 392 * y = 5488

    Now, we can subtract equation 1 from equation 2 to eliminate y:

    4984 * p - 140 * p = 5488 - 630
    4844 * p = 4858
    p = 4858 / 4844
    p = 1.0029

    Now that we have the value of p, we can substitute it back into either equation 1 or 2 to find the value of y. Let's use equation 1:

    20 * 1.0029 + 56 * y = 90
    20.058 + 56 * y = 90
    56 * y = 90 - 20.058
    56 * y = 69.942
    y = 69.942 / 56
    y = 1.249

    ### Answer: p = 1.0029, y = 1.249

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:

     Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method.
    First, we can multiply the first equation by 3 and the second equation by 2 to make the coefficients of d equal:
    216*k - 66*d = 51
    12*d +138*k = 192
    Next, we can add the two equations to eliminate d:
    216*k - 66*d + 12*d +138*k = 51 + 192
    354*k = 243
    k = 243/354 = <<243/354=0.6864406779661017>>0.6864406779661017
    Now, we can substitute the value of k back into one of the original equations to find the value of d:
    72*0.6864406779661017 - 22*d = 17
    49.31355932203389 - 22*d = 17
    22*d = 49.31355932203389 - 17
    22*d = 32.31355932203389
    d = 32.31355932203389/22 = <<32.31355932203389/22=1.4688072419106314>>1.4688072419106314
    ### Answer: k = 0.6864406779661017, d = 1.4688072419106314

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:

     Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method.

    First, we can multiply the second equation by 33/10 to make the coefficients of d in both equations equal:

    (33/10) * (10 * d + 14 * h) = (33/10) * 91

    33 * d + 462 * h = 2997

    Now, we can subtract the first equation from this new equation to eliminate d:

    (33 * d + 462 * h) - (73 * h - 33 * d) = 2997 - 48

    33 * d + 462 * h - 73 * h + 33 * d = 2949

    66 * d + 389 * h = 2949

    Next, we can solve for d in terms of h:

    66 * d = 2949 - 389 * h

    d = (2949 - 389 * h) / 66

    Now, we can substitute this expression for d into the second equation:

    10 * ((2949 - 389 * h) / 66) + 14 * h = 91

    (29490 - 3890 * h) / 66 + 14 * h = 91

    29490 - 3890 * h + 924 * h = 6006

    -2966 * h = -23484

    h = 23484 / 2966

    h = 8

    Now that we have the value of h, we can substitute it back into the expression for d:

    d = (2949 - 389 * 8) / 66

    d = (2949 - 3112) / 66

    d = -163 / 66

    d = -2.46969696969697

    ### Answer: h = 8, d = -2.46969696969697

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we will multiply the first equation by 43 and the second equation by 44 to make the coefficients of e in both equations equal:

    1) 43 * (43 * q - 44 * e) = 43 * 29
    2) 44 * (30 * q + 43 * e) = 44 * 130

    Simplifying the equations, we get:

    1) 1849 * q - 1892 * e = 1247
    2) 1320 * q + 1892 * e = 5720

    Now, we can add the two equations to eliminate the e term:

    1849 * q - 1892 * e + 1320 * q + 1892 * e = 1247 + 5720

    Simplifying the equation, we get:

    3169 * q = 6967

    Dividing both sides by 3169, we get:

    q = 6967 / 3169
    q = 2.2

    Now, we can substitute the value of q in any of the original equations to find the value of e. Let's substitute it in the first equation:

    43 * 2.2 - 44 * e = 29

    Simplifying the equation, we get:

    94.6 - 44 * e = 29

    Subtracting 94.6 from both sides, we get:

    -44 * e = -65.6

    Dividing both sides by -44, we get:

    e = 65.6 / 44
    e = 1.5

    Therefore, the values of q and e are 2.2 and 1.5, respectively.

    ### Answer: q = 2.2, e = 1.5

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the second equation by 89 and the first equation by 37 to make the coefficients of r in both equations equal:

    (89 * 109) = (89 * ((37 * a) + (6 * r)))
    (37 * 53) = (37 * ((89 * r) - (74 * a)))

    Simplifying these equations, we get:

    9681 = (3293 * a) + (534 * r)
    1961 = (3293 * r) - (2738 * a)

    Next, we can subtract the second equation from the first equation to eliminate the variable a:

    (9681 - 1961) = ((3293 * a) + (534 * r)) - ((3293 * r) - (2738 * a))
    7720 = (534 * r) + (2738 * a) - (3293 * r) + (2738 * a)
    7720 = (534 * r) - (3293 * r) + (2738 * a) + (2738 * a)
    7720 = (-2759 * r) + (5476 * a)

    Now, we can solve for r by dividing both sides of the equation by -2759:

    r = (7720 / -2759) + (5476 * a / -2759)
    r = -2.80 + (-2 * a)

    Substituting this expression for r into one of the original equations, we can solve for a:

    53 = ((89 * (-2.80 + (-2 * a))) - (74 * a))
    53 = (-249.20 + (-178 * a)) - (74 * a)
    53 = -249.20 - (252 * a)

    Now, we can solve for a by dividing both sides of the equation by -252:

    a = (53 + 249.20) / -252
    a = 302.20 / -252
    a = -1.20

    Finally, we can substitute the value of a back into the expression for r to find the value of r:

    r = -2.80 + (-2 * (-1.20))
    r = -2.80 + 2.40
    r = -0.40

    Therefore, the values of r and a in the system of equations are r = -0.40 and a = -1.20.
    ### Answer: r = -0.40, a = -1.20

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:

     Reasoning: Let's solve the system of equations step by step.

    From the first equation:
    42g + 12q = 169

    From the second equation:
    12q - 50g = 19

    We can solve this system using substitution or elimination. Let's use elimination for simplicity. Multiply the first equation by 50 and the second equation by 42 to align the coefficients of 'g':

    50(42g + 12q) = 50(169)
    42(12q - 50g) = 42(19)

    This gives us:
    2100g + 600q = 8450
    504q - 2100g = 798

    Adding these two equations to eliminate 'g':
    600q + 504q = 8450 + 798
    1104q = 9248
    q = 9248 / 1104
    q = 8.375

    Substitute q back into one of the original equations to solve for 'g':
    42g + 12(8.375) = 169
    42g + 100.5 = 169
    42g = 68.5
    g = 68.5 / 42
    g = 1.631

    ### Answer: g = 1.631 kg, q = 8.375 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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 2 to make the coefficients of v the same:

    74 * s - 22 * v = 108

    Now, we can subtract the second equation from this new equation:

    (74 * s - 22 * v) - (82 * s - 62 * v) = 108 - 41

    This simplifies to:

    -8 * s + 40 * v = 67

    Now, we can solve for s:

    s = (40 * v - 67) / 8

    Next, we can substitute this expression for s into the first equation:

    37 * ((40 * v - 67) / 8) - 11 * v = 54

    Simplifying this equation, we get:

    1480 * v - 2479 - 88 * v = 432

    1392 * v = 2911

    v = 2911 / 1392

    Now, we can substitute this value of v back into the expression for s:

    s = (40 * (2911 / 1392) - 67) / 8

    Simplifying this expression, we get:

    s = (116440 / 1392 - 67) / 8

    s = (83.6 - 67) / 8

    s = 16.6 / 8

    s = 2.075

    Therefore, the values of s and v are:

    s = 2.075

    v = 2911 / 1392

    ### Answer: s = 2.075, v = 2911 / 1392

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:

     Reasoning: To solve for c and z, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 48 and the second equation by 41 to make the coefficients of c in both equations equal:

    48 * (41 * c + 65 * z) = 48 * 151
    41 * (78 * z - 48 * c) = 41 * 48

    This gives us:

    1968 * c + 3120 * z = 7248
    3198 * z - 1968 * c = 1968

    Now, we can add the two equations to eliminate c:

    (1968 * c + 3120 * z) + (3198 * z - 1968 * c) = 7248 + 1968
    3120 * z + 3198 * z = 9216
    6318 * z = 9216
    z = 9216 / 6318
    z = 1.458

    Now that we have the value of z, we can substitute it back into one of the original equations to solve for c. Let's use the first equation:

    41 * c + 65 * 1.458 = 151
    41 * c + 94.77 = 151
    41 * c = 151 - 94.77
    41 * c = 56.23
    c = 56.23 / 41
    c = 1.371

    ### Answer: c = 1.371, z = 1.458

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:

     Reasoning: Let's solve the system of equations using the substitution method.

    From the first equation, we can express c in terms of i:

    c = (7 + 91 * i) / 60

    Substitute this expression for c into the second equation:

    7 * i + 76 * ((7 + 91 * i) / 60) = 148

    Simplify and solve for i:

    i = 1

    Substitute i = 1 into the expression for c:

    c = (7 + 91 * 1) / 60

    c = 1.5

    ### Answer: c = 1.5, i = 1

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 41 and the second equation by 37 to make the coefficients of e in both equations equal:

    1) 2911*v - 1517*e = 2829
    2) 1517*e + 2812*v = 6808

    Now, we can add the two equations to eliminate e:

    2911*v + 2812*v = 2829 + 6808
    5723*v = 9637
    v = 9637/5723 = <<9637/5723=1.6837>>1.6837

    Now that we have the value of v, we can substitute it back into one of the original equations to find the value of e. Let's use the first equation:

    71*v - 37*e = 69
    71*1.6837 - 37*e = 69
    119.6427 - 37*e = 69
    37*e = 119.6427 - 69
    37*e = 50.6427
    e = 50.6427/37 = <<50.6427/37=1.3687>>1.3687

    ### Answer: e = 1.3687, v = 1.6837

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:

     Reasoning: Let's solve the system of equations:

    1) 25v + 69d = 113
    2) 26d + 42v = 85

    To eliminate 'v', we can multiply equation 1 by 42 and equation 2 by 25, then subtract the second equation from the first:

    42(25v + 69d) = 42(113)
    25(26d + 42v) = 25(85)

    1050v + 2898d = 4746
    650d + 1050v = 2125

    Subtracting the second equation from the first:

    2898d - 650d = 4746 - 2125
    2248d = 2621

    Now, divide both sides by 2248 to find 'd':

    d = 2621 / 2248
    d = 1.165

    Substitute 'd' back into equation 1 to find 'v':

    25v + 69(1.165) = 113
    25v + 80.385 = 113
    25v = 113 - 80.385
    25v = 32.615

    Divide both sides by 25 to find 'v':

    v = 32.615 / 25
    v = 1.3046

    ### Answer: d = 1.165, v = 1.3046

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:

     Reasoning: Multiply the first equation by 3 and the second equation by 43 to eliminate z:

    129 * z - 273 * g = 99
    3096 * g - 129 * z = 2924

    Add the two equations:

    2823 * g = 3023
    g = 3023 / 2823 = <<3023/2823=1.0708>>1.0708

    Substitute g back into the first equation:

    43 * z - 91 * 1.0708 = 33
    43 * z = 33 + 91 * 1.0708 = 129.8588
    z = 129.8588 / 43 = <<129.8588/43=3.0200>>3.0200

    ### Answer: g = 1.0708, z = 3.0200

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 21 and the second equation by 12 to make the coefficients of f the same:

    1050c + 252f = 1932
    432c - 252f = 408

    Next, we can add the two equations together to eliminate f:

    1482c = 2340

    Now, we can solve for c:

    c = 2340 / 1482 = 1.57894736842

    Since c must be a whole number, we can round it to the nearest whole number, which is 2.

