🔗 Problem

Problem: Let's find the values of variables q and s by solving the following system of equations: 1. (40 * s) - (91 * q) = 37 2. (30 * s) + (12 * q) = 121

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the second equation, and then substitute it into the first equation.

    From the second equation, let's solve for s in terms of q:
    (30 * s) + (12 * q) = 121
    30s = 121 - 12q
    s = (121 - 12q) / 30
    s = 121/30 - 12q/30
    s = 121/30 - 2q/5
    <<121/30=121/30=121/30=4.033333333333333>><<12/30=12/30=2/5=0.4>>
    s = 4.033333333333333 - 0.4q

    Now, substitute this expression for s into the first equation:
    (40 * s) - (91 * q) = 37
    40 * (4.033333333333333 - 0.4q) - 91q = 37
    161.33333333333333 - 16q - 91q = 37
    161.33333333333333 - 107q = 37
    -107q = -124.33333333333333
    <<-124.33333333333333/-107=-124.33333333333333/-107=1.162790697674419>>
    q = 1.162790697674419

    Now that we have the value of q, we can substitute it back into the equation for s:
    s = 4.033333333333333 - 0.4q
    s = 4.033333333333333 - 0.4 * 1.162790697674419
    <<4.033333333333333-0.4*1.162790697674419=4.033333333333333-0.465116279069767=3.568217054263566>>
    s = 3.568217054263566

    Therefore, the solution to the system of equations is:
    q ≈ 1.163 (rounded to 3 decimal places)
    s ≈ 3.568 (rounded to 3 decimal places)

    ### Answer: q ≈ 1.163, s ≈ 3.568

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of b and m: 4 * b + 44 * m = 180 and 79 * m - 6 * b = 48.

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    Given equations:
    1. 4b + 44m = 180
    2. 79m - 6b = 48

    Step 1: Solve for b in terms of m using equation 1.
    4b + 44m = 180
    4b = 180 - 44m
    b = (180 - 44m) ÷ 4
    b = 45 - 11m

    Step 2: Substitute the expression for b into equation 2.
    79m - 6(45 - 11m) = 48
    79m - 270 + 66m = 48
    145m - 270 = 48
    145m = 318
    m = 318 ÷ 145
    <<145m = 318
    m = 318 ÷ 145
    m = 2.19310344827586>>m ≈ 2.19

    Step 3: Substitute the value of m into the expression for b from step 1.
    b = 45 - 11m
    b = 45 - 11(2.19)
    b = 45 - 24.09
    <<b = 45 - 11(2.19)
    b = 45 - 24.09
    b = 20.91>>b ≈ 20.91

    Therefore, the solution to the system of equations is:
    m ≈ 2.19
    b ≈ 20.91

    ### Answer: m ≈ 2.19, b ≈ 20.91

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations to find the values of o and g: 42 * o - 15 * g = 28 47 * o + 29 * g = 141.

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, let's solve for o:
    42 * o - 15 * g = 28
    42 * o = 28 + 15 * g
    o = (28 + 15 * g) / 42
    o = (2 * g) / 3 + 2 / 3

    Now, substitute the expression for o into the second equation:
    47 * ((2 * g) / 3 + 2 / 3) + 29 * g = 141
    (94 * g) / 3 + 94 / 3 + 29 * g = 141
    (94 * g) / 3 + 29 * g = 141 - 94 / 3
    (94 * g) / 3 + 29 * g = 423 / 3 - 94 / 3
    (94 * g) / 3 + 29 * g = 329 / 3
    (94 * g + 87 * g) / 3 = 329 / 3
    181 * g / 3 = 329 / 3
    181 * g = 329
    g = 329 / 181
    g = 1.81767955801105

    Now that we have the value of g, we can substitute it back into the expression for o:
    o = (2 * 1.81767955801105) / 3 + 2 / 3
    o = 3.63535911602209 / 3 + 2 / 3
    o = 1.21178637200736 + 0.66666666666667
    o = 1.87845303867403

    Therefore, the solution to the system of equations is:
    g ≈ 1.81768
    o ≈ 1.87845

    ### Answer: g ≈ 1.81768, o ≈ 1.87845

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

Correct? No

🔗 Problem

Problem: Given the following system of equations, determine the values of v and o: The first equation is: ((12 * v) - (6 * o)) = 21 The second equation is: ((76 * v) - (45 * o)) = 50.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for v in the first equation and then substitute it into the second equation to solve for o.

    First equation: ((12 * v) - (6 * o)) = 21
    12v - 6o = 21
    12v = 21 + 6o
    v = (21 + 6o) / 12
    v = (7/4) + (o/2) ... (1)

    Now, substitute (1) into the second equation:
    ((76 * v) - (45 * o)) = 50
    76 * ((7/4) + (o/2)) - 45o = 50
    133 + 38o - 45o = 50
    133 - 7o = 50
    -7o = -83
    o = 83/7
    o = 11.857... ≈ 11.86

    Now that we have the value of o, we can substitute it back into equation (1) to find v:
    v = (7/4) + (o/2)
    v = 1.75 + (11.86/2)
    v = 1.75 + 5.93
    v = 7.68

    ### Answer: v ≈ 7.68, o ≈ 11.86

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

Correct? No

🔗 Problem

Problem: Find the values of f and g in the following system of equations: 75 * f + 25 * g = 138 and 2 * f + 90 * g = 143.

Gemma 2B:

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

    1. Solve the first equation for f in terms of g:
       75 * f + 25 * g = 138
       75 * f = 138 - 25 * g
       f = (138 - 25 * g) / 75
       f = (138 - 25 * g) / 75 = <<(138 - 25 * g) / 75 = (138/75) - (25/75)*g>>  (138/75) - (25/75)*g

    2. Substitute the expression for f into the second equation:
       2 * f + 90 * g = 143
       2 * ((138/75) - (25/75)*g) + 90 * g = 143
       (276/75) - (50/75)*g + 90 * g = 143
       (276/75) + (6750/75)*g - (50/75)*g = 143
       (276/75) + (6700/75)*g = 143

    3. Solve the equation for g:
       (276/75) + (6700/75)*g = 143
       (6700/75)*g = 143 - (276/75)
       (6700/75)*g = (10725/75) - (276/75) = <<(10725/75) - (276/75) = 10449/75>>10449/75
       g = (10449/75) / (6700/75)
       g = <<(10449/75) / (6700/75) = 10449/6700>>10449/6700

    4. Substitute the value of g into the expression for f:
       f = (138/75) - (25/75)*g
       f = (138/75) - (25/75)*(10449/6700)
       f = (138/75) - (261225/502500) = <<(138/75) - (261225/502500) = (69000/502500) - (261225/502500) = -192225/502500>>-192225/502500

    Therefore, the values of f and g are:
    g = 10449/6700
    f = -192225/502500

    ### Answer: f = -192225/502500, g = 10449/6700

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations for variables y and n: the first equation is 194 = 4 * y + 14 * n and the second equation is 7 = 65 * n - 79 * y.

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    First equation: 194 = 4y + 14n
    194 - 14n = 4y
    (194 - 14n) / 4 = y
    y = (194 - 14n) / 4 ... (1)

    Second equation: 7 = 65n - 79y

    Substitute (1) into the second equation:
    7 = 65n - 79((194 - 14n) / 4)
    7 = 65n - 79(194 - 14n) / 4
    7 = 65n - (15286 - 1106n) / 4
    28 = 260n - 15286 + 1106n
    28 = 1366n - 15286
    15314 = 1366n
    n = 15314 / 1366
    n = 11.2116251831 ... (2)

    Substitute (2) into (1) to find y:
    y = (194 - 14 × 11.2116251831) / 4
    y = (194 - 156.962752564) / 4
    y = 37.037247436 / 4
    y = 9.25931185898

    ### Answer: y ≈ 9.25931185898, n ≈ 11.2116251831

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

Correct? No

🔗 Problem

Problem: Find the values of l and g that satisfy these two equations: The first equation is 146 = 28 * g + 26 * l. The second equation is 6 = 41 * l - 53 * g.

Gemma 2B:

    To solve for l and g, we can use substitution or elimination methods. Let's use the substitution method here.

    From the second equation, we can express l in terms of g:
    6 = 41 * l - 53 * g
    41 * l = 53 * g + 6
    l = (53 * g + 6) / 41

    Now, substitute this expression for l into the first equation:
    146 = 28 * g + 26 * ((53 * g + 6) / 41)

    Simplify:
    146 = 28 * g + (1378 * g + 156) / 41
    146 = 28 * g + (1378 * g) / 41 + 156 / 41
    146 = 28 * g + 33.6097560976 * g + 3.8048780488
    146 = 61.6097560976 * g + 3.8048780488

    Subtract 3.8048780488 from both sides:
    142.1951219512 = 61.6097560976 * g

    Divide both sides by 61.6097560976:
    g = 142.1951219512 / 61.6097560976
    g ≈ 2.3081395348

    Now that we have the value of g, we can substitute it back into the expression for l:
    l = (53 * 2.3081395348 + 6) / 41
    l ≈ 3.0000000001

    Therefore, the values of l and g that satisfy the given equations are approximately:
    g ≈ 2.3081395348
    l ≈ 3.0000000001

    ### Answer: g ≈ 2.3081395348, l ≈ 3.0000000001

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

Correct? No

🔗 Problem

Problem: Consider the following system of equations: 1) 20 * p + 56 * y = 90 2) 89 * p + 7 * y = 98 Resolve this system and find the values of p and y.

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one of the variables in the first equation and substitute it into the second equation.

    From equation 1, solve for y:
    20 * p + 56 * y = 90
    56 * y = 90 - 20 * p
    y = (90 - 20 * p) / 56
    y = 45/28 - (5/14) * p

    Substitute the expression for y into equation 2:
    89 * p + 7 * (45/28 - (5/14) * p) = 98
    89 * p + 315/28 - 35/14 * p = 98
    89 * p - 35/14 * p = 98 - 315/28
    (89 - 35/14) * p = 98 - 315/28
    (1246/14 - 35/14) * p = 98 - 315/28
    (1211/14) * p = 98 - 315/28
    1211 * p = 1372 - 315
    1211 * p = 1057
    p = 1057/1211

    Substitute p = 1057/1211 into the expression for y:
    y = 45/28 - (5/14) * (1057/1211)
    y = 45/28 - 5285/16954
    y = 12135/16954 - 5285/16954
    y = 6850/16954

    Therefore, the solution to the system of equations is:
    p = 1057/1211
    y = 6850/16954

    ### Answer: p = 1057/1211, y = 6850/16954

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

Correct? No

🔗 Problem

Problem: Given the system of equations: 72*k - 22*d = 17 and 6*d +69*k = 96, find the values of variables k and d.