    Now that we have the value of c, we can substitute it back into one of the original equations to solve for f. Let's use the first equation:

    50(2) + 12f = 92
    100 + 12f = 92
    12f = -8
    f = -8 / 12 = -0.66666666667

    Since f must also be a whole number, we can round it to the nearest whole number, which is -1.

    However, since the number of figs cannot be negative, we must have made a mistake in our calculations. Let's go back and check our work.

    Upon reviewing our work, we see that we made a mistake when we rounded c to the nearest whole number. Since c must be a whole number, we should have rounded it to the nearest whole number that makes sense in the context of the problem. In this case, the only whole number that makes sense for c is 1.

    Now that we have the correct value of c, we can substitute it back into one of the original equations to solve for f. Let's use the first equation:

    50(1) + 12f = 92
    50 + 12f = 92
    12f = 42
    f = 42 / 12 = 3.5

    Since f must also be a whole number, we can round it to the nearest whole number, which is 4.

    Therefore, the quantity of cherries (c) is 1 and the quantity of figs (f) is 4.
    ### Answer: c = 1, f = 4

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:

     Reasoning: Multiply equation 1 by 82 and equation 2 by 87 to eliminate u:

    1) 7134u - 6150d = 574
    2) 7395d - 7134u = 4698

    Add the two equations to eliminate u:

    7395d - 6150d = 4698 + 574
    1245d = 5272
    d = 5272/1245 = <<5272/1245=4.23>>4.23

    Substitute d = 4.23 into equation 1:

    87u - 75(4.23) = 7
    87u - 317.25 = 7
    87u = 324.25
    u = 324.25/87 = <<324.25/87=3.73>>3.73

    ### Answer: u = 3.73, d = 4.23

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:

     Reasoning: Let's solve the system of equations:
    75a - 4i = 42
    10i + 13a = 160
    Multiplying the first equation by 10 and the second by 4, we get:
    750a - 40i = 420
    40i + 52a = 640
    Adding these two equations, we eliminate i:
    802a = 1060
    a = 1060 / 802 = <<1060/802=1.3216957625937675>>1.3216957625937675
    Substituting a back into one of the original equations:
    10i + 13(1.3216957625937675) = 160
    10i + 17.182044913718977 = 160
    10i = 142.81795508628102
    i = 14.281795508628102
    ### Answer: a = 1.3216957625937675, i = 14.281795508628102

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the second equation by 2 to make the coefficients of p in both equations equal:

    2 * (27 * c - 18 * p) = 2 * 13
    54 * c - 36 * p = 26

    Now, we can add the two equations together to eliminate p:

    (37 * p + 74 * c) + (54 * c - 36 * p) = 183 + 26
    37 * p - 36 * p + 74 * c + 54 * c = 209
    p + 128 * c = 209

    Next, we can solve for p:

    p = 209 - 128 * c

    Now, we can substitute this expression for p into one of the original equations to solve for c. Let's use the first equation:

    37 * (209 - 128 * c) + 74 * c = 183
    7733 - 4736 * c + 74 * c = 183
    -4662 * c = -7550
    c = 7550 / 4662
    c = 1.62

    Finally, we can substitute this value of c back into the expression for p to find its value:

    p = 209 - 128 * 1.62
    p = 209 - 207.36
    p = 1.64

    Therefore, the solution to the system of equations is p = 1.64 and c = 1.62.
    ### Answer: p = 1.64, c = 1.62

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:

     Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.
    First, we can multiply the first equation by 16 and the second equation by 30 to make the coefficients of p the same:
    (48 * k) * 16 - (30 * p) * 16 = 37 * 16
    (16 * p) * 30 + (33 * k) * 30 = 105 * 30
    This gives us:
    768 * k - 480 * p = 592
    480 * p + 990 * k = 3150
    Next, we can add the two equations together to eliminate p:
    (768 * k - 480 * p) + (480 * p + 990 * k) = 592 + 3150
    This simplifies to:
    1758 * k = 3742
    Dividing both sides by 1758 gives us:
    k = 3742 / 1758 = 2.128
    Now that we have the value of k, we can substitute it back into one of the original equations to solve for p. Using the first equation:
    (48 * 2.128) - (30 * p) = 37
    Simplifying this gives us:
    102.144 - 30 * p = 37
    Subtracting 102.144 from both sides gives us:
    -30 * p = -65.144
    Dividing both sides by -30 gives us:
    p = 65.144 / 30 = 2.171
    ### Answer: k = 2.128, p = 2.171

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we'll use elimination.

    First, we'll multiply the first equation by 46 and the second equation by 69 to make the coefficients of f in both equations equal:

    1. 3450v - 3174f = 644
    2. 3174f + 3657v = 7690

    Now, we'll subtract the first equation from the second equation to eliminate f:

    3657v - 3450v = 7690 - 644
    207v = 7046

    Now, we'll solve for v:

    v = 7046 / 207
    v = <<7046/207=34>>34

    Now that we have the value of v, we can substitute it back into one of the original equations to solve for f. Let's use the first equation:

    75(34) - 69f = 14
    2550 - 69f = 14
    -69f = 14 - 2550
    -69f = -2536

    Now, we'll solve for f:

    f = -2536 / -69
    f = <<-2536/-69=36.75>>36.75

    ### Answer: v = 34, f = 36.75

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:

     Reasoning: Multiply the first equation by 70 and the second equation by 13 to get:
    490 * o + 910 * g = 9170
    1157 * g - 910 * o = 481
    Add the two equations to get:
    2067 * g = 9651
    g = 9651/2067 = <<9651/2067=4.666666666666667>>4.666666666666667
    Substitute g = 4.666666666666667 into the first equation to get:
    7 * o + 13 * 4.666666666666667 = 131
    7 * o = 131 - 13 * 4.666666666666667 = 131 - 60.666666666666664 = 70.33333333333334
    o = 70.33333333333334/7 = <<70.33333333333334/7=10.047619047619048>>10.047619047619048
    ### Answer: o = 10.047619047619048, g = 4.666666666666667

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:

     Reasoning: To solve the system of equations, we can use the substitution or elimination method. Let's use the elimination method.
    First, we can multiply the second equation by 3 to make the coefficients of k the same:
    99 * k - 33 * d = 138
    Now, we can add the two equations to eliminate k:
    88 * d - 60 * k + 99 * k - 33 * d = 38 + 138
    55 * d + 39 * k = 176
    Next, we can solve for d:
    d = (176 - 39 * k) / 55
    Now, we can substitute this expression for d into one of the original equations to solve for k:
    88 * ((176 - 39 * k) / 55) - 60 * k = 38
    Simplifying this equation, we get:
    1568 - 3432 * k / 55 - 60 * k = 38
    Multiplying through by 55 to clear the fraction, we get:
    86240 - 3432 * k - 3300 * k = 2090
    Combining like terms, we get:
    -6732 * k = -84150
    Dividing both sides by -6732, we get:
    k = 12.5
    Now, we can substitute this value of k back into the expression for d to find its value:
    d = (176 - 39 * 12.5) / 55
    d = (176 - 487.5) / 55
    d = -211.5 / 55
    d = -3.84545454545
    ### Answer: d = -3.84545454545, k = 12.5

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.
    First, we can multiply the second equation by 12 to make the coefficients of z in both equations equal:
    24 * z + 72 * y = 139
    24 * z + 984 * y = 1152
    Next, we can subtract the first equation from the second equation to eliminate z:
    (24 * z + 984 * y) - (24 * z + 72 * y) = 1152 - 139
    912 * y = 1013
    Now, we can solve for y by dividing both sides of the equation by 912:
    y = 1013 / 912
    y = 1.110745862
    Next, we can substitute the value of y back into one of the original equations to solve for z. Let's use the first equation:
    24 * z + 72 * y = 139
    24 * z + 72 * 1.110745862 = 139
    24 * z + 79.97343222 = 139
    24 * z = 59.02656778
    z = 59.02656778 / 24
    z = 2.459440324
    ### Answer: y = 1.110745862, z = 2.459440324

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:

     Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method.
    First, we can multiply the second equation by 29 to make the coefficients of v in both equations equal:
    (87 * o) - (29 * v) = 1
    (232 * v) + (870 * o) = 2639
    Next, we can add the two equations to eliminate v:
    (87 * o) - (29 * v) + (232 * v) + (870 * o) = 1 + 2639
    (957 * o) + (203 * v) = 2640
    Now, we can solve for o:
    o = (2640 - 203 * v) / 957
    Substituting this expression for o into the first equation, we can solve for v:
    (87 * ((2640 - 203 * v) / 957)) - (29 * v) = 1
    Simplifying this equation, we get:
    (230880 - 17701 * v) / 957 - 29 * v = 1
    Multiplying both sides by 957, we get:
    230880 - 17701 * v - 27753 * v = 957
    Combining like terms, we get:
    -45454 * v = -229923
    Dividing both sides by -45454, we get:
    v = 5
    Substituting this value of v back into the expression for o, we get:
    o = (2640 - 203 * 5) / 957
    o = 2
    ### Answer: v = 5, o = 2

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:

     Reasoning: Let's solve the system of equations step by step. We have:
    (23 * h) - (46 * p) = 58
    (11 * p) + (23 * h) = 124

    First, we can multiply the second equation by 2 to make the coefficients of h the same in both equations:
    (23 * h) - (46 * p) = 58
    (22 * p) + (46 * h) = 248

    Now, we can subtract the first equation from the second equation to eliminate h:
    (22 * p) + (46 * h) - ((23 * h) - (46 * p)) = 248 - 58
    (22 * p) + (46 * h) - (23 * h) + (46 * p) = 190
    (68 * p) + (23 * h) = 190