Gemma 2B:

    To solve the given system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    Given equations:
    1. 72*k - 22*d = 17
    2. 6*d + 69*k = 96

    Step 1: Solve for d in terms of k using equation 1.
    72*k - 22*d = 17
    -22*d = -72*k + 17
    d = (72*k - 17) / 22
    d = (72*k - 17) / 22

    Step 2: Substitute the expression for d into equation 2.
    6*d + 69*k = 96
    6*((72*k - 17) / 22) + 69*k = 96
    (432*k - 102) / 22 + 69*k = 96
    432*k - 102 + 1518*k = 2112
    1950*k - 102 = 2112
    1950*k = 2214
    k = 2214 / 1950
    k = <<2214/1950=1.135384615384615>>1.135384615384615

    Step 3: Substitute the value of k into the expression for d.
    d = (72*k - 17) / 22
    d = (72 * 1.135384615384615 - 17) / 22
    d = (81.74769230769231 - 17) / 22
    d = 64.74769230769231 / 22
    d = <<64.74769230769231/22=2.943076923076923>>2.943076923076923

    Therefore, the values of the variables are:
    k ≈ 1.135384615384615
    d ≈ 2.943076923076923

    ### Answer: k ≈ 1.135384615384615, d ≈ 2.943076923076923

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations to find the values of d and h: 73 * h - 33 * d = 48 10 * d + 14 * h = 91

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

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

    Step 1: Solve the first equation for h in terms of d.
    73 * h - 33 * d = 48
    73 * h = 33 * d + 48
    h = (33 * d + 48) / 73

    Step 2: Substitute the expression for h into the second equation.
    10 * d + 14 * ((33 * d + 48) / 73) = 91

    Step 3: Simplify the equation and solve for d.
    10 * d + (462 * d + 672) / 73 = 91
    730 * d + 462 * d + 672 = 6643
    1192 * d + 672 = 6643
    1192 * d = 5971
    d = 5971 / 1192
    d = 5

    Step 4: Substitute the value of d into the expression for h from Step 1.
    h = (33 * 5 + 48) / 73
    h = 213 / 73
    h = 2.9178082191780823 (rounded to 16 decimal places)

    Therefore, the solution to the system of equations is:
    d = 5
    h ≈ 2.9178082191780823

    ### Answer: d = 5, h ≈ 2.9178082191780823

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of q and e: 1) 43 * q - 44 * e = 29 2) 30 * q + 43 * e = 130

Gemma 2B:

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

    From equation 1, we can express q in terms of e:
    43 * q - 44 * e = 29
    43 * q = 29 + 44 * e
    q = (29 + 44 * e) / 43

    Now, substitute this expression for q into equation 2:
    30 * ((29 + 44 * e) / 43) + 43 * e = 130

    Simplify:
    (870 + 1320 * e) / 43 + 43 * e = 130
    870 / 43 + 1320 * e / 43 + 43 * e = 130
    20.23 + 30.70 * e + 43 * e = 130
    73.70 * e = 109.77
    e = 109.77 / 73.70
    e ≈ 1.49

    Now that we have the value of e, we can substitute it back into the expression for q:
    q = (29 + 44 * 1.49) / 43
    q = (29 + 65.56) / 43
    q = 94.56 / 43
    q ≈ 2.20

    ### Answer: q ≈ 2.20, e ≈ 1.49

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the second equation, then substitute it into the first equation to solve for the remaining variable.

    From the second equation, let's solve for r in terms of a:
    109 = 37a + 6r
    6r = 109 - 37a
    r = (109 - 37a) / 6
    r = (109/6) - (37/6)a
    r = 18.1666... - 6.1666...a

    Now, substitute this expression for r into the first equation:
    53 = 89r - 74a
    53 = 89(18.1666... - 6.1666...a) - 74a
    53 = 1616.8333... - 549.8333...a - 74a
    53 = 1616.8333... - 623.8333...a
    623.8333...a = 1616.8333... - 53
    623.8333...a = 1563.8333...
    a = 1563.8333... / 623.8333...
    a = 2.5

    Now that we have the value of a, we can substitute it back into the expression for r:
    r = 18.1666... - 6.1666...(2.5)
    r = 18.1666... - 15.4166...
    r = 2.75

    Therefore, the solution to the system of equations is:
    a = 2.5
    r = 2.75

    ### Answer: a = 2.5, r = 2.75

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

Correct? No

🔗 Problem

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:

    Let's solve this step by step. We need to find the weight of one unit of 'g' and one unit of 'q'.

    From equation 1:
    42g + 12q = 169 ... (1)

    From equation 2:
    12q - 50g = 19 ... (2)

    Step 1: Multiply equation (2) by -1 to eliminate the negative sign.
    50g - 12q = -19 ... (3)

    Step 2: Add equations (1) and (3) to eliminate 'q'.
    42g + 12q = 169
    50g - 12q = -19
    _______________
    92g = 150

    Step 3: Solve for 'g'.
    g = 150 ÷ 92
    g = 1.63043478 ... (4)

    Step 4: Substitute the value of 'g' in equation (1) to solve for 'q'.
    42(1.63043478) + 12q = 169
    68.47826076 + 12q = 169
    12q = 169 - 68.47826076
    12q = 100.5217392
    q = 100.5217392 ÷ 12
    q = 8.37681160 ... (5)

    Therefore, the weight of one unit of grain 'g' is approximately 1.63 kg, and the weight of one unit of grain 'q' is approximately 8.38 kg.

    ### Answer: g ≈ 1.63 kg, q ≈ 8.38 kg

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for s in the first equation and substitute it into the second equation to solve for v.

    1. Solve for s in the first equation:
       37 * s - 11 * v = 54
       37 * s = 54 + 11 * v
       s = (54 + 11 * v) / 37

    2. Substitute the expression for s into the second equation:
       82 * s - 62 * v = 41
       82 * ((54 + 11 * v) / 37) - 62 * v = 41

    3. Simplify the equation:
       (82 * (54 + 11 * v)) / 37 - 62 * v = 41
       (4428 + 902 * v) / 37 - 62 * v = 41

    4. Multiply both sides by 37 to eliminate fractions:
       4428 + 902 * v - 2294 * v = 1517
       4428 - 1392 * v = 1517

    5. Subtract 4428 from both sides:
       -1392 * v = -2911

    6. Divide both sides by -1392 to solve for v:
       v = 2911 / 1392
       v ≈ 2.09

    7. Substitute v ≈ 2.09 into the equation from step 1 to solve for s:
       s = (54 + 11 * 2.09) / 37
       s = (54 + 22.99) / 37
       s = 76.99 / 37
       s ≈ 2.08

    Therefore, the approximate values for the system of equations are:
    s ≈ 2.08
    v ≈ 2.09

    ### Answer: s ≈ 2.08, v ≈ 2.09

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

Correct? No

🔗 Problem

Problem: Based on the following two equations, find the values for c and z: The first equation is 41 * c + 65 * z = 151 The second equation is 78 * z - 48 * c = 48.

Gemma 2B:

    To solve for c and z, we can use the substitution method. Let's solve for c in the first equation and substitute it into the second equation to solve for z.

    From the first equation, 41 * c + 65 * z = 151, we can express c in terms of z:
    41 * c = 151 - 65 * z
    c = (151 - 65 * z) / 41

    Now, let's substitute this expression for c into the second equation, 78 * z - 48 * c = 48:
    78 * z - 48 * ((151 - 65 * z) / 41) = 48
    78 * z - (48 * 151 - 48 * 65 * z) / 41 = 48
    78 * z - (7248 - 3120 * z) / 41 = 48
    3198 * z - 7248 / 41 + 3120 * z / 41 = 48
    (3198 * 41 * z - 7248 + 3120 * z) / 41 = 48
    (131118 * z - 7248 + 3120 * z) / 41 = 48
    (134238 * z - 7248) / 41 = 48
    134238 * z - 7248 = 1968
    134238 * z = 9216
    z = 9216 / 134238
    z = <<9216/134238=0.0686275>>0.0686275

    Now that we have the value of z, we can substitute it back into the equation for c:
    c = (151 - 65 * 0.0686275) / 41
    c = (151 - 4.460788) / 41
    c = 146.539212 / 41
    c = <<146.539212/41=3.574371>>3.574371

    Therefore, the values for c and z are:
    z ≈ 0.0686275
    c ≈ 3.574371

    ### Answer: c ≈ 3.574371, z ≈ 0.0686275

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

Correct? No

🔗 Problem

Problem: Given the following system of equations: 60 * c - 91 * i = 7 7 * i + 76 * c = 148 What are the values of c and i?

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    Given equations:
    60c - 91i = 7 ... (1)
    7i + 76c = 148 ... (2)

    Step 1: Solve for c in terms of i using equation (1).
    60c - 91i = 7
    60c = 91i + 7
    c = (91i + 7) / 60 ... (3)

    Step 2: Substitute the expression for c from equation (3) into equation (2).
    7i + 76c = 148
    7i + 76 * ((91i + 7) / 60) = 148
    7i + (76 * (91i + 7)) / 60 = 148
    7i + (6916i + 532) / 60 = 148
    7i + 115.2667i + 8.8667 = 148
    122.2667i + 8.8667 = 148
    122.2667i = 139.1333
    i = 139.1333 / 122.2667
    i ≈ 1.1385

    Step 3: Substitute the value of i into equation (3) to solve for c.
    c = (91i + 7) / 60
    c = (91 * 1.1385 + 7) / 60
    c = (103.6035 + 7) / 60
    c = 110.6035 / 60
    c ≈ 1.8434

    Therefore, the values of c and i are:
    i ≈ 1.1385
    c ≈ 1.8434

    ### Answer: i ≈ 1.1385, c ≈ 1.8434

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

Correct? No

🔗 Problem

Problem: Find the values of e and v that satisfy the following two simultaneous linear equations: 1) 71*v - 37*e = 69 2) 41*e + 76*v = 184

Gemma 2B:

    To solve the given simultaneous linear equations, we can use the substitution method.

    From equation 1, we can express v in terms of e:
    71*v - 37*e = 69
    71*v = 37*e + 69
    v = (37*e + 69) / 71

    Now, substitute this expression for v into equation 2:
    41*e + 76*v = 184
    41*e + 76*((37*e + 69) / 71) = 184

    Simplify the equation:
    41*e + (76*37*e + 76*69) / 71 = 184
    41*e + (2812*e + 5244) / 71 = 184

    Multiply both sides by 71:
    2911*e + 2812*e + 5244 = 13064
    5723*e + 5244 = 13064
    5723*e = 7820
    e = 7820 / 5723

    Solve for e:
    e = <<7820/5723=1.3664>>1.3664 (rounded to 4 decimal places)

    Substitute e = 1.3664 into the expression for v:
    v = (37*1.3664 + 69) / 71
    v = 119.5568 / 71

    Solve for v:
    v = <<119.5568/71=1.6839>>1.6839 (rounded to 4 decimal places)

    ### Answer: e = 1.3664, v = 1.6839

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

Correct? No

🔗 Problem

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:

    Let's solve this system of equations step by step using the substitution method.

    Given equations:
    1. 25v + 69d = 113
    2. 42v + 26d = 85

    Step 1: Solve equation 1 for v in terms of d.
    25v + 69d = 113
    25v = 113 - 69d
    v = (113 - 69d) / 25

    Step 2: Substitute the expression for v into equation 2.
    42((113 - 69d) / 25) + 26d = 85

    Step 3: Simplify the equation and solve for d.
    (4746 - 2898d) / 25 + 26d = 85
    4746 - 2898d + 650d = 2125
    4746 - 2248d = 2125
    -2248d = -2621
    d = -2621 / -2248
    d = <<-2621/-2248=1.165925266903915>>1.165925266903915

    Step 4: Substitute the value of d into the expression for v from step 1.
    v = (113 - 69(1.165925266903915)) / 25
    v = (113 - 80.44834334623217) / 25
    v = 32.55165665376783 / 25
    v = <<32.55165665376783/25=1.302066266150713>>1.302066266150713

    Therefore, the values of d and v are:
    d ≈ 1.165925266903915
    v ≈ 1.302066266150713

    ### Answer: d ≈ 1.165925266903915, v ≈ 1.302066266150713

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

Correct? No

🔗 Problem

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

Gemma 2B:

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

    From the first equation, we can express z in terms of g:
    43 * z - 91 * g = 33
    43 * z = 91 * g + 33
    z = (91 * g + 33) / 43

    Now, substitute this expression for z into the second equation:
    72 * g - 3 * ((91 * g + 33) / 43) = 68

    Simplify:
    72 * g - (273 * g + 99) / 43 = 68
    3096 * g - 273 * g - 99 = 2924
    2823 * g - 99 = 2924
    2823 * g = 3023
    g = 3023 / 2823

    Now that we have the value of g, we can substitute it back into the expression for z:
    z = (91 * (3023 / 2823) + 33) / 43

    Simplify:
    z = (273993 / 2823 + 33) / 43
    z = (273993 + 93159) / 121969
    z = 367152 / 121969

    Therefore, the solutions for g and z are:
    g = 3023 / 2823
    z = 367152 / 121969

    ### Answer: g = 3023 / 2823, z = 367152 / 121969

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

Correct? No

🔗 Problem

Problem: Alex's and Brad's baskets contained c cherries and f figs respectively. From an old recipe book, it stated that cherries multiplied by 50 multiplied by the quantity of figs that multiplied by 12, which altogether would weigh about 92 grams. Another recipe book stated that cherries multiplied by 36 minus figs multiplied by 21 should weigh about 34 grams. Can you find the quantity of cherries (c) and figs (f)? Use the following system of equations to solve: 50c + 12f = 92 36c - 21f = 34.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for c in the second equation and substitute it into the first equation to solve for f.

    From the second equation:
    36c - 21f = 34
    36c = 34 + 21f
    c = (34 + 21f) / 36
    c = (17 + 21f) / 18 (simplifying the fraction)

    Substituting this expression for c into the first equation:
    50c + 12f = 92
    50((17 + 21f) / 18) + 12f = 92
    (850 + 1050f) / 18 + 12f = 92
    850 + 1050f + 216f = 1656
    1266f = 806
    f = 806 / 1266
    f = 62 / 79

    Now that we have the value of f, we can substitute it back into the equation for c:
    c = (17 + 21f) / 18
    c = (17 + 21(62/79)) / 18
    c = (1343 + 1302) / 1422
    c = 2645 / 1422
    c = 1 76/79

    Therefore, the quantity of cherries (c) is 1 76/79, and the quantity of figs (f) is 62/79.

    ### Answer: c = 1 76/79, f = 62/79

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

Correct? No

🔗 Problem

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:

    Great! Let's solve this system of equations step by step to find the values of u and d.

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

    Step 1: Multiply equation (1) by 82 and equation (2) by 87 to eliminate the variable u.
    Equation (1): 82(87u - 75d = 7) → 7134u - 6150d = 574
    Equation (2): 87(85d - 82u = 54) → 7395d - 7134u = 4698

    Step 2: Add the resulting equations from step 1 to eliminate the variable u.
    7134u - 6150d = 574
    7395d - 7134u = 4698
    1245d = 5272

    Step 3: Solve for d by dividing both sides by 1245.
    d = 5272 ÷ 1245
    d = 4.23

    Step 4: Substitute the value of d into equation (1) to solve for u.
    87u - 75(4.23) = 7
    87u - 317.25 = 7
    87u = 324.25
    u = 324.25 ÷ 87
    u = 3.73

    Therefore, the values of u and d are:
    u = 3.73
    d = 4.23

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

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

Correct? No

🔗 Problem

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:

    Let's solve this problem step by step using a system of equations. We have two equations with two unknown variables, a and i.

    Equation 1: (75 * a) - (4 * i) = 42
    Equation 2: (10 * i) + (13 * a) = 160

    Step 1: Simplify Equation 1 by dividing both sides by 3.
    (25 * a) - (4/3 * i) = 14

    Step 2: Substitute the value of a from Equation 2 into the simplified Equation 1.
    (25 * ((160 - 10 * i) / 13)) - (4/3 * i) = 14

    Step 3: Solve for i.
    (25 * (160 - 10 * i)) / 13 - (4/3 * i) = 14
    (4000 - 250 * i) / 13 - (4/3 * i) = 14
    (4000 - 250 * i) / 13 - (52/3 * i) = 14
    (12000 - 750 * i) / 39 - (52/3 * i) = 14
    (12000 - 750 * i) / 39 - (156/9 * i) = 14
    (108000 - 6750 * i) / 351 - (156/9 * i) = 14
    (108000 - 6750 * i) / 351 - (6084/351 * i) = 14
    (108000 - 6750 * i - 6084 * i) / 351 = 14
    (108000 - 12834 * i) / 351 = 14
    108000 - 12834 * i = 4914
    -12834 * i = -103086
    i = 8

    Step 4: Substitute the value of i into Equation 2 to solve for a.
    (10 * 8) + (13 * a) = 160
    80 + 13 * a = 160
    13 * a = 80
    a = 80 / 13

    Therefore, the values are:
    i = 8
    a = 80 / 13

    ### Answer: i = 8, a = 80 / 13

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    Given equations:
    37 * p + 74 * c = 183 ... (1)
    27 * c - 18 * p = 13 ... (2)

    Step 1: Solve equation (1) for p in terms of c.
    37 * p + 74 * c = 183
    37 * p = 183 - 74 * c
    p = (183 - 74 * c) / 37

    Step 2: Substitute the expression for p into equation (2).
    27 * c - 18 * ((183 - 74 * c) / 37) = 13

    Step 3: Solve the equation for c.
    27 * c - (3294 - 1332 * c) / 37 = 13
    999 * c - (3294 - 1332 * c) / 37 = 481
    36963 * c - 3294 + 1332 * c = 17797
    38295 * c = 21091
    c = 21091 / 38295
    c = 0.5508 (rounded to 4 decimal places)

    Step 4: Substitute the value of c into the expression for p from Step 1.
    p = (183 - 74 * 0.5508) / 37
    p = (183 - 40.7592) / 37
    p = 142.2408 / 37
    p = 3.8444 (rounded to 4 decimal places)

    Therefore, the solution to the system of equations is:
    p ≈ 3.8444
    c ≈ 0.5508

    ### Answer: p ≈ 3.8444, c ≈ 0.5508

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve for k and p, we can use the substitution method. Let's solve for k in the first equation and substitute it into the second equation to solve for p.

    From the first equation:
    (48 * k) - (30 * p) = 37
    48k - 30p = 37
    48k = 37 + 30p
    k = (37 + 30p) / 48

    Now, substitute the expression for k into the second equation:
    (16 * p) + (33 * k) = 105
    16p + 33 * ((37 + 30p) / 48) = 105
    16p + (1221 + 990p) / 48 = 105
    16p + 1221/48 + 990p/48 = 105
    16p + 25.4375 + 20.625p = 105
    36.625p + 25.4375 = 105
    36.625p = 79.5625
    p = 79.5625 / 36.625
    p = 2.1725

    Now that we have the value of p, we can substitute it back into the expression for k:
    k = (37 + 30p) / 48
    k = (37 + 30 * 2.1725) / 48
    k = (37 + 65.175) / 48
    k = 102.175 / 48
    k = 2.1286

    Therefore, the values of k and p are approximately:
    p ≈ 2.1725
    k ≈ 2.1286

    ### Answer: k ≈ 2.1286, p ≈ 2.1725

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

Correct? No

🔗 Problem

Problem: Consider the following pair of equations. 1. 75v - 69f = 14 2. 46f + 53v = 110 Determine the values for v and f that satisfy these simultaneous equations.

Gemma 2B:

    To solve for v and f in the given pair of simultaneous equations, we can use the substitution method.

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

    Step 1: Solve equation 2 for f in terms of v.
    46f + 53v = 110
    46f = 110 - 53v
    f = (110 - 53v) ÷ 46
    f = 110/46 - 53v/46
    f = 110/46 - 53/46 × v
    <<110/46=2.391304348, 53/46=1.152173913>>
    f = 2.391304348 - 1.152173913v

    Step 2: Substitute the expression for f into equation 1.
    75v - 69f = 14
    75v - 69(2.391304348 - 1.152173913v) = 14
    75v - 165 + 79.5v = 14
    154.5v - 165 = 14
    154.5v = 179
    v = 179 ÷ 154.5
    <<179/154.5=1.158576052>>
    v = 1.158576052

    Step 3: Substitute the value of v into the expression for f from step 1.
    f = 2.391304348 - 1.152173913v
    f = 2.391304348 - 1.152173913 × 1.158576052
    <<2.391304348-1.152173913*1.158576052=1.057471264>>
    f = 1.057471264

    Therefore, the values that satisfy the given simultaneous equations are:
    v ≈ 1.158576052
    f ≈ 1.057471264

    ### Answer: v ≈ 1.158576052, f ≈ 1.057471264

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, we can solve for o:
    7 * o + 13 * g = 131
    7 * o = 131 - 13 * g
    o = (131 - 13 * g) / 7

    Now, let's substitute this expression for o into the second equation:
    89 * g - 70 * ((131 - 13 * g) / 7) = 37

    Simplify the equation:
    89 * g - 10 * (131 - 13 * g) = 37
    89 * g - 1310 + 130 * g = 37
    219 * g - 1310 = 37
    219 * g = 1347
    g = 1347 / 219
    g = <<1347/219=6.15068493150685>>6.15068493150685

    Now that we have the value of g, we can substitute it back into the expression for o:
    o = (131 - 13 * 6.15068493150685) / 7
    o = (131 - 80) / 7
    o = 51 / 7
    o = <<51/7=7.28571428571429>>7.28571428571429

    ### Answer: o = 7.28571428571429, g = 6.15068493150685

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for d in terms of k in the first equation, then substitute it into the second equation to solve for k. Finally, we'll substitute the value of k back into the equation for d to find its value.

    1. Solve the first equation for d in terms of k:
       88 * d - 60 * k = 38
       88 * d = 60 * k + 38
       d = (60 * k + 38) / 88

    2. Substitute the expression for d into the second equation:
       33 * k - 11 * ((60 * k + 38) / 88) = 46
       33 * k - (660 * k + 418) / 88 = 46
       2904 * k - 660 * k - 418 = 4048
       2244 * k - 418 = 4048
       2244 * k = 4466
       k = 4466 / 2244
       <<4466/2244=1.99>>k = 1.99

    3. Substitute k = 1.99 into the equation for d:
       d = (60 * 1.99 + 38) / 88
       d = 157.4 / 88
       <<157.4/88=1.79>>d = 1.79

    Therefore, the solution to the system of equations is:
    ### Answer: d = 1.79, k = 1.99

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for z in terms of y using the second equation and then substitute it into the first equation to solve for y.