    Now, we can substitute the value of (23 * h) from the first equation into the new equation:
    (68 * p) + 58 = 190
    (68 * p) = 132
    p = 132 / 68
    p = 1.9411764705882353

    Now, we can substitute the value of p back into one of the original equations to solve for h. Let's use the second equation:
    (11 * p) + (23 * h) = 124
    (11 * 1.9411764705882353) + (23 * h) = 124
    21.352941176470588 + (23 * h) = 124
    (23 * h) = 102.64705882352941
    h = 102.64705882352941 / 23
    h = 4.463

    ### Answer: h = 4.463, p = 1.941

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:

     Reasoning: Multiply the first equation by 7 to get 133d + 42f = 840. Add this to the second equation to eliminate f: 133d + 42f + 90d - 42f = 840 + 69, which simplifies to 223d = 909. Solving for d gives d = 909/223 = <<909/223=4>>4. Substitute d = 4 into the first equation to find f: 19(4) + 6f = 120, which simplifies to 76 + 6f = 120. Solving for f gives f = (120 - 76)/6 = <<(120-76)/6=7>>7.
    ### Answer: d = 4, f = 7

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:

     Reasoning: To solve this system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we will multiply the first equation by 52 and the second equation by 55 to make the coefficients of 'r' in both equations equal:

    1) 42 * m + 55 * r = 127 => 2184 * m + 2860 * r = 6604
    2) 66 * m - 52 * r = 44 => 3630 * m - 2860 * r = 2420

    Now, we will add the two equations to eliminate 'r':

    2184 * m + 2860 * r + 3630 * m - 2860 * r = 6604 + 2420
    5814 * m = 9024
    m = 9024 / 5814
    m = 1.55

    Now, we will substitute the value of 'm' in the first equation to find the value of 'r':

    42 * 1.55 + 55 * r = 127
    65.1 + 55 * r = 127
    55 * r = 127 - 65.1
    55 * r = 61.9
    r = 61.9 / 55
    r = 1.125

    ### Answer: m = 1.55, r = 1.125

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:

     Reasoning: To solve this system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 71 and the second equation by 31 to make the coefficients of n the same:

    71(31n + 64w) = 71(120)

    31(57w + 71n) = 31(173)

    Simplifying these equations, we get:

    2201n + 4544w = 8520

    1767w + 2201n = 5363

    Next, we can subtract the second equation from the first equation to eliminate n:

    (2201n + 4544w) - (1767w + 2201n) = 8520 - 5363

    Simplifying this equation, we get:

    2777w = 3157

    Dividing both sides by 2777, we get:

    w = 1.136

    Now that we have the value of w, we can substitute it back into one of the original equations to solve for n. Let's use the first equation:

    31n + 64(1.136) = 120

    Simplifying this equation, we get:

    31n + 72.96 = 120

    Subtracting 72.96 from both sides, we get:

    31n = 47.04

    Dividing both sides by 31, we get:

    n = 1.517

    Therefore, the values of n and w that satisfy the system of linear equations simultaneously are n = 1.517 and w = 1.136.
    ### Answer: n = 1.517, w = 1.136

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:

     Reasoning: To solve this system of linear equations, we can use the substitution or elimination method. Here, we will use the elimination method.

    First, we can multiply equation (1) by 76 and equation (2) by 26 to make the coefficients of u in both equations equal:

    76 * (72 * a + 26 * u) = 76 * 112
    26 * (76 * u - 77 * a) = 26 * 31

    This gives us:

    5472 * a + 1976 * u = 8432
    1976 * u - 2002 * a = 806

    Now, we can subtract the second equation from the first to eliminate u:

    (5472 * a + 1976 * u) - (1976 * u - 2002 * a) = 8432 - 806
    5472 * a + 1976 * u - 1976 * u + 2002 * a = 7626
    7474 * a = 7626

    Now, we can solve for a:

    a = 7626 / 7474
    a = 1.02

    Now that we have the value of a, we can substitute it back into either equation (1) or (2) to solve for u. Let's use equation (1):

    72 * 1.02 + 26 * u = 112
    73.44 + 26 * u = 112
    26 * u = 112 - 73.44
    26 * u = 38.56
    u = 38.56 / 26
    u = 1.48

    Therefore, the values of u and a are:

    u = 1.48
    a = 1.02
    ### Answer: u = 1.48, a = 1.02

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:

     Reasoning: Multiply the first equation by 3 to get 162n + 54b = 429. Multiply the second equation by 2 to get 174n - 186b = 96. Add the two equations to eliminate b: 336n = 525. Divide both sides by 336 to get n = 525/336 = 1.5625. Substitute n = 1.5625 into the first equation to get 54(1.5625) + 18b = 143. Simplify to get 84.375 + 18b = 143. Subtract 84.375 from both sides to get 18b = 58.625. Divide both sides by 18 to get b = 58.625/18 = 3.2569444444444446. Round b to the nearest hundredth to get b = 3.26.
    ### Answer: n = 1.5625, b = 3.26

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we'll use elimination to make the coefficients of p the same in both equations. Multiply the first equation by 88 and the second equation by 7 to align the coefficients of p:

    (53 * e + 7 * p) * 88 = 103 * 88
    (69 * e - 88 * p) * 7 = 6 * 7

    This gives us:
    4664 * e + 616 * p = 9064
    483 * e - 616 * p = 42

    Adding these two equations to eliminate p, we get:
    5147 * e = 9106

    Solving for e, we find:
    e = 9106 / 5147 = <<9106/5147=1.766>>1.766 (approximately)

    Substituting e back into one of the original equations to solve for p, we use the first equation:
    53 * 1.766 + 7 * p = 103
    93.598 + 7 * p = 103
    7 * p = 103 - 93.598
    7 * p = 9.402
    p = 9.402 / 7 = <<9.402/7=1.343>>1.343 (approximately)

    ### Answer: e ≈ 1.766, p ≈ 1.343

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:

     Reasoning: To solve this system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 24 and the second equation by 18 to make the coefficients of p in both equations equal:

    First equation: 24 * ((18 * p) - (7 * s)) = 24 * 21
    Second equation: 18 * ((51 * s) - (24 * p)) = 18 * 14

    Simplifying these equations, we get:

    First equation: 432 * p - 168 * s = 504
    Second equation: 918 * s - 432 * p = 252

    Now, we can add the two equations to eliminate the variable p:

    (432 * p - 168 * s) + (-432 * p + 918 * s) = 504 + 252

    Simplifying this equation, we get:

    750 * s = 756

    Dividing both sides by 750, we get:

    s = 756 / 750 = 1.008

    Now that we have the value of s, we can substitute it back into one of the original equations to solve for p. Let's use the first equation:

    (18 * p) - (7 * 1.008) = 21

    Simplifying this equation, we get:

    18 * p - 7.056 = 21

    Adding 7.056 to both sides, we get:

    18 * p = 28.056

    Dividing both sides by 18, we get:

    p = 28.056 / 18 = 1.5586666666666667

    Therefore, the values of p and s that satisfy both equations are:

    p = 1.5586666666666667
    s = 1.008
    ### Answer: p = 1.5586666666666667, s = 1.008

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:

     Reasoning: Let's solve the system of equations:
    59c - 34w = 84
    11c + 42w = 98
    Multiplying the second equation by 34/42 to eliminate w:
    (11c + 42w) * (34/42) = 98 * (34/42)
    (11c * 34/42) + (42w * 34/42) = 98 * 34/42
    (11c * 34/42) + 34w = 98 * 34/42
    Now, subtract the first equation from this new equation:
    (11c * 34/42) + 34w - (59c - 34w) = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c * 34/42) - 59c = 98 * 34/42 - 84
    (11c

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:

     Reasoning: Multiply the first equation by 42 and the second equation by 45 to eliminate f:
    42 * (45 * f + 42 * z) = 42 * 101,
    45 * (60 * z - 42 * f) = 45 * 13.

    Simplify the equations:
    1890 * f + 1764 * z = 4242,
    2700 * z - 1890 * f = 585.

    Add the two equations to eliminate f:
    1764 * z + 2700 * z = 4242 + 585,
    4464 * z = 4827.

    Divide both sides by 4464 to solve for z:
    z = 4827 / 4464,
    z = 1.08.

    Substitute z = 1.08 into the first equation to solve for f:
    45 * f + 42 * 1.08 = 101,
    45 * f + 45.36 = 101.

    Subtract 45.36 from both sides:
    45 * f = 101 - 45.36,
    45 * f = 55.64.

    Divide both sides by 45 to solve for f:
    f = 55.64 / 45,
    f = 1.24.

    ### Answer: f = 1.24, z = 1.08

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:

     Reasoning: Multiply the first equation by 15 and the second equation by 26 to eliminate p:
    15 * (26 * p - 21 * u) = 15 * 6
    26 * (15 * p + 58 * u) = 26 * 83
    Simplify the equations:
    390 * p - 315 * u = 90
    390 * p + 1508 * u = 2158
    Subtract the first equation from the second equation to eliminate p:
    (390 * p + 1508 * u) - (390 * p - 315 * u) = 2158 - 90
    Simplify the equation:
    1823 * u = 2068
    Divide both sides by 1823 to solve for u:
    u = 2068 / 1823
    u = 1.134
    Substitute the value of u into one of the original equations to solve for p:
    26 * p - 21 * 1.134 = 6
    Simplify the equation:
    26 * p - 23.814 = 6
    Add 23.814 to both sides:
    26 * p = 29.814
    Divide both sides by 26 to solve for p:
    p = 29.814 / 26
    p = 1.147
    ### 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:

     Reasoning: To solve for n and s, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the second equation by 38/8 to make the coefficient of s in both equations the same:

    (59 * n + 8 * s) * (38/8) = 75 * (38/8)
    (59 * n * 38/8) + (8 * s * 38/8) = 75 * (38/8)
    (59 * n * 38/8) + 38 * s = 75 * (38/8)

    Now, we can subtract the first equation from this new equation to eliminate s:

    (59 * n * 38/8) + 38 * s - (92 * n - 38 * s) = 75 * (38/8) - 62
    (59 * n * 38/8) - 92 * n = 75 * (38/8) - 62
    (59 * n * 38/8) - 92 * n = 3450/8 - 62
    (59 * n * 38/8) - 92 * n = 3450/8 - 496/8
    (59 * n * 38/8) - 92 * n = 2954/8
    (59 * n * 38/8) - 92 * n = 369.25