    Given equations:
    1. 24 * z + 72 * y = 139
    2. 2 * z + 82 * y = 96

    Step 1: Solve for z in terms of y using the second equation.
    2 * z + 82 * y = 96
    2 * z = 96 - 82 * y
    z = (96 - 82 * y) / 2
    z = 48 - 41 * y

    Step 2: Substitute the expression for z into the first equation and solve for y.
    24 * z + 72 * y = 139
    24 * (48 - 41 * y) + 72 * y = 139
    1152 - 984 * y + 72 * y = 139
    1152 - 912 * y = 139
    -912 * y = -1013
    y = -1013 / -912
    y = <<-1013/-912=1.11>>1.11 (rounded to two decimal places)

    Step 3: Substitute the value of y into the expression for z from Step 1.
    z = 48 - 41 * y
    z = 48 - 41 * 1.11
    z = 48 - 45.51
    z = <<48-45.51=2.49>>2.49 (rounded to two decimal places)

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

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    Given equations:
    1. (87 * o) - (29 * v) = 1
    2. (8 * v) + (30 * o) = 91

    Step 1: Solve for v in terms of o using equation 1.
    (87 * o) - (29 * v) = 1
    -29 * v = 1 - (87 * o)
    v = (1 - (87 * o)) / (-29)
    v = (87 * o - 1) / 29

    Step 2: Substitute the expression for v into equation 2.
    (8 * ((87 * o - 1) / 29)) + (30 * o) = 91

    Step 3: Solve for o.
    (8 * (87 * o - 1) / 29) + (30 * o) = 91
    (696 * o - 8) / 29 + (30 * o) = 91
    (696 * o - 8 + 870 * o) / 29 = 91
    (1566 * o - 8) / 29 = 91
    1566 * o - 8 = 91 * 29
    1566 * o - 8 = 2639
    1566 * o = 2647
    o = 2647 / 1566
    o = 1.69 (rounded to two decimal places)

    Step 4: Substitute the value of o into the expression for v from step 1.
    v = (87 * 1.69 - 1) / 29
    v = (147.03 - 1) / 29
    v = 146.03 / 29
    v = 5.04 (rounded to two decimal places)

    Therefore, the solution to the system of equations is:
    o ≈ 1.69
    v ≈ 5.04

    ### Answer: o ≈ 1.69, v ≈ 5.04

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations for h and p: The first equation is (23 * h) - (46 * p) = 58 The second equation is (11 * p) + (23 * h) = 124.

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    First equation: (23 * h) - (46 * p) = 58
    23h - 46p = 58
    23h = 58 + 46p
    h = (58 + 46p) / 23
    h = 58/23 + 2p

    Now, substitute the expression for h into the second equation:
    Second equation: (11 * p) + (23 * h) = 124
    11p + 23(58/23 + 2p) = 124
    11p + 58 + 46p = 124
    57p + 58 = 124
    57p = 66
    p = 66/57

    Now that we have the value of p, we can substitute it back into the expression for h:
    h = 58/23 + 2p
    h = 58/23 + 2(66/57)
    h = 58/23 + 132/57
    h = (58*57 + 132*23) / (23*57)
    h = 3306 + 3036 / 1311
    h = 6342 / 1311

    Therefore, the solution to the system of equations is:
    p = 66/57
    h = 6342/1311

    ### Answer: p = 66/57, h = 6342/1311

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

Correct? Yes

🔗 Problem

Problem: If the following system of equations represents the relationship between integers d and f, determine their values: 1) 19d + 6f = 120, 2) 90d - 42f = 69.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve equation 1) for d in terms of f, then substitute the expression for d into equation 2) to solve for f. Finally, we'll substitute the value of f back into equation 1) to find the value of d.

    Step 1: Solve equation 1) for d in terms of f.
    19d + 6f = 120
    19d = 120 - 6f
    d = (120 - 6f) ÷ 19
    d = (120 - 6f)/19

    Step 2: Substitute the expression for d into equation 2) and solve for f.
    90d - 42f = 69
    90((120 - 6f)/19) - 42f = 69
    (10800 - 540f)/19 - 42f = 69
    10800/19 - 540f/19 - 42f = 69
    (10800 - 540f - 798f)/19 = 69
    (10800 - 1338f)/19 = 69
    10800 - 1338f = 1311
    -1338f = -9489
    f = -9489/-1338
    f = <<-9489/-1338=7.1>>7

    Step 3: Substitute the value of f into equation 1) to find the value of d.
    19d + 6f = 120
    19d + 6(7) = 120
    19d + 42 = 120
    19d = 78
    d = 78/19
    d = <<78/19=4.1>>4

    Therefore, the values of d and f are:
    d = 4
    f = 7

    ### Answer: d = 4, f = 7

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

Correct? No

🔗 Problem

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:

    Sure, I can solve this system of linear equations using the substitution method. Let's solve for one of the variables in the first equation and substitute it into the second equation.

    Given equations:
    1) 42 * m + 55 * r = 127
    2) 66 * m - 52 * r = 44

    Step 1: Solve equation 1 for m in terms of r.
    42 * m + 55 * r = 127
    42 * m = 127 - 55 * r
    m = (127 - 55 * r) / 42

    Step 2: Substitute the expression for m into equation 2.
    66 * ((127 - 55 * r) / 42) - 52 * r = 44
    (66 * (127 - 55 * r)) / 42 - 52 * r = 44
    (8382 - 3630 * r) / 42 - 52 * r = 44
    199.5714 - 86.4286 * r - 52 * r = 44
    199.5714 - 138.4286 * r = 44
    -138.4286 * r = -155.5714
    r = 1.1238

    Step 3: Substitute the value of r into the expression for m.
    m = (127 - 55 * 1.1238) / 42
    m = (127 - 61.8095) / 42
    m = 65.1905 / 42
    m = 1.5522

    Therefore, the solution to the system of linear equations is:
    r ≈ 1.1238
    m ≈ 1.5522

    ### Answer: r ≈ 1.1238, m ≈ 1.5522

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of linear equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, let's solve for n:
    31n + 64w = 120
    31n = 120 - 64w
    n = (120 - 64w) / 31

    Now, substitute this expression for n into the second equation:
    57w + 71n = 173
    57w + 71((120 - 64w) / 31) = 173

    Simplify:
    57w + (8520 - 4544w) / 31 = 173
    1767w + 8520 - 4544w = 5363
    1767w - 4544w = 5363 - 8520
    -2777w = -3157
    w = 3157 / 2777
    w = <<3157/2777=1.136693554>>1.136693554...

    Now that we have the value of w, we can substitute it back into the expression for n:
    n = (120 - 64w) / 31
    n = (120 - 64(1.136693554)) / 31
    n = (120 - 72.748387456) / 31
    n = 47.251612544 / 31
    n = <<47.251612544/31=1.524245243>>1.524245243...

    Therefore, the values of n and w that satisfy the system of linear equations simultaneously are:
    w ≈ 1.136693554
    n ≈ 1.524245243

    ### Answer: w ≈ 1.136693554, n ≈ 1.524245243

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

Correct? No

🔗 Problem

Problem: Determine the values of u and a from the following system of linear equations: 1) 72 * a + 26 * u = 112 2) 76 * u - 77 * a = 31.

Gemma 2B:

    To solve this system of linear equations, we can use the substitution method. Let's solve equation (1) for u in terms of a, then substitute the result into equation (2) to solve for a. Finally, we'll substitute the value of a back into equation (1) to find u.

    From equation (1), we have:
    72 * a + 26 * u = 112
    26 * u = 112 - 72 * a
    u = (112 - 72 * a) / 26
    u = 56/13 - 36/13 * a

    Now, substitute this expression for u into equation (2):
    76 * (56/13 - 36/13 * a) - 77 * a = 31
    4256/13 - 2736/13 * a - 77 * a = 31
    4256/13 - (2736/13 + 1001) * a = 31
    4256/13 - (3737/13) * a = 31
    4256 - 3737 * a = 403
    519 - 3737 * a = 403
    -3737 * a = -116
    a = 116/3737
    a = 31/1000

    Now that we have the value of a, let's substitute it back into the equation for u:
    u = 56/13 - 36/13 * (31/1000)
    u = 56/13 - 1116/(13000)
    u = 728/169 - 93/1690
    u = 635/169

    Therefore, the values of u and a are:
    a = 31/1000
    u = 635/169

    ### Answer: a = 31/1000, u = 635/169

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of b and n: the first equation is 54n + 18b = 143 and the second equation is 87n - 93b = 48.

Gemma 2B:

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

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

    Step 1: Solve equation 1 for n in terms of b.
    54n + 18b = 143
    54n = 143 - 18b
    n = (143 - 18b) / 54
    n = (143/54) - (1/3)b

    Step 2: Substitute the expression for n into equation 2.
    87n - 93b = 48
    87((143/54) - (1/3)b) - 93b = 48
    (87 × 143/54) - (87/3)b - 93b = 48
    (87 × 143/54) - (87/3 + 93)b = 48
    (87 × 143/54) - (29 + 93)b = 48
    (87 × 143/54) - 122b = 48

    Step 3: Solve the equation for b.
    (87 × 143/54) - 122b = 48
    -122b = 48 - (87 × 143/54)
    -122b = 48 - 231
    -122b = -183
    b = -183 / -122
    b = 3/2 or 1.5

    Step 4: Substitute the value of b into the expression for n from step 1.
    n = (143/54) - (1/3)b
    n = (143/54) - (1/3) × (3/2)
    n = (143/54) - (1/2)
    n = (143/54) - (27/54)
    n = (143 - 27)/54
    n = 116/54

    Therefore, the solution to the system of equations is:
    b = 3/2 or 1.5
    n = 116/54

    ### Answer: b = 3/2 or 1.5, n = 116/54

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

Correct? No

🔗 Problem

Problem: Given the following system of equations about a company's production economics, where e represents the cost of electricity and p is the cost of petroleum: 53 * e + 7 * p = 103 69 * e - 88 * p = 6 What are the costs of electricity (e) and petroleum (p)?

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    From the first equation, we can solve for p in terms of e:
    53 * e + 7 * p = 103
    7 * p = 103 - 53 * e
    p = (103 - 53 * e) / 7

    Now, substitute this expression for p into the second equation:
    69 * e - 88 * p = 6
    69 * e - 88 * ((103 - 53 * e) / 7) = 6
    69 * e - 88 * (103 / 7) + 88 * (53 * e / 7) = 6
    69 * e - 1298 / 7 + 4664 * e / 7 = 6
    483 * e / 7 + 4664 * e / 7 - 1298 / 7 = 6
    5147 * e / 7 - 1298 / 7 = 6
    5147 * e = 42 + 1298
    5147 * e = 1340
    e = 1340 / 5147
    e = <<1340/5147=0.2603>>0.2603

    Now that we have the value of e, we can substitute it back into the equation for p:
    p = (103 - 53 * e) / 7
    p = (103 - 53 * 0.2603) / 7
    p = (103 - 13.7959) / 7
    p = 89.2041 / 7
    p = <<89.2041/7=12.7434>>12.7434

    ### Answer: The cost of electricity (e) is approximately 0.2603, and the cost of petroleum (p) is approximately 12.7434.

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

Correct? No

🔗 Problem

Problem: Consider the following pair of linear equations. First equation: ((18 * p) - (7 * s)) = 21 Second equation: ((51 * s) - (24 * p)) = 14 Find the values of p and s that satisfy both equations.

Gemma 2B:

    To solve this system of linear equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    First equation: ((18 * p) - (7 * s)) = 21
    18p - 7s = 21
    18p = 21 + 7s
    p = (21 + 7s) / 18
    p = (7s + 21) / 18

    Now, substitute the expression for p into the second equation:

    Second equation: ((51 * s) - (24 * p)) = 14
    51s - 24 * ((7s + 21) / 18) = 14
    51s - (24 * (7s + 21)) / 18 = 14
    51s - (168s + 504) / 18 = 14
    51s - (168s/18 + 504/18) = 14
    51s - (28s/3 + 84/3) = 14
    153s/3 - 28s/3 - 84/3 = 14
    (153s - 28s - 84) / 3 = 14
    (125s - 84) / 3 = 14
    125s - 84 = 42
    125s = 126
    s = 126 / 125

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

    p = (7s + 21) / 18
    p = (7 * (126/125) + 21) / 18
    p = (882/125 + 21) / 18
    p = (882/125 + 2625/125) / 18
    p = 3507 / (125 * 18)
    p = 3507 / 2250

    Therefore, the values of p and s that satisfy both equations are:
    s = 126 / 125 = 1.008
    p = 3507 / 2250 ≈ 1.558

    ### Answer: p ≈ 1.558, s = 1.008

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

Correct? No

🔗 Problem

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:

    Let's solve this step by step using the substitution method.