    Now, we can solve for n:

    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) = 92 * n + 369.25
    (59 * n * 38/8) = 92 * n + 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n * 38/8) - 92 * n = 369.25
    (59 * n *

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we'll use elimination. First, we'll multiply the first equation by 58 and the second equation by 35 to make the coefficients of e the same:
    58 * (87 * z - 35 * e) = 58 * 63
    35 * (81 * z + 58 * e) = 35 * 154

    This gives us:
    5046 * z - 1990 * e = 3654
    2835 * z + 2030 * e = 5390

    Now, we'll add the two equations to eliminate e:
    (5046 * z - 1990 * e) + (2835 * z + 2030 * e) = 3654 + 5390
    7881 * z = 9044

    Now, we'll solve for z:
    z = 9044 / 7881
    z = <<9044/7881=1.1475>>1.1475

    Next, we'll substitute the value of z back into one of the original equations to solve for e. Using the first equation:
    87 * 1.1475 - 35 * e = 63
    99.9375 - 35 * e = 63

    Now, we'll solve for e:
    35 * e = 99.9375 - 63
    35 * e = 36.9375
    e = 36.9375 / 35
    e = <<36.9375/35=1.0553>>1.0553

    ### Answer: z = 1.1475, e = 1.0553

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 36 and the second equation by 44 to make the coefficients of h the same:

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

    This simplifies to:

    396 * t + 1584 * h = 3960
    4312 * t - 1584 * h = 792

    Next, we can add the two equations together to eliminate h:

    (396 * t) + (4312 * t) = 3960 + 792
    4708 * t = 4752

    Now, we can solve for t:

    t = 4752 / 4708
    t = 1

    Now that we have the value of t, we can substitute it back into one of the original equations to solve for h. Let's use the first equation:

    (11 * 1) + (44 * h) = 110
    11 + 44 * h = 110
    44 * h = 99
    h = 99 / 44
    h = 9 / 4

    ### Answer: t = 1, h = 9/4

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:

     Reasoning: ((69 * b) - (50 * o)) = 55
    ((41 * o) + (18 * b)) = 85
    Solving the system of linear equations, we get:
    b = 1
    o = 2
    ### Answer: b = 1, o = 2

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.
    First, we can multiply the second equation by 16 to make the coefficients of q in both equations equal:
    67 * q - 80 * p = 65,
     and 96 * q + 80 * p = 2832.
    Next, we can add the two equations to eliminate p:
    163 * q = 2897.
    Dividing both sides by 163, we get:
    q = 17.77.
    Substituting this value of q into the second equation, we get:
    6 * 17.77 + 5 * p = 177.
    Simplifying, we get:
    106.62 + 5 * p = 177.
    Subtracting 106.62 from both sides, we get:
    5 * p = 70.38.
    Dividing both sides by 5, we get:
    p = 14.076.
    Therefore, the values of p and q are p = 14.076 and q = 17.77.
    ### Answer: p = 14.076, q = 17.77

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:

     Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method.
    First, we can multiply the first equation by 41 and the second equation by 48 to make the coefficients of f the same:
    410 * k + 1968 * f = 6581
    1584 * k - 1968 * f = 1488
    Now, we can add the two equations to eliminate f:
    2094 * k = 8069
    k = 8069 / 2094 = <<8069/2094=3.85>>3.85
    Now that we have the value of k, we can substitute it back into one of the original equations to find the value of f:
    10 * 3.85 + 48 * f = 161
    38.5 + 48 * f = 161
    48 * f = 122.5
    f = 122.5 / 48 = <<122.5/48=2.55>>2.55
    ### Answer: k = 3.85, f = 2.55

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:

     Reasoning:
    First, we can solve the first equation for n: n = 58 * l - 31
    Then, we can substitute this expression for n into the second equation: 103 = (58 * l - 31) + 73 * l
    Simplifying the equation, we get: 103 = 131 * l - 31
    Adding 31 to both sides, we get: 134 = 131 * l
    Dividing both sides by 131, we get: l = 134 / 131 = <<134/131=1.023>>1.023
    Substituting this value of l back into the expression for n, we get: n = 58 * 1.023 - 31 = <<58*1.023-31=28.934>>28.934
    ### Answer: l = 1.023, n = 28.934

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 37 and the second equation by 28 to make the coefficients of j equal:

    37(29y - 28j) = 37(39)
    28(25y + 37j) = 28(116)

    This gives us:

    1073y - 1036j = 1443
    700y + 1036j = 3248

    Now, we can add the two equations to eliminate j:

    1073y + 700y = 1443 + 3248
    1773y = 4691

    Dividing both sides by 1773, we get:

    y = 4691 / 1773
    y = 2.64

    Now that we have the value of y, we can substitute it back into one of the original equations to find the value of j. Let's use the first equation:

    29(2.64) - 28j = 39
    76.56 - 28j = 39

    Subtracting 76.56 from both sides, we get:

    -28j = 39 - 76.56
    -28j = -37.56

    Dividing both sides by -28, we get:

    j = -37.56 / -28
    j = 1.34

    Therefore, the values of y and j that satisfy the system of equations are y = 2.64 and j = 1.34.

    ### Answer: y = 2.64, j = 1.34

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 17 and the second equation by 85 to make the coefficients of a in both equations equal:

    17*(93u - 85a) = 17*74
    85*(96u + 17a) = 85*186

    This gives us:

    1581u - 1445a = 1258
    8160u + 1445a = 15810

    Adding these two equations, we get:

    9741u = 17068

    Dividing both sides by 9741, we get:

    u = 17068/9741 = 1.75

    Substituting this value of u into the first equation, we get:

    93*1.75 - 85a = 74

    Solving for a, we get:

    a = (93*1.75 - 74)/85 = 1.00

    ### Answer: u = 1.75, a = 1.00

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:

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

    First, we can multiply the first equation by 23 and the second equation by 61 to make the coefficients of g the same:

    23 * (97 * u - 61 * g) = 23 * 15
    61 * (85 * u + 23 * g) = 61 * 186

    This gives us:

    2231 * u - 1403 * g = 345
    5185 * u + 1403 * g = 11346

    Now, we can add these two equations to eliminate g:

    2231 * u + 5185 * u = 345 + 11346
    7416 * u = 11691

    Now, we can solve for u:

    u = 11691 / 7416
    u = 1.576

    Now that we have the value of u, we can substitute it back into one of the original equations to find the value of g. Let's use the first equation:

    97 * 1.576 - 61 * g = 15
    152.872 - 61 * g = 15
    -61 * g = 15 - 152.872
    -61 * g = -137.872

    Now, we can solve for g:

    g = -137.872 / -61
    g = 2.257

    ### Answer: u = 1.576, g = 2.257

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:

     Reasoning: Subtracting the first equation from the second equation, we get (85 * r) - (17 * r) = 187 - 103, which simplifies to 68 * r = 84. Dividing both sides by 68, we get r = 84/68 = <<84/68=1.2352941176470588>>1.2352941176470588. Substituting r = 1.2352941176470588 into the first equation, we get (17 * 1.2352941176470588) + (67 * b) = 103, which simplifies to 21 + 67 * b = 103. Subtracting 21 from both sides, we get 67 * b = 82. Dividing both sides by 67, we get b = 82/67 = <<82/67=1.2238805970149254>>1.2238805970149254.
    ### Answer: r = 1.2352941176470588, b = 1.2238805970149254

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:

     Reasoning: To solve this system of equations, we can use the substitution or elimination method. Here, we'll use the elimination method.

    First, we'll multiply the first equation by 30 and the second equation by 54 to make the coefficients of n equal:

    1. (30 * 85 * n) - (30 * 54 * t) = 30 * 41
    2. (54 * 38 * t) + (54 * 30 * n) = 54 * 126

    This simplifies to:

    1. 2550n - 1620t = 1230
    2. 2052t + 1620n = 6744

    Next, we'll subtract the first equation from the second to eliminate n:

    2052t + 1620n - (2550n - 1620t) = 6744 - 1230

    This simplifies to:

    3672t = 5514

    Now, we can solve for t:

    t = 5514 / 3672
    t = 1.5

    Substitute t = 1.5 into the first original equation to solve for n:

    (85 * n) - (54 * 1.5) = 41
    85n - 81 = 41
    85n = 122
    n = 122 / 85
    n = 1.435

    ### Answer: t = 1.5, n = 1.435

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:

     Reasoning: Let's solve the system of equations step by step.

    1) From the first equation, we have:
    75u - 69q = 5

    2) From the second equation, we have:
    48u + 25q = 140

    3) To eliminate q, we can multiply the first equation by 25 and the second equation by 69:
    (75u - 69q) * 25 = 5 * 25
    (48u + 25q) * 69 = 140 * 69

    4) This gives us:
    1875u - 1725q = 125
    3312u + 1725q = 9660

    5) Adding these two equations, we get:
    5187u = 9785

    6) Solving for u, we have:
    u = 9785 / 5187 = 1.886

    7) Substituting u = 1.886 into the first equation, we get:
    75 * 1.886 - 69q = 5

    8) Solving for q, we have:
    69q = 75 * 1.886 - 5
    q = (75 * 1.886 - 5) / 69 = 2.000

    ### Answer: u = 1.886, q = 2.000

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:

     Reasoning: ((20 * v) + (17 * x)) = 149 and ((24 * x) - (49 * v)) = 22.
    Solving the system of equations, we get v = 1 and x = 7.
    ### Answer: v = 1, x = 7

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 91 and the second equation by 41 to make the coefficients of g in both equations equal:

    39 * t * 91 + 41 * g * 91 = 147 * 91
    94 * t * 41 - 91 * g * 41 = 42 * 41

    This simplifies to:

    3549 * t + 3731 * g = 13377
    3854 * t - 3731 * g = 1722

    Now, we can add the two equations to eliminate g:

    3549 * t + 3731 * g + 3854 * t - 3731 * g = 13377 + 1722
    7403 * t = 15109

    Now, we can solve for t:

    t = 15109 / 7403
    t = <<15109/7403=2.04>>2.04

    Now that we have the value of t, we can substitute it back into one of the original equations to solve for g. Let's use the first equation:

    39 * t + 41 * g = 147
    39 * 2.04 + 41 * g = 147
    79.56 + 41 * g = 147

    Now, we can solve for g:

    41 * g = 147 - 79.56
    41 * g = 67.44
    g = 67.44 / 41
    g = <<67.44/41=1.64>>1.64

    ### Answer: t = 2.04, g = 1.64

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:

     Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Let's use elimination for this problem. First, we can multiply the first equation by 55 and the second equation by 6 to make the coefficients of c the same:
    55(6c + 55o) = 55(112)
    6(55c - 48o) = 6(36)
    This simplifies to:
    330c + 3025o = 6160
    330c - 288o = 216
    Now, we can subtract the second equation from the first to eliminate c:
    (330c + 3025o) - (330c - 288o) = 6160 - 216
    3025o + 288o = 5944
    3313o = 5944
    o = 5944 / 3313
    o = 1.794 (approximately)
    Now, we can substitute the value of o back into one of the original equations to solve for c. Let's use the first equation:
    6c + 55(1.794) = 112
    6c + 98.67 = 112
    6c = 112 - 98.67
    6c = 13.33
    c = 13.33 / 6
    c = 2.2217 (approximately)
    ### Answer: c ≈ 2.2217, o ≈ 1.794

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:

     Reasoning: Multiply the first equation by 94 and the second equation by 68 to eliminate u:
     752w + 6392u = 15980,
     4828w - 6392u = 536.
    Add the two equations to eliminate u:
     5580w = 16516.
    Divide both sides by 5580 to solve for w:
     w = 16516/5580 = <<16516/5580=3>>3.
    Substitute w = 3 into the first equation to solve for u:
     8(3) + 68u = 170,
     24 + 68u = 170,
     68u = 146,
     u = 146/68 = <<146/68=2>>2.
    ### Answer: u = 2, w = 3

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 38 and the second equation by 68 to make the coefficients of r the same:

    (68*r + 8*f) * 38 = 92 * 38
    (97*f + 38*r) * 68 = 182 * 68

    This gives us:

    2584*r + 304*f = 3496
    6616*f + 2584*r = 12376

    Next, we can subtract the first equation from the second equation to eliminate r:

    (6616*f + 2584*r) - (2584*r + 304*f) = 12376 - 3496

    This simplifies to:

    6312*f = 8880

    Now, we can solve for f:

    f = 8880 / 6312
    f = 1.4

    Now that we have the value of f, we can substitute it back into one of the original equations to solve for r. Let's use the first equation:

    68*r + 8*1.4 = 92

    This simplifies to:

    68*r + 11.2 = 92

    Now, we can solve for r:

    68*r = 92 - 11.2
    68*r = 80.8
    r = 80.8 / 68
    r = 1.19

    ### Answer: r = 1.19, f = 1.4

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:

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

    First, we can multiply the first equation by 39 and the second equation by 67 to make the coefficients of r in both equations equal:

    39(67d - 37r) = 39(50)
    67(39r - 21d) = 67(16)

    This gives us:

    2613d - 1443r = 1950
    2613r - 1407d = 1072

    Now, we can add these two equations to eliminate r:

    (2613d - 1443r) + (2613r - 1407d) = 1950 + 1072
    2613d - 1407d = 3022
    1206d = 3022
    d = 3022/1206
    d = 2.5

    Now that we have the value of d, we can substitute it back into one of the original equations to find the value of r. Let's use the first equation:

    67d - 37r = 50
    67(2.5) - 37r = 50
    167.5 - 37r = 50
    -37r = 50 - 167.5
    -37r = -117.5
    r = -117.5/-37
    r = 3.175675675675676

    ### Answer: d = 2.5, r = 3.175675675675676

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:

     Reasoning: To solve the system, we can use the substitution or elimination method. Here, we'll use the elimination method.

    First, we'll multiply the first equation by 84 to align the coefficients of a:
    84 * (163 = 50x + a) => 13752 = 4200x + 84a

    Now, we'll add this to the second equation to eliminate a:
    13752 + 24 = 4200x + 84a + 98x - 84a
    13776 = 4298x

    Solving for x:
    x = 13776 / 4298 = <<13776/4298=3.2>>3.2

    Substitute x back into the first equation to find a:
    163 = 50 * 3.2 + a
    163 = 160 + a
    a = 163 - 160 = 3

    ### Answer: x = 3.2, a = 3

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:

     Reasoning: Multiply the first equation by 44 and the second equation by 67 to eliminate p:
    44 * (67 * p + 18 * b) = 44 * 110
    67 * (82 * b - 44 * p) = 67 * 58
    Simplify the equations:
    2948 * p + 792 * b = 4840
    5494 * b - 2948 * p = 3816
    Add the two equations to eliminate p:
    (2948 * p + 792 * b) + (5494 * b - 2948 * p) = 4840 + 3816
    Simplify the equation:
    6286 * b = 8656
    Divide both sides by 6286 to solve for b:
    b = 8656 / 6286
    b = 1.375
    Substitute the value of b into one of the original equations to solve for p:
    67 * p + 18 * 1.375 = 110
    Simplify the equation:
    67 * p + 24.75 = 110
    Subtract 24.75 from both sides:
    67 * p = 85.25
    Divide both sides by 67 to solve for p:
    p = 85.25 / 67
    p = 1.2727272727272727
    ### Answer: p = 1.2727272727272727, b = 1.375

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:

     Reasoning:
    1. Set up the equations:
       \[
       \begin{align*}
       63c - 36e &= 27 \quad \text{(1)} \\
       8c + 32e &= 69 \quad \text{(2)}
       \end{align*}
       \]

    2. Simplify equation (2) by dividing all terms by 8:
       \[
       c + 4e = \frac{69}{8} \quad \text{(3)}
       \]

    3. Solve equation (3) for c:
       \[
       c = \frac{69}{8} - 4e \quad \text{(4)}
       \]

    4. Substitute equation (4) into equation (1):
       \[
       63\left(\frac{69}{8} - 4e\right) - 36e = 27
       \]

    5. Simplify and solve for e:
       \[
       \frac{4293}{8} - 252e - 36e = 27
       \]
       \[
       -288e = 27 - \frac{4293}{8}
       \]
       \[
       -288e = -\frac{3969}{8}
       \]
       \[
       e = \frac{3969}{8 \cdot 288}
       \]
       \[
       e = \frac{3969}{2304}
       \]
       \[
       e = \frac{1323}{768}
       \]
       \[
       e = \frac{441}{256}
       \]

    6. Substitute e back into equation (4) to find c:
       \[
       c = \frac{69}{8} - 4\left(\frac{441}{256}\right)
       \]
       \[
       c = \frac{69}{8} - \frac{1764}{256}
       \]
       \[
       c = \frac{552}{256} - \frac{1764}{256}
       \]
       \[
       c = \frac{-1212}{256}
       \]
       \[
       c = -\frac{303}{64}
       \]

    ### Answer: c = -303/64, e = 441/256

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:

     Reasoning: To solve the system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 51 and the second equation by 29 to make the coefficients of p the same:

    51 * (26 * x + 29 * p) = 51 * 107

    29 * (61 * x - 51 * p) = 29 * 4

    This gives us:

    1326 * x + 1519 * p = 5457

    1769 * x - 1479 * p = 116

    Next, we can add the two equations together to eliminate p:

    1326 * x + 1519 * p + 1769 * x - 1479 * p = 5457 + 116

    This simplifies to:

    3095 * x = 5573

    Dividing both sides by 3095 gives us:

    x = 5573 / 3095 = <<5573/3095=1.8>>1.8

    Now that we have the value of x, we can substitute it back into one of the original equations to solve for p. Using the first equation:

    26 * 1.8 + 29 * p = 107

    46.8 + 29 * p = 107

    Subtracting 46.8 from both sides gives us:

    29 * p = 60.2

    Dividing both sides by 29 gives us:

    p = 60.2 / 29 = <<60.2/29=2.1>>2.1

    ### Answer: x = 1.8, p = 2.1

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:

     Reasoning: Let's solve the system of equations using substitution or elimination method. We can use the elimination method by multiplying the first equation by 10 and the second equation by 29 to eliminate l.

    First equation multiplied by 10: 290l + 170g = 680
    Second equation multiplied by 29: 899g - 290l = 493

    Adding the two equations: 1069g = 1173
    g = 1173/1069 = <<1173/1069=1.1>>1.1

    Substituting g = 1.1 in the first equation: 29l + 17(1.1) = 68
    29l + 18.7 = 68
    29l = 49.3
    l = 49.3/29 = <<49.3/29=1.7>>1.7

    ### Answer: g = 1.1, l = 1.7

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:

     Reasoning: Multiply the first equation by 24 and the second equation by 33 to eliminate a:

    24*(33*a - 54*c) = 24*20
    33*(24*a - 26*c) = 33*29

    Simplify:

    792*a - 1296*c = 480
    792*a - 858*c = 957

    Subtract the second equation from the first:

    (792*a - 1296*c) - (792*a - 858*c) = 480 - 957
    -438*c = -477

    Divide by -438:

    c = 477/438 = 1.089

    Substitute c = 1.089 into the first equation:

    33*a - 54*1.089 = 20
    33*a - 58.806 = 20
    33*a = 78.806
    a = 78.806/33 = 2.388

    ### Answer: a = 2.388, c = 1.089

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:

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

    First, we can multiply the first equation by 84 and the second equation by 71 to make the coefficients of d the same:

    84(71d + 9j) = 84(132)
    71(43j - 84d) = 71(91)

    This gives us:

    5964d + 756j = 11088
    3053j - 5964d = 6461

    Now, we can add these two equations to eliminate d:

    5964d + 756j + 3053j - 5964d = 11088 + 6461

    This simplifies to:

    3809j = 17549

    Now, we can solve for j:

    j = 17549 / 3809
    j = 4.608

    Next, we can substitute the value of j back into one of the original equations to solve for d. Let's use the first equation:

    71d + 9(4.608) = 132

    This simplifies to:

    71d + 41.472 = 132

    Now, we can solve for d:

    71d = 132 - 41.472
    71d = 90.528

    d = 90.528 / 71
    d = 1.275

    Therefore, the solution to the system of equations is d = 1.275 and j = 4.608.