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

    Step 1: Solve equation 1 for c in terms of w.
    59c - 34w = 84
    59c = 84 + 34w
    c = (84 + 34w) / 59
    c = 84/59 + (34/59)w

    Step 2: Substitute the expression for c into equation 2.
    11c + 42w = 98
    11(84/59 + (34/59)w) + 42w = 98
    (924/59 + (374/59)w) + 42w = 98
    924/59 + (374/59)w + 42w = 98
    924/59 + (374/59 + 42)w = 98
    924/59 + (2478/59 + 42)w = 98
    924/59 + (2520/59)w = 98

    Step 3: Solve the equation for w.
    924/59 + (2520/59)w = 98
    (2520/59)w = 98 - 924/59
    w = (98 - 924/59) / (2520/59)
    w = (5782 - 924) / 2520
    w = 4858 / 2520
    w = 1.927

    Step 4: Substitute the value of w into the expression for c.
    c = 84/59 + (34/59)w
    c = 84/59 + (34/59)(1.927)
    c = 1.424 + 1.111
    c = 2.535

    Therefore, the solution to the system of equations is:
    c ≈ 2.535
    w ≈ 1.927

    ### Answer: c ≈ 2.535, w ≈ 1.927

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of f and z: First equation is 45 * f + 42 * z = 101, Second equation is 60 * z - 42 * f = 13.

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

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

    Step 1: Solve for f in terms of z using the first equation.
    45f + 42z = 101
    45f = 101 - 42z
    f = (101 - 42z) ÷ 45
    f = (101 - 42z) / 45

    Step 2: Substitute the expression for f into the second equation.
    60z - 42((101 - 42z) / 45) = 13

    Step 3: Solve for z.
    60z - 42(101 - 42z) / 45 = 13
    60z - (42 × 101 - 42 × 42z) / 45 = 13
    60z - (4242 - 1764z) / 45 = 13
    2700z - 4242 + 1764z = 585
    4464z - 4242 = 585
    4464z = 4827
    z = 4827 / 4464
    z = 1.0813 (rounded to 4 decimal places)

    Step 4: Substitute the value of z into the expression for f from Step 1.
    f = (101 - 42 × 1.0813) / 45
    f = (101 - 45.4146) / 45
    f = 55.5854 / 45
    f = 1.2352 (rounded to 4 decimal places)

    ### Answer: f = 1.2352, z = 1.0813

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

Correct? No

🔗 Problem

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

Gemma 2B:

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

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

    Step 1: Solve the first equation for p in terms of u.
    26 * p - 21 * u = 6
    26 * p = 6 + 21 * u
    p = (6 + 21 * u) / 26
    p = (3 + 21 * u) / 13

    Step 2: Substitute the expression for p into the second equation.
    15 * p + 58 * u = 83
    15 * ((3 + 21 * u) / 13) + 58 * u = 83
    (45 + 315 * u) / 13 + 58 * u = 83

    Step 3: Solve the equation for u.
    (45 + 315 * u) / 13 + 58 * u = 83
    45 / 13 + 315 * u / 13 + 58 * u = 83
    45 / 13 + (315 / 13 + 58) * u = 83
    45 / 13 + (315 / 13 + 754 / 13) * u = 83
    45 / 13 + 1069 * u / 13 = 83
    1069 * u / 13 = 83 - 45 / 13
    1069 * u / 13 = 1034 / 13
    u = 1034 / 1069
    u = <<1034/1069=0.9672>>0.9672

    Step 4: Substitute the value of u into the expression for p.
    p = (3 + 21 * u) / 13
    p = (3 + 21 * 0.9672) / 13
    p = 23.3112 / 13
    p = <<23.3112/13=1.7932>>1.7932

    Therefore, the values of p and u are:
    u = 0.9672
    p = 1.7932

    ### Answer: p = 1.7932, u = 0.9672

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

Correct? No

🔗 Problem

Problem: Solve for the values of n and s using these two simultaneous equations: 1) 92 * n - 38 * s = 62 2) 59 * n + 8 * s = 75

Gemma 2B:

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

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

    Step 1: Solve equation (2) for n in terms of s.
    59 * n + 8 * s = 75
    59 * n = 75 - 8 * s
    n = (75 - 8 * s) / 59

    Step 2: Substitute the expression for n from step 1 into equation (1).
    92 * ((75 - 8 * s) / 59) - 38 * s = 62
    (92 * (75 - 8 * s)) / 59 - 38 * s = 62
    (6900 - 736 * s) / 59 - 38 * s = 62
    6900 / 59 - (736 * s) / 59 - 38 * s = 62
    6900 / 59 - (736 * s + 2242 * s) / 59 = 62
    6900 / 59 - (2978 * s) / 59 = 62
    6900 - 2978 * s = 3658
    3922 = 2978 * s
    s = 3922 / 2978

    Step 3: Calculate the value of s.
    s = 3922 / 2978 ≈ 1.32 (rounded to two decimal places)

    Step 4: Substitute the value of s into the expression for n from step 1 to find the value of n.
    n = (75 - 8 * 1.32) / 59
    n = (75 - 10.56) / 59
    n = 64.44 / 59 ≈ 1.09 (rounded to two decimal places)

    Therefore, the values of n and s that satisfy the given simultaneous equations are approximately:
    n ≈ 1.09
    s ≈ 1.32

    ### Answer: n ≈ 1.09, s ≈ 1.32

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

Correct? No

🔗 Problem

Problem: Solving the following pairs of simultaneous equations, what would be the values for e and z? The equations are: 87 * z - 35 * e = 63 and 81 * z + 58 * e = 154.

Gemma 2B:

    To solve for e and z in the given simultaneous equations, we can use the substitution method.

    Given equations:
    87 * z - 35 * e = 63 ... (1)
    81 * z + 58 * e = 154 ... (2)

    Step 1: Solve equation (1) for z in terms of e.
    87 * z - 35 * e = 63
    87 * z = 63 + 35 * e
    z = (63 + 35 * e) / 87 ... (3)

    Step 2: Substitute the expression for z from equation (3) into equation (2).
    81 * ((63 + 35 * e) / 87) + 58 * e = 154
    (81 * (63 + 35 * e)) / 87 + 58 * e = 154
    (5103 + 2835 * e) / 87 + 58 * e = 154
    5103 / 87 + (2835 * e) / 87 + 58 * e = 154
    58.66 + 32.59 * e + 58 * e = 154
    58.66 + 90.59 * e = 154
    90.59 * e = 95.34
    e = 95.34 / 90.59
    e ≈ 1.05

    Step 3: Substitute the value of e into equation (3) to find z.
    z = (63 + 35 * 1.05) / 87
    z = (63 + 36.75) / 87
    z = 99.75 / 87
    z ≈ 1.15

    Therefore, the values for e and z are approximately:
    e ≈ 1.05
    z ≈ 1.15

    ### Answer: e ≈ 1.05, z ≈ 1.15

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

Correct? No

🔗 Problem

Problem: To find the values of h and t, solve the following system of simultaneous linear equations: (11 * t) + (44 * h) = 110 (98 * t) - (36 * h) = 18.

Gemma 2B:

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

    Given equations:
    (11 * t) + (44 * h) = 110 ... (1)
    (98 * t) - (36 * h) = 18 ... (2)

    Step 1: Solve equation (1) for t in terms of h.
    (11 * t) + (44 * h) = 110
    11 * t = 110 - 44 * h
    t = (110 - 44 * h) / 11
    t = 10 - 4 * h ... (3)

    Step 2: Substitute the expression for t from equation (3) into equation (2).
    (98 * t) - (36 * h) = 18
    98 * (10 - 4 * h) - 36 * h = 18
    980 - 392 * h - 36 * h = 18
    980 - 428 * h = 18
    -428 * h = -962
    h = 962 / 428
    h = 2.25

    Step 3: Substitute the value of h into equation (3) to find the value of t.
    t = 10 - 4 * h
    t = 10 - 4 * 2.25
    t = 10 - 9
    t = 1

    Therefore, the solution to the system of simultaneous linear equations is:
    h = 2.25
    t = 1

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

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

Correct? No

🔗 Problem

Problem: Solve the following system of linear equations to find the values of b and o: ((69 * b) - (50 * o)) = 55 ((41 * o) + (18 * b)) = 85.

Gemma 2B:

    To solve the system of linear equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    From the first equation: ((69 * b) - (50 * o)) = 55
    69b - 50o = 55
    69b = 55 + 50o
    b = (55 + 50o) / 69

    Now, substitute the expression for b into the second equation:
    ((41 * o) + (18 * b)) = 85
    41o + 18 * ((55 + 50o) / 69) = 85
    41o + (990 + 900o) / 69 = 85
    41o + (1890o + 990) / 69 = 85
    2769o + 990 = 5865
    2769o = 4875
    o = 4875 / 2769

    Now that we have the value of o, we can substitute it back into the expression for b:
    b = (55 + 50o) / 69
    b = (55 + 50 * (4875 / 2769)) / 69
    b = (55 + 243750 / 2769) / 69
    b = (151745 / 2769) / 69

    Simplifying the fractions:
    o = 4875 / 2769 ≈ 1.76
    b = (151745 / 2769) / 69 ≈ 0.79

    ### Answer: o ≈ 1.76, b ≈ 0.79

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations to find the values of p and q: 67 * q - 80 * p = 65, and 6 * q + 5 * p = 177.

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the second equation and then substitute it into the first equation.

    From the second equation, 6 * q + 5 * p = 177, let's solve for p:
    6 * q + 5 * p = 177
    5 * p = 177 - 6 * q
    p = (177 - 6 * q) / 5

    Now, substitute the expression for p into the first equation:
    67 * q - 80 * p = 65
    67 * q - 80 * ((177 - 6 * q) / 5) = 65
    67 * q - 80 * (177 / 5 - 6 * q / 5) = 65
    67 * q - 80 * (35.4 - 1.2 * q) = 65
    67 * q - 2832 + 96 * q = 65
    163 * q - 2832 = 65
    163 * q = 2897
    q = 2897 / 163
    q = <<2897/163=17.77>>17.77 (rounded to two decimal places)

    Now that we have the value of q, we can substitute it into the equation for p:
    p = (177 - 6 * q) / 5
    p = (177 - 6 * 17.77) / 5
    p = (177 - 106.62) / 5
    p = 70.38 / 5
    p = <<70.38/5=14.08>>14.08 (rounded to two decimal places)

    ### Answer: p = 14.08, q = 17.77

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

Correct? No

🔗 Problem

Problem: Find the values of k and f that satisfy the following system of equations: 10 * k + 48 * f = 161 33 * k - 41 * f = 31.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for k in the second equation and substitute it into the first equation to solve for f.