    ### Answer: d = 1.275, j = 4.608

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:

     Reasoning: Let's solve this system of equations using the substitution method. First, we can solve the first equation for o:

    o = (163 - 26n) / 81

    Next, we can substitute this expression for o into the second equation:

    87n + 23((163 - 26n) / 81) = 143

    Multiplying both sides by 81 to eliminate the fraction, we get:

    7047n + 3749 - 598n = 11583

    Combining like terms, we get:

    6449n = 7834

    Dividing both sides by 6449, we get:

    n = 7834 / 6449 = 1.214

    Substituting this value of n back into the expression for o, we get:

    o = (163 - 26(1.214)) / 81 = 1.506

    Therefore, the values of o and n are approximately 1.506 and 1.214, respectively.
    ### Answer: o = 1.506, n = 1.214

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:

     Reasoning: Multiply the first equation by 61 and the second equation by 46 to eliminate s:

    61 * (46 * s - 61 * w) = 61 * 2
    46 * (61 * s + 67 * w) = 46 * 178

    Simplify:

    2796 * s - 3721 * w = 122
    2806 * s + 3082 * w = 8188

    Subtract the first equation from the second equation to eliminate s:

    (2806 * s + 3082 * w) - (2796 * s - 3721 * w) = 8188 - 122

    Simplify:

    10 * s + 6803 * w = 8066

    Divide by 10:

    s + 680.3 * w = 806.6

    Substitute s = 806.6 - 680.3 * w into the first equation:

    46 * (806.6 - 680.3 * w) - 61 * w = 2

    Simplify:

    37097.6 - 31293.8 * w - 61 * w = 2

    Combine like terms:

    -31354.8 * w = -37095.6

    Divide by -31354.8:

    w = 1.183

    Substitute w = 1.183 into the equation s = 806.6 - 680.3 * w:

    s = 806.6 - 680.3 * 1.183

    Simplify:

    s = 806.6 - 806.5

    s = 0.1

    ### Answer: w = 1.183, s = 0.1

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:

     Reasoning: Let's solve the system of equations using the substitution method. From the first equation, we can express k in terms of i: k = (86 + 8*i) / 51. Substituting this expression into the second equation, we get: 39*i + 8*((86 + 8*i) / 51) = 193. Simplifying this equation, we get: 39*i + (688 + 64*i) / 51 = 193. Multiplying both sides by 51, we get: 1989*i + 688 + 64*i = 9843. Combining like terms, we get: 2053*i = 9155. Dividing both sides by 2053, we get: i = 4.46. Substituting this value of i back into the expression for k, we get: k = (86 + 8*4.46) / 51 = 2.04.
    ### Answer: i = 4.46, k = 2.04

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:

     Reasoning: Multiply the second equation by 17 to get 1581m + 17l = 2142. Subtract the first equation from this to eliminate l: 1581m + 17l - (26m + 17l) = 2142 - 146, which simplifies to 1555m = 1996. Dividing both sides by 1555 gives m = 1996/1555 = 1.2836. Substitute this value of m into the second equation to get 93(1.2836) + l = 126, which simplifies to 119.2368 + l = 126. Subtracting 119.2368 from both sides gives l = 126 - 119.2368 = 6.7632.
    ### Answer: l = 6.7632, m = 1.2836

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we'll use elimination.

    First, we'll multiply equation 1 by 29 and equation 2 by 8 to make the coefficients of 'f' the same:

    1) (29 * 8 * f) + (29 * 39 * h) = 29 * 66

    2) (8 * 29 * f) + (8 * 46 * h) = 8 * 119

    Simplifying, we get:

    1) 232f + 1131h = 1914

    2) 232f + 368h = 952

    Now, we'll subtract equation 2 from equation 1 to eliminate 'f':

    (232f + 1131h) - (232f + 368h) = 1914 - 952

    Simplifying, we get:

    763h = 962

    Dividing both sides by 763, we get:

    h = 962 / 763 = 1.26

    Now, we can substitute the value of 'h' back into one of the original equations to solve for 'f'. Let's use equation 1:

    (8 * f) + (39 * 1.26) = 66

    Simplifying, we get:

    8f + 49.14 = 66

    Subtracting 49.14 from both sides, we get:

    8f = 16.86

    Dividing both sides by 8, we get:

    f = 16.86 / 8 = 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:

     Reasoning: Let's solve the system of equations step by step. First, we can multiply the first equation by 3 and the second equation by 1 to make the coefficients of b in both equations equal. This gives us: 66 * b + 69 * k = 327 and 62 * k - 69 * b = 14. Now, we can add the two equations together to eliminate b: (66 * b + 69 * k) + (62 * k - 69 * b) = 327 + 14, which simplifies to 131 * k = 341. Dividing both sides by 131 gives us k = 341 / 131 = 2.6. Substituting k = 2.6 into the first equation gives us 22 * b + 23 * 2.6 = 109, which simplifies to 22 * b + 59.8 = 109. Subtracting 59.8 from both sides gives us 22 * b = 49.2, and dividing both sides by 22 gives us b = 49.2 / 22 = 2.2. Therefore, the values of k and b that satisfy the system of equations are k = 2.6 and b = 2.2.
    ### Answer: k = 2.6, b = 2.2

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

Correct? No

🔗 Problem

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:

     Reasoning: To solve the system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 2 and the second equation by 55 to make the coefficients of s the same:

    1. 168m - 110s = 2
    2. 3465m - 110s = 5390

    Next, we can subtract the first equation from the second equation to eliminate s:

    3465m - 168m = 5390 - 2
    3297m = 5388

    Now, we can solve for m:

    m = 5388 / 3297
    m = 1.63

    Next, we can substitute the value of m back into one of the original equations to solve for s. Let's use the first equation:

    84(1.63) - 55s = 1
    136.92 - 55s = 1
    -55s = 1 - 136.92
    -55s = -135.92

    Now, we can solve for s:

    s = -135.92 / -55
    s = 2.47

    Therefore, the solution to the system of linear equations is m = 1.63 and s = 2.47.

    ### Answer: m = 1.63, s = 2.47

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:

     Reasoning: To solve this system of equations, we can use the substitution or elimination method. Here, we'll use the elimination method.

    First, we can multiply the first equation by 51 and the second equation by 6 to make the coefficients of b the same:

    1. (51 * 6 * b) + (51 * 5 * k) = 51 * 105
    2. (6 * 95 * k) - (6 * 51 * b) = 6 * 67

    This simplifies to:

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

    Now, we can add these two equations to eliminate b:

    306b + 255k + 570k - 306b = 5355 + 402

    This simplifies to:

    825k = 5757

    Now, we can solve for k:

    k = 5757 / 825
    k = 7

    Next, we can substitute the value of k back into one of the original equations to solve for b. Let's use the first equation:

    (6 * b) + (5 * 7) = 105

    This simplifies to:

    6b + 35 = 105

    Now, we can solve for b:

    6b = 105 - 35
    6b = 70
    b = 70 / 6
    b = 11.666...

    Since b must be a whole number (as it represents the number of items), we can round b to the nearest whole number, which is 12.

    ### Answer: b = 12, k = 7

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:

     Reasoning:
    First, we can solve for d in terms of u using the first equation:
    33d = 114 - 62u
    d = (114 - 62u) / 33

    Next, we can substitute this expression for d into the second equation:
    32u + 40((114 - 62u) / 33) = 83

    Simplifying this equation, we get:
    32u + (40 * 114) / 33 - (40 * 62u) / 33 = 83
    32u + 1360 / 33 - 2480u / 33 = 83
    32u - 2480u / 33 = 83 - 1360 / 33
    (32u * 33) - 2480u = (83 * 33) - 1360
    1056u - 2480u = 2739 - 1360
    -1424u = 1379
    u = -1379 / 1424

    Now that we have the value of u, we can substitute it back into the expression for d:
    d = (114 - 62(-1379 / 1424)) / 33
    d = (114 + 85618 / 1424) / 33
    d = (114 * 1424 + 85618) / (33 * 1424)
    d = (162456 + 85618) / 46992
    d = 248074 / 46992
    d = 5.28

    Therefore, the values of d and u are d = 5.28 and u = -1379 / 1424.
    ### Answer: d = 5.28, u = -1379 / 1424

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:

     Reasoning: Let's solve this system of equations using the substitution method. First, we can solve the first equation for c:

    5 * c + 28 * p = 102
    5 * c = 102 - 28 * p
    c = (102 - 28 * p) / 5

    Now, we can substitute this expression for c into the second equation:

    20 * c - 67 * p = 13
    20 * ((102 - 28 * p) / 5) - 67 * p = 13
    4 * (102 - 28 * p) - 67 * p = 13
    408 - 112 * p - 67 * p = 13
    408 - 179 * p = 13
    -179 * p = 13 - 408
    -179 * p = -395
    p = -395 / -179
    p = 2.2

    Now that we have the value of p, we can substitute it back into the expression for c:

    c = (102 - 28 * p) / 5
    c = (102 - 28 * 2.2) / 5
    c = (102 - 61.6) / 5
    c = 40.4 / 5
    c = 8.08

    ### Answer: c = 8.08, p = 2.2

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:

     Reasoning: Let's solve the system of equations using the substitution method. First, we can solve the first equation for r:
    58 * r = 147 - 12 * q
    r = (147 - 12 * q) / 58
    Now, we can substitute this expression for r into the second equation:
    97 * q - 76 * ((147 - 12 * q) / 58) = 76
    Simplifying this equation, we get:
    97 * q - (147 * 76 / 58) + (12 * 76 * q / 58) = 76
    97 * q - 186 + 152 * q / 7 = 76
    Multiplying both sides by 7 to eliminate the fraction, we get:
    679 * q - 1302 + 152 * q = 532
    Combining like terms, we get:
    831 * q = 1834
    Dividing both sides by 831, we get:
    q = 1834 / 831
    q = 2.206
    Now, we can substitute this value of q back into the expression for r:
    r = (147 - 12 * 2.206) / 58
    r = (147 - 26.472) / 58
    r = 120.528 / 58
    r = 2.078
    ### Answer: r = 2.078, q = 2.206

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:

     Reasoning: Multiply the first equation by 2 and the second equation by 3 to eliminate k:
    94 * v - 6 * k = 20
    219 * v + 6 * k = 384
    Add the two equations to eliminate k:
    313 * v = 404
    v = 404 / 313 = <<404/313=1.2907348242805112>>1.2907348242805112
    Substitute v into the first equation to solve for k:
    47 * 1.2907348242805112 - 3 * k = 10
    60.66453674018392 - 3 * k = 10
    3 * k = 50.66453674018392
    k = 50.66453674018392 / 3 = <<50.66453674018392/3=16.88817891339464>>16.88817891339464
    ### Answer: v = 1.2907348242805112, k = 16.88817891339464

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:

     Reasoning: Let's solve the system of equations step by step.
    First, we can multiply the first equation by 27 and the second equation by 89 to eliminate p:
    27*(89p - 41q) = 27*60
    89*(27p + 91q) = 89*124
    This gives us:
    2393p - 1107q = 1620
    2403p + 8099q = 11036
    Subtracting the first equation from the second gives:
    10p + 9206q = 9416
    Solving for p gives:
    p = (9416 - 9206q)/10
    Substituting this into the first equation gives:
    89*((9416 - 9206q)/10) - 41q = 60
    Simplifying this gives:
    84744 - 81954q - 410q = 600
    Solving for q gives:
    q = (84744 - 600)/(81954 + 410) = 1
    Substituting q = 1 into the equation for p gives:
    p = (9416 - 9206*1)/10 = 21
    ### Answer: p = 21, q = 1

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:

     Reasoning: Let's solve the system of equations step by step.

    First, we can multiply the second equation by 3 to make the coefficients of d in both equations the same:

    (81 * u) + (108 * d) = 213

    Now, we can subtract the first equation from this new equation to eliminate d:

    (81 * u) + (108 * d) - ((88 * u) - (96 * d)) = 213 - 12

    Simplifying this, we get:

    (81 * u) + (108 * d) - (88 * u) + (96 * d) = 201

    Combining like terms, we get:

    (81 * u - 88 * u) + (108 * d + 96 * d) = 201

    -7 * u + 204 * d = 201

    Now, we can solve for u:

    u = (201 - 204 * d) / -7

    Substituting this expression for u into the first equation, we get:

    (88 * ((201 - 204 * d) / -7)) - (96 * d) = 12

    Simplifying this, we get:

    (88 * (201 - 204 * d)) / -7 - 96 * d = 12

    Multiplying both sides by -7 to get rid of the fraction, we get:

    88 * (201 - 204 * d) - 672 * d = -84

    Expanding this, we get:

    17728 - 17952 * d - 672 * d = -84

    Combining like terms, we get:

    17728 - 18624 * d = -84

    Now, we can solve for d:

    d = (17728 + 84) / 18624

    d = 17812 / 18624

    d = 0.956

    Substituting this value of d back into the expression for u, we get:

    u = (201 - 204 * 0.956) / -7

    u = (201 - 194.904) / -7

    u = 6.096 / -7

    u = -0.871

    ### Answer: u = -0.871, d = 0.956

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:

     Reasoning: Multiply the second equation by 9 to get 27 * n - 18 * f = 630.
    Subtract the first equation from this to get 7 * n = 621.
    Divide both sides by 7 to get n = 88.71428571428571.
    Substitute this value of n into the first equation to get 20 * 88.71428571428571 - 18 * f = 9.
    Solve for f to get f = 97.5.
    ### Answer: n = 88.71428571428571, f = 97.5

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:

     Reasoning: Multiply the first equation by 22 and the second equation by 97 to eliminate j:
    154 * l + 2134 * j = 2926
    2134 * j + 3599 * l = 7372
    Subtract the first equation from the second equation to eliminate j:
    3599 * l - 154 * l = 7372 - 2926
    3445 * l = 4446
    l = 4446 / 3445 = 1.29
    Substitute l = 1.29 into the first equation to solve for j:
    7 * 1.29 + 97 * j = 133
    9.03 + 97 * j = 133
    97 * j = 123.97
    j = 123.97 / 97 = 1.28
    ### Answer: j = 1.28, l = 1.29

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:

     Reasoning: Multiply the first equation by 21 and the second equation by 46 to eliminate f:

    441d - 966f = 1113
    368d + 3864f = 6118

    Add the two equations:

    809d = 7231
    d = 7231/809 = <<7231/809=9>>9

    Substitute d = 9 into the first equation:

    21(9) - 46f = 53
    189 - 46f = 53
    -46f = 53 - 189
    -46f = -136
    f = -136/-46 = <<-136/-46=3>>3

    ### Answer: d = 9, f = 3

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:

     Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.
    First, we can multiply the first equation by 68 and the second equation by 11 to make the coefficients of w equal:
    68(74u + 11w) = 68(116)
    11(66u - 68w) = 11(21)
    Simplifying these equations, we get:
    5032u + 748w = 7888
    726u - 748w = 231
    Next, we can add these two equations to eliminate w:
    (5032u + 748w) + (726u - 748w) = 7888 + 231
    5758u = 8119
    Dividing both sides by 5758, we get:
    u = 8119/5758
    Simplifying this fraction, we get:
    u = 1.41
    Now that we have the value of u, we can substitute it back into one of the original equations to solve for w. Let's use the first equation:
    74(1.41) + 11w = 116
    104.34 + 11w = 116
    Subtracting 104.34 from both sides, we get:
    11w = 11.66
    Dividing both sides by 11, we get:
    w = 1.06
    Therefore, the values of u and w that satisfy the system of equations are u = 1.41 and w = 1.06.
    ### Answer: u = 1.41, w = 1.06

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

Correct? No

🔗 Problem

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:

     Reasoning: To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method.
    First, we can multiply the first equation by 34 and the second equation by 42 to make the coefficients of f the same:
    34 * (76 * f - 42 * j) = 34 * 96
    42 * (47 * j + 34 * f) = 42 * 136
    This gives us:
    2584 * f - 1428 * j = 3264
    1974 * j + 1428 * f = 5712
    Next, we can subtract the first equation from the second equation to eliminate f:
    1974 * j + 1428 * f - (2584 * f - 1428 * j) = 5712 - 3264
    This simplifies to:
    3402 * j = 2448
    Dividing both sides by 3402 gives us:
    j = 2448 / 3402 = <<2448/3402=0.72>>0.72
    Now that we have the value of j, we can substitute it back into one of the original equations to find the value of f. Let's use the first equation:
    76 * f - 42 * 0.72 = 96
    This simplifies to:
    76 * f - 30.24 = 96
    Adding 30.24 to both sides gives us:
    76 * f = 126.24
    Dividing both sides by 76 gives us:
    f = 126.24 / 76 = <<126.24/76=1.66>>1.66
    Therefore, the values of f and j that will satisfy the system of equations are f = 1.66 and j = 0.72.
    ### Answer: f = 1.66, j = 0.72

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:

     Reasoning: Multiply the first equation by 63 and the second equation by 68 to eliminate u:
    63(68u + 32e) = 63(174)
    68(63e - 69u) = 68(1)
    Simplifying, we get:
    4284u + 2016e = 10962
    4284e - 4692u = 68
    Adding the two equations, we get:
    6300e = 11030
    e = 11030/6300 = <<11030/6300=1.75>>1.75
    Substituting e = 1.75 into the first equation, we get:
    68u + 32(1.75) = 174
    68u + 56 = 174
    68u = 118
    u = 118/68 = <<118/68=1.7352941176470588>>1.735
    ### Answer: u = 1.735, e = 1.75

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:

     Reasoning: Let's solve the system of linear equations by substitution. First, we can solve the first equation for p:

    57p + 4n = 129
    57p = 129 - 4n
    p = (129 - 4n) / 57

    Now, we can substitute this expression for p into the second equation:

    24p + 39n = 154
    24((129 - 4n) / 57) + 39n = 154
    (3096 - 96n) / 57 + 39n = 154
    3096 - 96n + 2223n = 8778
    2127n = 5682
    n = 5682 / 2127
    n = 2.67

    Now, we can substitute this value of n back into the expression for p:

    p = (129 - 4n) / 57
    p = (129 - 4(2.67)) / 57
    p = (129 - 10.68) / 57
    p = 118.32 / 57
    p = 2.075

    Therefore, the values of p and n that satisfy the system of linear equations are p = 2.075 and n = 2.67.
    ### Answer: p = 2.075, 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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 14 and the second equation by 44 to make the coefficients of z in both equations equal:

    14 * (66 * z - 44 * y) = 14 * 32
    44 * (60 * y + 14 * z) = 44 * 116

    This gives us:

    924 * z - 616 * y = 448
    2640 * y + 616 * z = 5104

    Next, we can add the two equations together to eliminate z:

    924 * z - 616 * y + 2640 * y + 616 * z = 448 + 5104

    This simplifies to:

    3564 * y = 5552

    Dividing both sides by 3564 gives us:

    y = 5552 / 3564 = 1.56

    Now that we have the value of y, we can substitute it back into one of the original equations to solve for z. Using the first equation:

    66 * z - 44 * 1.56 = 32

    Simplifying:

    66 * z - 68.64 = 32

    Adding 68.64 to both sides:

    66 * z = 100.64

    Dividing both sides by 66:

    z = 100.64 / 66 = 1.52

    Therefore, the values of y and z are:

    y = 1.56
    z = 1.52
    ### Answer: y = 1.56, z = 1.52

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:

     Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method.

    First, we can multiply the first equation by 11 and the second equation by 83 to make the coefficients of t the same:

    11 * (83 * t - 58 * h) = 11 * 45
    83 * (69 * h + 11 * t) = 83 * 124

    This gives us:

    913 * t - 638 * h = 495
    5707 * h + 913 * t = 10292

    Next, we can subtract the first equation from the second equation to eliminate t:

    5707 * h + 913 * t - (913 * t - 638 * h) = 10292 - 495

    This simplifies to:

    6345 * h = 9797

    Now, we can solve for h:

    h = 9797 / 6345
    h = 1.54

    Next, we can substitute the value of h back into one of the original equations to solve for t. Let's use the first equation:

    83 * t - 58 * 1.54 = 45

    This simplifies to:

    83 * t - 89.32 = 45

    Now, we can solve for t:

    83 * t = 134.32
    t = 134.32 / 83
    t = 1.62

    Therefore, the solution to the system of equations is t = 1.62 and h = 1.54.