    From the second equation:
    33 * k - 41 * f = 31
    33 * k = 31 + 41 * f
    k = (31 + 41 * f) / 33

    Substitute this value of k into the first equation:
    10 * k + 48 * f = 161
    10 * ((31 + 41 * f) / 33) + 48 * f = 161
    (310 + 410 * f) / 33 + 48 * f = 161
    310 / 33 + 410 * f / 33 + 48 * f = 161
    310 / 33 + (410 / 33 + 48) * f = 161
    310 / 33 + (410 / 33 + 1584 / 33) * f = 161
    310 / 33 + 1994 * f / 33 = 161
    310 + 1994 * f = 161 * 33
    310 + 1994 * f = 5313
    1994 * f = 5313 - 310
    1994 * f = 5003
    f = 5003 / 1994
    f = <<5003/1994=2.51>>2.51

    Now, substitute the value of f into the second equation to find k:
    33 * k - 41 * f = 31
    33 * k - 41 * 2.51 = 31
    33 * k - 102.91 = 31
    33 * k = 31 + 102.91
    33 * k = 133.91
    k = 133.91 / 33
    k = <<133.91/33=4.06>>4.06

    ### Answer: k = 4.06, f = 2.51

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of l and n: The first equation is 31 = 58 * l - n The second equation is 103 = n + 73 * l

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for n in the first equation and substitute it into the second equation to solve for l.

    1. Solve the first equation for n:
       31 = 58 * l - n
       31 - 58 * l = -n
       n = 58 * l - 31

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

    3. Solve the equation for l:
       103 + 31 = 131 * l
       134 = 131 * l
       l = 134 / 131
       <<134/131=1.022900763358779>>l ≈ 1.023

    4. Substitute the value of l into the expression for n:
       n = 58 * l - 31
       n = 58 * 1.023 - 31
       <<58*1.023-31=28.334>>n ≈ 28.334

    ### Answer: l ≈ 1.023, n ≈ 28.334

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

Correct? No

🔗 Problem

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

Gemma 2B:

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

    Step 1: Solve the first equation for y in terms of j.
    29y - 28j = 39
    29y = 39 + 28j
    y = (39 + 28j) / 29

    Step 2: Substitute the expression for y into the second equation.
    25y + 37j = 116
    25((39 + 28j) / 29) + 37j = 116

    Step 3: Simplify the equation and solve for j.
    25(39 + 28j) + 1073j = 3364
    975 + 700j + 1073j = 3364
    1773j = 2389
    j = 2389 / 1773
    j = 1.347 (rounded to three decimal places)

    Step 4: Substitute the value of j into the expression for y from Step 1.
    y = (39 + 28j) / 29
    y = (39 + 28(1.347)) / 29
    y = 76.716 / 29
    y = 2.645 (rounded to three decimal places)

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

    ### Answer: y ≈ 2.645, j ≈ 1.347

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

Correct? No

🔗 Problem

Problem: Given the following system of equations, solve for u and a: The first equation is 93u - 85a = 74 The second equation is 96u + 17a = 186.

Gemma 2B:

    To solve for u and a, we can use the substitution method or the elimination method. Let's use the substitution method here.

    Step 1: Solve the first equation for u in terms of a.
    93u - 85a = 74
    93u = 74 + 85a
    u = (74 + 85a) ÷ 93
    u = (74 + 85a) / 93

    Step 2: Substitute the expression for u into the second equation.
    96u + 17a = 186
    96((74 + 85a) / 93) + 17a = 186

    Step 3: Simplify the equation and solve for a.
    (96(74 + 85a) / 93) + 17a = 186
    (7104 + 8160a) / 93 + 17a = 186
    7104 / 93 + 8160a / 93 + 17a = 186
    76.387 + 87.742a + 17a = 186
    104.742a = 109.613
    a = 109.613 / 104.742
    a ≈ 1.046

    Step 4: Substitute the value of a into the expression for u from Step 1.
    u = (74 + 85a) / 93
    u = (74 + 85(1.046)) / 93
    u = (74 + 88.91) / 93
    u = 162.91 / 93
    u ≈ 1.752

    Therefore, the solution to the system of equations is:
    a ≈ 1.046
    u ≈ 1.752

    ### Answer: u ≈ 1.752, a ≈ 1.046

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

Correct? No

🔗 Problem

Problem: Given the following pair of simultaneous equations, please find out the values of u and g: 97 * u - 61 * g = 15 85 * u + 23 * g = 186

Gemma 2B:

    To solve for u and g, we can use the substitution method. Let's solve for u in the first equation and substitute it into the second equation to solve for g.

    From the first equation, we have:
    97 * u - 61 * g = 15
    97 * u = 61 * g + 15
    u = (61 * g + 15) / 97

    Now, let's substitute this expression for u into the second equation:
    85 * u + 23 * g = 186
    85 * ((61 * g + 15) / 97) + 23 * g = 186

    Simplify:
    (5185 * g + 1275) / 97 + 23 * g = 186
    5185 * g + 1275 + 2231 * g = 18062
    7416 * g + 1275 = 18062
    7416 * g = 16787
    g = 16787 / 7416
    g = <<16787/7416=2.26>>2.26 (rounded to two decimal places)

    Now that we have the value of g, we can substitute it back into the equation for u:
    u = (61 * g + 15) / 97
    u = (61 * 2.26 + 15) / 97
    u = 152.86 / 97
    u = <<152.86/97=1.58>>1.58 (rounded to two decimal places)

    ### Answer: u = 1.58, g = 2.26

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

Correct? No

🔗 Problem

Problem: Determine the values of r and b that satisfy the following system of equations. The first equation is (17 * r) + (67 * b) = 103. The second equation is (85 * r) + (67 * b) = 187.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one of the variables in the first equation and substitute it into the second equation.

    From the first equation, let's solve for r:
    (17 * r) + (67 * b) = 103
    17r = 103 - 67b
    r = (103 - 67b) / 17

    Now, substitute this expression for r into the second equation:
    (85 * r) + (67 * b) = 187
    85 * ((103 - 67b) / 17) + 67b = 187

    Simplify:
    (8755 - 5695b) / 17 + 67b = 187
    8755 - 5695b + 1139b = 187 * 17
    8755 - 4556b = 3179
    -4556b = -5576
    b = -5576 / -4556
    b = <<-5576/-4556=1.224>>1.224

    Now that we have the value of b, we can substitute it back into the expression for r:
    r = (103 - 67b) / 17
    r = (103 - 67 * 1.224) / 17
    r = (103 - 82.008) / 17
    r = 20.992 / 17
    r = <<20.992/17=1.235>>1.235

    ### Answer: r = 1.235, b = 1.224

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

Correct? No

🔗 Problem

Problem: Given the following system of equations, solve for the values of t and n: 1. (85 * n) - (54 * t) = 41 2. (38 * t) + (30 * n) = 126.

Gemma 2B:

    To solve for the values of t and n, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    From equation 1, solve for n in terms of t:
    (85 * n) - (54 * t) = 41
    (85 * n) = (54 * t) + 41
    n = ((54 * t) + 41) / 85

    Now, substitute the expression for n into equation 2:
    (38 * t) + (30 * (((54 * t) + 41) / 85)) = 126

    Simplify:
    (38 * t) + ((1620 * t) + 1230) / 85 = 126
    (3230 * t) + 1230 = 10710
    3230t = 9480
    t = 9480 / 3230
    t ≈ 2.9350931677018633540372670807453 <<38*2.9350931677018633540372670807453+30*((54*2.9350931677018633540372670807453+41)/85)≈126>>

    Now that we have the value of t, we can substitute it back into the equation for n:
    n = ((54 * 2.9350931677018633540372670807453) + 41) / 85
    n ≈ 2.0117647058823529411764705882353 <<85*2.0117647058823529411764705882353-54*2.9350931677018633540372670807453≈41>>

    Therefore, the solution to the system of equations is:
    t ≈ 2.9350931677018633540372670807453
    n ≈ 2.0117647058823529411764705882353

    ### Answer: t ≈ 2.9350931677018633540372670807453, n ≈ 2.0117647058823529411764705882353

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

Correct? No

🔗 Problem

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:

    Let's solve this system of equations step by step:

    1) 75u - 69q = 5
    2) 48u + 25q = 140

    Step 1: Multiply equation (1) by 48 and equation (2) by 75.
    48(75u - 69q = 5) → 3600u - 3312q = 240
    75(48u + 25q = 140) → 3600u + 1875q = 10500

    Step 2: Subtract the new equation (1) from the new equation (2) to eliminate the variable u.
    (3600u + 1875q = 10500) - (3600u - 3312q = 240)
    1875q + 3312q = 10500 - 240
    5187q = 10260

    Step 3: Solve for q by dividing both sides by 5187.
    q = 10260 ÷ 5187
    q = 1.9779... ≈ 1.98

    Step 4: Substitute the value of q into equation (1) to solve for u.
    75u - 69(1.98) = 5
    75u - 136.62 = 5
    75u = 141.62
    u = 141.62 ÷ 75
    u = 1.8882... ≈ 1.89

    Therefore, the solution to the system of equations is:
    u ≈ 1.89
    q ≈ 1.98

    ### Answer: u ≈ 1.89, q ≈ 1.98

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

Correct? No

🔗 Problem

Problem: Determine the values of v and x from the following system of equations: ((20 * v) + (17 * x)) = 149 and ((24 * x) - (49 * v)) = 22.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for v in the first equation and substitute it into the second equation to solve for x.

    1. Solve for v in the first equation:
       ((20 * v) + (17 * x)) = 149
       (20 * v) = 149 - (17 * x)
       v = (149 - (17 * x)) / 20

    2. Substitute the expression for v into the second equation:
       ((24 * x) - (49 * ((149 - (17 * x)) / 20))) = 22
       ((24 * x) - (49 * (149 - (17 * x)) / 20)) = 22
       ((24 * x) - ((7301 - (833 * x)) / 20)) = 22
       (480 * x) - (7301 - (833 * x)) = 440
       (480 * x) - 7301 + (833 * x) = 440
       (1313 * x) - 7301 = 440
       (1313 * x) = 7741
       x = 7741 / 1313
       x = 5.896 (rounded to three decimal places)

    3. Substitute the value of x into the expression for v:
       v = (149 - (17 * 5.896)) / 20
       v = (149 - 100.232) / 20
       v = 48.768 / 20
       v = 2.438 (rounded to three decimal places)

    Therefore, the values of v and x are approximately:
    v ≈ 2.438
    x ≈ 5.896

    ### Answer: v ≈ 2.438, x ≈ 5.896

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

Correct? No

🔗 Problem

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

Gemma 2B:

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

    Given equations:
    39 * t + 41 * g = 147 ... (1)
    94 * t - 91 * g = 42 ... (2)

    Step 1: Solve equation (1) for t in terms of g.
    39 * t + 41 * g = 147
    39 * t = 147 - 41 * g
    t = (147 - 41 * g) / 39

    Step 2: Substitute the expression for t into equation (2).
    94 * ((147 - 41 * g) / 39) - 91 * g = 42

    Step 3: Simplify the equation and solve for g.
    (94 * (147 - 41 * g)) / 39 - 91 * g = 42
    (13818 - 3854 * g) / 39 - 91 * g = 42
    354.3076923 - 98.82051282 * g - 91 * g = 42
    354.3076923 - 189.8205128 * g = 42
    -189.8205128 * g = -312.3076923
    g = 1.645390071

    Step 4: Substitute the value of g into the expression for t from Step 1.
    t = (147 - 41 * 1.645390071) / 39
    t = (147 - 67.46099291) / 39
    t = 79.53900709 / 39
    t = 2.039974797

    Therefore, the solution to the system of equations is:
    g = 1.645390071
    t = 2.039974797

    ### Answer: t = 2.039974797, g = 1.645390071

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other in the first equation, then substitute it into the second equation.

    1. Solve the first equation for c in terms of o:
       6c + 55o = 112
       6c = 112 - 55o
       c = (112 - 55o) / 6
       c = 56/3 - 55o/6

    2. Substitute the expression for c into the second equation:
       55c - 48o = 36
       55(56/3 - 55o/6) - 48o = 36
       (55 × 56/3) - (55 × 55o/6) - 48o = 36
       1026.67 - 504.17o - 48o = 36
       1026.67 - 552.17o = 36
       -552.17o = -990.67
       o = 990.67/552.17
       o = 1.79 (rounded to two decimal places)

    3. Substitute o = 1.79 into the expression for c:
       c = 56/3 - 55o/6
       c = 56/3 - 55 × 1.79/6
       c = 18.67 - 16.41
       c = 2.26 (rounded to two decimal places)

    Therefore, the values that satisfy the system of equations are approximately:
    c = 2.26
    o = 1.79

    ### Answer: c = 2.26, o = 1.79

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    Given equations:
    8w + 68u = 170 ... (1)
    71w - 94u = 8 ... (2)

    Step 1: Solve for w in equation (1):
    8w + 68u = 170
    8w = 170 - 68u
    w = (170 - 68u) ÷ 8
    w = 21.25 - 8.5u ... (3)

    Step 2: Substitute w from equation (3) into equation (2):
    71(21.25 - 8.5u) - 94u = 8
    1508.75 - 603.5u - 94u = 8
    1508.75 - 697.5u = 8
    -697.5u = 8 - 1508.75
    -697.5u = -1500.75
    u = -1500.75 ÷ -697.5
    u = 2.15

    Step 3: Substitute u = 2.15 into equation (3) to find w:
    w = 21.25 - 8.5(2.15)
    w = 21.25 - 18.275
    w = 2.975

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

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

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, solve for f:
    68*r + 8*f = 92
    8*f = 92 - 68*r
    f = (92 - 68*r) / 8
    f = (92/8) - (68/8)*r
    f = 11.5 - 8.5*r

    Now, substitute the expression for f into the second equation:
    97*f + 38*r = 182
    97*(11.5 - 8.5*r) + 38*r = 182
    1115.5 - 824.5*r + 38*r = 182
    1115.5 - 786.5*r = 182
    -786.5*r = -933.5
    r = 933.5 / 786.5
    r = 1.1869436201780415 ≈ 1.19

    Now that we have the value of r, we can substitute it back into the expression for f:
    f = 11.5 - 8.5*r
    f = 11.5 - 8.5*1.1869436201780415
    f = 11.5 - 10.089020771513353
    f = 1.4109792284866471 ≈ 1.41

    Therefore, the solution to the system of equations is:
    r ≈ 1.19
    f ≈ 1.41

    ### Answer: r ≈ 1.19, f ≈ 1.41

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

Correct? No

🔗 Problem

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:

    Let's solve this problem step by step using the given equations:

    Equation 1: 67d - 37r = 50
    Equation 2: 39r - 21d = 16

    Step 1: Multiply equation 2 by 2 to eliminate the decimal fraction when adding the equations later.
    2 × (39r - 21d = 16)
    78r - 42d = 32

    Step 2: Add equation 1 and the modified equation 2 to eliminate the variable d.
    (67d - 37r = 50) + (78r - 42d = 32)
    25d + 41r = 82

    Step 3: Substitute the value of 25d from equation 1 into the equation from step 2.
    Equation 1: 67d - 37r = 50
    25d = 50 + 37r
    Substituting in the equation from step 2:
    (50 + 37r) + 41r = 82
    50 + 78r = 82

    Step 4: Solve for r.
    78r = 82 - 50
    78r = 32
    r = 32 ÷ 78
    r = 0.4102564...
    Rounded to the nearest cent, r ≈ $0.41

    Step 5: Substitute the value of r in equation 1 to solve for d.
    67d - 37 × 0.41 = 50
    67d - 15.17 = 50
    67d = 65.17
    d = 65.17 ÷ 67
    d = 0.9726865...
    Rounded to the nearest cent, d ≈ $0.97

    Therefore, the cost of each ragdoll (r) is approximately $0.41, and the cost of each apple (d) is approximately $0.97.

    ### Answer: r ≈ $0.41, d ≈ $0.97

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

Correct? No

🔗 Problem

Problem: To find the values of x and a, solve the following system of simultaneous linear equations: First equation: 163 equals 50 times x plus a Second equation: 24 equals 98 times x minus 84 times a.

Gemma 2B:

    To solve this system of simultaneous linear equations, we can use the substitution method. Let's solve for x in the second equation and substitute it into the first equation to find the value of a. Then, we can use the value of a to find the value of x.

    1. Solve the second equation for x:
       24 = 98x - 84a
       24 + 84a = 98x
       (24 + 84a) / 98 = x
       x = (24 + 84a) / 98

    2. Substitute the expression for x into the first equation:
       163 = 50x + a
       163 = 50((24 + 84a) / 98) + a
       163 = (1200 + 4200a) / 98 + a
       163 = (1200 / 98) + (4200a / 98) + a
       163 = 12.24 + 42.86a + a
       163 = 12.24 + 43.86a
       163 - 12.24 = 43.86a
       150.76 = 43.86a
       a = 150.76 / 43.86
       a ≈ 3.44

    3. Substitute the value of a into the expression for x:
       x = (24 + 84a) / 98
       x = (24 + 84 × 3.44) / 98
       x = (24 + 288.96) / 98
       x = 312.96 / 98
       x ≈ 3.19

    Therefore, the values of x and a that satisfy the given system of simultaneous linear equations are:
    ### Answer: x ≈ 3.19 and a ≈ 3.44

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    Given equations:
    67 * p + 18 * b = 110 ... (1)
    82 * b - 44 * p = 58 ... (2)

    Step 1: Solve for b in terms of p using equation (1).
    67 * p + 18 * b = 110
    18 * b = 110 - 67 * p
    b = (110 - 67 * p) / 18 ... (3)

    Step 2: Substitute the expression for b from equation (3) into equation (2).
    82 * ((110 - 67 * p) / 18) - 44 * p = 58
    (9020 - 5494 * p) / 18 - 44 * p = 58
    501.11 - 305.22 * p - 44 * p = 58
    501.11 - 349.22 * p = 58
    -349.22 * p = -443.11
    p = -443.11 / -349.22
    p = <<-443.11/-349.22=1.27>>1.27

    Step 3: Substitute the value of p into equation (3) to solve for b.
    b = (110 - 67 * 1.27) / 18
    b = (110 - 85.09) / 18
    b = 24.91 / 18
    b = <<24.91/18=1.38>>1.38

    Therefore, the solution to the system of equations is:
    p = 1.27
    b = 1.38

    ### Answer: p = 1.27, b = 1.38

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

Correct? No

🔗 Problem

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:

    Let's solve this step by step:

    1. Let's write the equations based on the given information:
       - 63c - 36e = 27
       - 8c + 32e = 69

    2. Multiply the second equation by 2 to make the coefficients of e equal:
       - 63c - 36e = 27
       - 16c + 64e = 138

    3. Add the two equations to eliminate e:
       - 79c = 165

    4. Solve for c:
       - c = 165 ÷ 79
       - c = 165/79 ≈ 2.089 (rounded to 3 decimal places)

    5. Substitute the value of c in one of the original equations to solve for e. Let's use the first equation:
       - 63 × 2.089 - 36e = 27
       - 131.607 - 36e = 27
       - -36e = -104.607
       - e = 104.607 ÷ 36
       - e = 104.607/36 ≈ 2.906 (rounded to 3 decimal places)

    Therefore, the values of c and e are:
    - c ≈ 2.089
    - e ≈ 2.906

    ### Answer: c ≈ 2.089, e ≈ 2.906

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

Correct? No

🔗 Problem

Problem: Solve the following system of linear equations to find the values of x and p: 26 * x + 29 * p = 107 61 * x - 51 * p = 4

Gemma 2B:

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

    Given equations:
    26 * x + 29 * p = 107 ... (1)
    61 * x - 51 * p = 4 ... (2)

    Step 1: Solve equation (1) for x in terms of p.
    26 * x + 29 * p = 107
    26 * x = 107 - 29 * p
    x = (107 - 29 * p) / 26

    Step 2: Substitute the expression for x into equation (2).
    61 * ((107 - 29 * p) / 26) - 51 * p = 4
    (6527 - 1769 * p) / 26 - 51 * p = 4
    6527 - 1769 * p - 1326 * p = 104
    6527 - 3095 * p = 104
    -3095 * p = -6423
    p = 2.076 ... (rounded to 3 decimal places)

    Step 3: Substitute the value of p into the expression for x from Step 1.
    x = (107 - 29 * 2.076) / 26
    x = (107 - 60.204) / 26
    x = 46.796 / 26
    x = 1.800 ... (rounded to 3 decimal places)

    Therefore, the solution to the system of linear equations is:
    x ≈ 1.800
    p ≈ 2.076

    ### Answer: x ≈ 1.800, p ≈ 2.076

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

Correct? No

🔗 Problem

Problem: Find the values for g and l from the system of equations below. The first equation is: ((29 * l) + (17 * g)) = 68. The second equation is: ((31 * g) - (10 * l)) = 17.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, let's solve for l:
    ((29 * l) + (17 * g)) = 68
    29l + 17g = 68
    29l = 68 - 17g
    l = (68 - 17g) / 29

    Now, substitute the expression for l into the second equation:
    ((31 * g) - (10 * ((68 - 17g) / 29))) = 17

    Simplify:
    31g - (10 * (68 - 17g) / 29) = 17
    31g - ((680 - 170g) / 29) = 17
    31g - (680/29 - 170g/29) = 17
    31g - 680/29 + 170g/29 = 17
    (31g + 170g/29) - 680/29 = 17
    (899g/29) - 680/29 = 17
    899g/29 = 17 + 680/29
    899g/29 = 697/29
    899g = 697
    g = 697/899
    g = <<697/899=0.7753059>>0.7753059

    Now, substitute the value of g back into the equation for l:
    l = (68 - 17g) / 29
    l = (68 - 17 * 0.7753059) / 29
    l = (68 - 13.1802) / 29
    l = 54.8198 / 29
    l = <<54.8198/29=1.8903379>>1.8903379

    Therefore, the values for g and l are:
    g ≈ 0.7753059
    l ≈ 1.8903379

    ### Answer: g ≈ 0.7753059, l ≈ 1.8903379

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

Correct? No

🔗 Problem

Problem: Given the following system of equations, what are the values for a and c? 33*a - 54*c = 20 24*a - 26*c = 29.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for 'a' in the first equation and substitute it into the second equation to solve for 'c'. Then, we can use the value of 'c' to find 'a' in the first equation.