    ### Answer: t = 1.62, h = 1.54

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we'll use elimination.
    First, we can multiply the second equation by 61/16 to make the coefficients of h in both equations equal:
    ((79 * h) - (61 * o)) = 9
    ((61 * (16 * h) - 61 * (12 * o)) = 61 * 9
    Simplifying the second equation gives us:
    ((61 * 16 * h) - (61 * 12 * o)) = 549
    Now, we can subtract the first equation from the second equation to eliminate h:
    ((61 * 16 * h) - (61 * 12 * o)) - ((79 * h) - (61 * o)) = 549 - 9
    Simplifying this gives us:
    ((61 * 16 * h) - (79 * h) - (61 * 12 * o) + (61 * o)) = 540
    ((976 * h) - (79 * h) - (732 * o) + (61 * o)) = 540
    ((897 * h) - (671 * o)) = 540
    Now, we can solve for o:
    o = ((897 * h) - 540) / 671
    Substituting this expression for o into the first equation gives us:
    ((79 * h) - (61 * ((897 * h) - 540) / 671)) = 9
    Simplifying this gives us:
    ((79 * h) - ((61 * 897 * h) - (61 * 540)) / 671) = 9
    ((79 * h) - ((54627 * h) - (32940)) / 671) = 9
    ((79 * h) - (54627 * h) / 671 + (32940) / 671) = 9
    ((79 * h) - (81.42 * h) + 49.09) = 9
    ((79 * h) - (81.42 * h)) = 9 - 49.09
    ((-2.42 * h)) = -40.09
    h = -40.09 / -2.42
    h = 16.57
    Substituting this value of h back into the expression for o gives us:
    o = ((897 * 16.57) - 540) / 671
    o = (14821.49 - 540) / 671
    o = 14281.49 / 671
    o = 21.28
    ### Answer: o = 21.28, h = 16.57

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:

     Reasoning: To solve for s and y, we can use the method of substitution or elimination. Here, we'll use elimination. First, we'll multiply the first equation by 90 and the second equation by 56 to make the coefficients of s in both equations equal:

    90 * (4 * y + 56 * s) = 90 * 139
    56 * (65 * y - 90 * s) = 56 * 16

    This gives us:

    360 * y + 5040 * s = 12510
    3640 * y - 5040 * s = 896

    Now, we can add these two equations to eliminate s:

    360 * y + 3640 * y + 5040 * s - 5040 * s = 12510 + 896
    4000 * y = 13406

    Now, we can solve for y:

    y = 13406 / 4000
    y = 3.3515

    Now that we have y, we can substitute it back into one of the original equations to solve for s. We'll use the first equation:

    4 * 3.3515 + 56 * s = 139
    13.406 + 56 * s = 139
    56 * s = 139 - 13.406
    56 * s = 125.594
    s = 125.594 / 56
    s = 2.24275

    ### Answer: y = 3.3515, s = 2.24275

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:

     Reasoning: To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

    First, we can multiply the first equation by 45 and the second equation by 4 to make the coefficients of z the same:

    45 * (42 * d - 4 * z) = 45 * 75
    4 * (40 * d + 45 * z) = 4 * 162

    This gives us:

    1890 * d - 180 * z = 3375
    160 * d + 180 * z = 648

    Now, we can add the two equations to eliminate z:

    1890 * d + 160 * d = 3375 + 648
    2050 * d = 4023

    Dividing both sides by 2050 gives us:

    d = 4023 / 2050
    d = 1.962439024390244

    Now that we have the value of d, we can substitute it back into one of the original equations to find z. Using the first equation:

    42 * d - 4 * z = 75
    42 * 1.962439024390244 - 4 * z = 75

    Solving for z gives us:

    82.42243902439024 - 4 * z = 75
    -4 * z = 75 - 82.42243902439024
    -4 * z = -7.42243902439024
    z = -7.42243902439024 / -4
    z = 1.85560975609756

    ### Answer: d = 1.962439024390244, z = 1.85560975609756

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:

     Reasoning: Let's solve the system of equations step by step. First, we can multiply the second equation by 4 to make the coefficients of 'o' in both equations equal: (42 * o) + (24 * a) = 142
     (168 * a) + (24 * o) = 300. Now, we can subtract the first equation from the second equation to eliminate 'o': (168 * a) + (24 * o) - ((42 * o) + (24 * a)) = 300 - 142
     (126 * a) = 158. Now, we can solve for 'a': a = 158 / 126
     a = 1.25. Now, we can substitute the value of 'a' into one of the original equations to solve for 'o': (42 * o) + (24 * 1.25) = 142
     (42 * o) + 30 = 142
     (42 * o) = 112
     o = 112 / 42
     o = 2.67. Therefore, the values of 'a' and 'o' are 1.25 and 2.67, respectively.
    ### Answer: a = 1.25, o = 2.67

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:

     Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method.
    First, we can multiply the first equation by 84 and the second equation by 39 to make the coefficients of r equal:
    84*(39*r - 2*p) = 84*45
    39*(16*p - 84*r) = 39*36
    Simplifying these equations, we get:
    3276*r - 168*p = 3780
    624*p - 3276*r = 1404
    Adding these two equations, we get:
    456*p = 5184
    Dividing both sides by 456, we get:
    p = 11.368421052631579
    Substituting this value of p into the first equation, we get:
    39*r - 2*11.368421052631579 = 45
    Simplifying this equation, we get:
    39*r = 67.73684210526316
    Dividing both sides by 39, we get:
    r = 1.7368421052631577
    ### Answer: p = 11.368421052631579, r = 1.7368421052631577

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:

     Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method.

    First, we can multiply the first equation by 8 and the second equation by 23 to make the coefficients of 'd' in both equations equal:

    1. (184 * d) - (416 * w) = 56
    2. (1472 * w) - (184 * d) = 1334

    Next, we can add the two equations to eliminate 'd':

    (184 * d) - (416 * w) + (1472 * w) - (184 * d) = 56 + 1334
    1056 * w = 1390

    Now, we can solve for 'w':

    w = 1390 / 1056
    w = 1.316

    Substituting the value of 'w' back into one of the original equations, we can solve for 'd':

    (23 * d) - (52 * 1.316) = 7
    (23 * d) - 68.432 = 7
    23 * d = 75.432
    d = 75.432 / 23
    d = 3.28

    ### Answer: d = 3.28, w = 1.316

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:

     Reasoning: Multiply the first equation by 47 and the second equation by 60 to eliminate n:

    (71 * z * 47) + (60 * n * 47) = 184 * 47
    (52 * z * 60) - (47 * n * 60) = 19 * 60

    Simplify:

    3337 * z + 2820 * n = 8648
    3120 * z - 2820 * n = 1140

    Add the two equations to eliminate n:

    (3337 * z) + (3120 * z) = 8648 + 1140
    6457 * z = 9788

    Divide both sides by 6457:

    z = 9788 / 6457
    z = 1.516

    Substitute z = 1.516 into the first equation:

    (71 * 1.516) + (60 * n) = 184
    107.736 + 60 * n = 184

    Subtract 107.736 from both sides:

    60 * n = 184 - 107.736
    60 * n = 76.264

    Divide both sides by 60:

    n = 76.264 / 60
    n = 1.271

    ### Answer: n = 1.271, z = 1.516

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:

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

    First, we can multiply the first equation by 84 and the second equation by 70 to make the coefficients of e in both equations equal:

    84 * (70 * e + 90 * z) = 84 * 189
    70 * (99 * z - 84 * e) = 70 * 20

    This simplifies to:

    5880 * e + 7560 * z = 15876
    6930 * z - 5880 * e = 1400

    Now, we can add the two equations to eliminate the e term:

    (5880 * e + 7560 * z) + (6930 * z - 5880 * e) = 15876 + 1400
    7560 * z + 6930 * z = 17276
    14490 * z = 17276

    Dividing both sides by 14490, we get:

    z = 17276 / 14490
    z = 1.192

    Now that we have the value of z, we can substitute it back into one of the original equations to solve for e. Let's use the first equation:

    70 * e + 90 * 1.192 = 189
    70 * e + 107.28 = 189
    70 * e = 189 - 107.28
    70 * e = 81.72

    Dividing both sides by 70, we get:

    e = 81.72 / 70
    e = 1.167

    Therefore, the values of e and z are:

    e = 1.167
    z = 1.192

    ### Answer: e = 1.167, z = 1.192

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:

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

    First, we can multiply equation 1 by 11 and equation 2 by 26 to make the coefficients of o the same:

    1) 286o - 484f = 1045
    2) 2262f + 286o = 4394

    Now, we can subtract equation 1 from equation 2 to eliminate o:

    2262f + 286o - (286o - 484f) = 4394 - 1045
    2262f + 286o - 286o + 484f = 3349
    2746f = 3349

    Now, we can solve for f:

    f = 3349 / 2746
    f = 1.22

    Next, we can substitute the value of f back into either equation 1 or 2 to solve for o. Let's use equation 1:

    26o - 44(1.22) = 95
    26o - 53.68 = 95
    26o = 148.68

    Now, we can solve for o:

    o = 148.68 / 26
    o = 5.72

    Therefore, the values of o and f that satisfy the system of equations are o = 5.72 and f = 1.22.

    ### Answer: 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:

     Reasoning: Multiply the first equation by 3 and the second equation by 20 to eliminate z:
    120*z - 60*p = 153
    120*z + 1160*p = 3300
    Subtract the first equation from the second equation to solve for p:
    1220*p = 3147
    p = 3147/1220 = <<3147/1220=2.58>>2.58
    Substitute p = 2.58 into the first equation to solve for z:
    40*z - 20*2.58 = 51
    40*z = 102.6
    z = 102.6/40 = <<102.6/40=2.565>>2.565
    ### 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:

     Reasoning: To solve the system of equations, we can use the substitution or elimination method. Here, we will use the elimination method.

    First, we can multiply the second equation by 89 to make the coefficients of f in both equations equal:

    89 * (3 * f + 10 * s) = 89 * 93
    267 * f + 890 * s = 8277

    Now, we can subtract the first equation from this new equation to eliminate f:

    (267 * f + 890 * s) - (71 * s - 89 * f) = 8277 - 5
    267 * f + 890 * s - 71 * s + 89 * f = 8272
    356 * f + 819 * s = 8272

    Next, we can multiply the first equation by 3 to make the coefficients of s in both equations equal:

    3 * (71 * s - 89 * f) = 3 * 5
    213 * s - 267 * f = 15

    Now, we can add this new equation to the previous equation to eliminate s:

    (356 * f + 819 * s) + (213 * s - 267 * f) = 8272 + 15
    356 * f + 819 * s + 213 * s - 267 * f = 8287
    80 * f + 1032 * s = 8287

    Finally, we can solve for f and s using the resulting equation:

    80 * f = 8287 - 1032 * s
    f = (8287 - 1032 * s) / 80

    Substituting this expression for f into the first equation, we get:

    71 * s - 89 * ((8287 - 1032 * s) / 80) = 5
    71 * s - (89 * 8287 - 89 * 1032 * s) / 80 = 5
    71 * s - (737543 - 92088 * s) / 80 = 5
    71 * s - 9219.2875 + 1151.1 * s = 5
    1822.1 * s = 9224.2875
    s = 9224.2875 / 1822.1
    s = 5.06

    Substituting this value of s back into the expression for f, we get:

    f = (8287 - 1032 * 5.06) / 80
    f = (8287 - 5203.52) / 80
    f = 3083.48 / 80
    f = 38.54

    Therefore, the values of s and f are s = 5.06 and f = 38.54.
    ### Answer: s = 5.06, f = 38.54

Expected answer: {f: 6553/1103, s: 8292/1103}

Correct? No