    1. Solve for 'a' in the first equation:
       33*a - 54*c = 20
       33*a = 54*c + 20
       a = (54*c + 20) / 33

    2. Substitute 'a' into the second equation:
       24*a - 26*c = 29
       24*((54*c + 20) / 33) - 26*c = 29

    3. Simplify the equation:
       24*(54*c + 20) / 33 - 26*c = 29
       (24*54*c + 24*20) / 33 - 26*c = 29
       (1296*c + 480) / 33 - 26*c = 29

    4. Multiply both sides by 33 to eliminate fractions:
       1296*c + 480 - 33*26*c = 29*33
       1296*c + 480 - 858*c = 957
       438*c + 480 = 957

    5. Subtract 480 from both sides:
       438*c = 477

    6. Divide both sides by 438 to solve for 'c':
       c = 477 / 438
       c ≈ 1.089 (rounded to three decimal places)

    7. Substitute 'c' into the first equation to solve for 'a':
       33*a - 54*1.089 = 20
       33*a - 58.806 = 20
       33*a = 78.806
       a = 78.806 / 33
       a ≈ 2.388 (rounded to three decimal places)

    ### Answer: a ≈ 2.388, c ≈ 1.089

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

Correct? No

🔗 Problem

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:

    Sure, I can help Kat solve this system of equations using the substitution method. Let's solve for one of the variables in the first equation and substitute it into the second equation.

    Given equations:
    71d + 9j = 132 ... (1)
    43j - 84d = 91 ... (2)

    Step 1: Solve equation (1) for j in terms of d.
    71d + 9j = 132
    9j = 132 - 71d
    j = (132 - 71d) ÷ 9
    j = (132 ÷ 9) - (71 ÷ 9)d
    j = 14 - (71 ÷ 9)d ... (3)

    Step 2: Substitute the expression for j from equation (3) into equation (2).
    43(14 - (71 ÷ 9)d) - 84d = 91
    602 - (1591 ÷ 3)d - 84d = 91
    602 - 530.33d - 84d = 91
    602 - 614.33d = 91

    Step 3: Solve the resulting equation for d.
    602 - 614.33d = 91
    -614.33d = 91 - 602
    -614.33d = -511
    d = -511 ÷ -614.33
    d ≈ 0.832

    Step 4: Substitute the value of d into equation (3) to find j.
    j = 14 - (71 ÷ 9)(0.832)
    j = 14 - 6.554
    j ≈ 7.446

    Therefore, the solution to the system of equations is:
    d ≈ 0.832 and j ≈ 7.446

    ### Answer: d ≈ 0.832, j ≈ 7.446

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

Correct? No

🔗 Problem

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

Gemma 2B:

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

    Given equations:
    (81 * o) + (26 * n) = 163 ... (1)
    (87 * n) + (23 * o) = 143 ... (2)

    Step 1: Solve equation (1) for o in terms of n.
    (81 * o) + (26 * n) = 163
    81 * o = 163 - 26 * n
    o = (163 - 26 * n) / 81 ... (3)

    Step 2: Substitute the expression for o from equation (3) into equation (2).
    (87 * n) + (23 * ((163 - 26 * n) / 81)) = 143

    Step 3: Simplify the equation and solve for n.
    (87 * n) + ((3749 - 598 * n) / 81) = 143
    7047 * n + 3749 - 598 * n = 11583
    6449 * n = 7834
    n = 7834 / 6449
    n ≈ 1.215 ... (4)

    Step 4: Substitute the value of n from equation (4) into equation (3) to find o.
    o = (163 - 26 * 1.215) / 81
    o = (163 - 31.59) / 81
    o = 131.41 / 81
    o ≈ 1.622

    Therefore, the approximate values of o and n are:
    o ≈ 1.622
    n ≈ 1.215

    ### Answer: o ≈ 1.622, n ≈ 1.215

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, let's solve for s:
    46 * s - 61 * w = 2
    46 * s = 61 * w + 2
    s = (61 * w + 2) / 46

    Now, substitute the expression for s into the second equation:
    61 * s + 67 * w = 178
    61 * ((61 * w + 2) / 46) + 67 * w = 178

    Simplify:
    (61 * (61 * w + 2)) / 46 + 67 * w = 178
    (3721 * w + 122) / 46 + 67 * w = 178

    Multiply both sides by 46:
    3721 * w + 122 + 3082 * w = 8188
    6803 * w + 122 = 8188

    Subtract 122 from both sides:
    6803 * w = 8066

    Divide both sides by 6803:
    w = 8066 / 6803
    w ≈ 1.1857

    Now that we have the value of w, we can substitute it back into the expression for s:
    s = (61 * w + 2) / 46
    s = (61 * 1.1857 + 2) / 46
    s ≈ 1.5857

    Therefore, the solution to the system of equations is:
    w ≈ 1.1857
    s ≈ 1.5857

    ### Answer: w ≈ 1.1857, s ≈ 1.5857

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

Correct? No

🔗 Problem

Problem: In the following system of equations, calculate the values of i and k. System of equations: 51*k - 8*i = 86 39*i + 8*k = 193.

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, we can solve for k:
    51*k - 8*i = 86
    51*k = 86 + 8*i
    k = (86 + 8*i) / 51
    k = (86/51) + (8/51)*i
    k = (86 + 8*i) / 51

    Now, substitute the expression for k into the second equation:
    39*i + 8*k = 193
    39*i + 8*((86 + 8*i) / 51) = 193
    39*i + (688 + 64*i) / 51 = 193
    39*i + (688/51) + (64/51)*i = 193
    39*i + (64/51)*i = 193 - (688/51)
    (1989/51)*i = 9073/51
    i = 9073/1989
    i = 91/21

    Now that we have the value of i, we can substitute it back into the expression for k:
    k = (86 + 8*i) / 51
    k = (86 + 8*(91/21)) / 51
    k = (86 + 364/21) / 51
    k = (1806 + 364) / (21*51)
    k = 2170 / 1071
    k = 10/49

    Therefore, the values of i and k are:
    i = 91/21
    k = 10/49

    ### Answer: i = 91/21, k = 10/49

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for l in the second equation and substitute it into the first equation to find the value of m. Then, we can use the value of m to find the value of l.

    1. Solve the second equation for l:
       93m + l = 126
       l = 126 - 93m

    2. Substitute the expression for l into the first equation:
       26m + 17l = 146
       26m + 17(126 - 93m) = 146
       26m + 2142 - 1581m = 146
       -1555m + 2142 = 146
       -1555m = -1996
       m = -1996 ÷ (-1555)
       m = 128/111

    3. Substitute the value of m into the second equation to find l:
       93m + l = 126
       93(128/111) + l = 126
       (93 × 128) ÷ 111 + l = 126
       11904 ÷ 111 + l = 126
       107 + l = 126
       l = 126 - 107
       l = 19

    Therefore, the solution to the system of equations is:
    m = 128/111
    l = 19

    ### Answer: m = 128/111, l = 19

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

Correct? No

🔗 Problem

Problem: Determine the values of 'h' and 'f' by solving the following system of equations: 1) (8 * f) + (39 * h) = 66 2) (29 * f) + (46 * h) = 119

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve equation (1) for f in terms of h, then substitute the expression for f into equation (2) to solve for h. Finally, we'll substitute the value of h back into equation (1) to find f.

    Step 1: Solve equation (1) for f in terms of h.
    (8 * f) + (39 * h) = 66
    8f = 66 - 39h
    f = (66 - 39h) / 8

    Step 2: Substitute the expression for f into equation (2) and solve for h.
    (29 * f) + (46 * h) = 119
    29 * ((66 - 39h) / 8) + 46h = 119
    (29 * (66 - 39h)) / 8 + 46h = 119
    (1914 - 1131h) / 8 + 46h = 119
    1914 - 1131h + 368h = 952
    1914 - 763h = 952
    -763h = -962
    h = 962 / 763
    h = 1.26079947651663

    Step 3: Substitute the value of h into the equation for f from Step 1.
    f = (66 - 39h) / 8
    f = (66 - 39 * 1.26079947651663) / 8
    f = (66 - 49.1711790841485) / 8
    f = 16.8288209158515 / 8
    f = 2.10360261448144

    Therefore, the values of h and f are:
    h ≈ 1.26079947651663
    f ≈ 2.10360261448144

    ### Answer: h ≈ 1.26079947651663, f ≈ 2.10360261448144

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

Correct? No

🔗 Problem

Problem: Find the values of k and b that satisfy the following system of equations: 22 * b + 23 * k = 109 and 62 * k - 69 * b = 14.

Gemma 2B:

    To solve for k and b in the given system of equations, we can use the substitution method.

    Given equations:
    1. 22 * b + 23 * k = 109
    2. 62 * k - 69 * b = 14

    Step 1: Solve equation 1 for b in terms of k.
    22 * b + 23 * k = 109
    22 * b = 109 - 23 * k
    b = (109 - 23 * k) / 22

    Step 2: Substitute the expression for b into equation 2.
    62 * k - 69 * ((109 - 23 * k) / 22) = 14

    Step 3: Simplify the equation and solve for k.
    62 * k - (3021 - 1587 * k) / 22 = 14
    1364 * k - 3021 + 1587 * k = 308
    2951 * k - 3021 = 308
    2951 * k = 3329
    k = 3329 / 2951
    <<3329/2951=1.128>>k ≈ 1.128

    Step 4: Substitute the value of k into the expression for b from step 1.
    b = (109 - 23 * 1.128) / 22
    b = (109 - 25.944) / 22
    b = 83.056 / 22
    <<83.056/22=3.775>>b ≈ 3.775

    Therefore, the values of k and b that satisfy the given system of equations are:
    ### Answer: k ≈ 1.128 and b ≈ 3.775

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve the system of linear equations, we can use the substitution method.

    Given equations:
    1. 84m - 55s = 1
    2. 63m - 2s = 98

    Step 1: Solve equation 2 for s in terms of m.
    63m - 2s = 98
    -2s = -63m + 98
    s = (63m - 98) / 2

    Step 2: Substitute the expression for s into equation 1.
    84m - 55((63m - 98) / 2) = 1
    84m - (3465m - 5390) / 2 = 1
    168m - 3465m + 5390 = 2
    -3297m + 5390 = 2
    -3297m = -5388
    m = -5388 / -3297
    m = 1.634 (rounded to three decimal places)

    Step 3: Substitute the value of m into the expression for s from step 1.
    s = (63(1.634) - 98) / 2
    s = (102.942 - 98) / 2
    s = 4.942 / 2
    s = 2.471 (rounded to three decimal places)

    Therefore, the solution to the system of linear equations is:
    m ≈ 1.634
    s ≈ 2.471

    ### Answer: m ≈ 1.634, s ≈ 2.471

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

Correct? No

🔗 Problem

Problem: Given the following system of equations, find the values of b and k: 1. (6 * b) + (5 * k) = 105 2. (95 * k) - (51 * b) = 67.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From equation 1, let's solve for b:
    (6 * b) + (5 * k) = 105
    6b = 105 - 5k
    b = (105 - 5k) / 6
    b = 17.5 - (5/6)k

    Now, substitute this expression for b into equation 2:
    (95 * k) - (51 * (17.5 - (5/6)k)) = 67
    95k - 892.5 + (255/6)k = 67
    95k + (255/6)k - 892.5 = 67
    (570/6)k + (255/6)k = 67 + 892.5
    (825/6)k = 959.5
    k = 959.5 / (825/6)
    k = 959.5 * (6/825)
    k = 6.98

    Now that we have the value of k, we can substitute it back into the expression for b:
    b = 17.5 - (5/6)k
    b = 17.5 - (5/6) * 6.98
    b = 17.5 - 5.82
    b = 11.68

    Therefore, the values of b and k are:
    b ≈ 11.68
    k ≈ 6.98

    ### Answer: b ≈ 11.68, k ≈ 6.98

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve for d and u, we can use the substitution method. Let's solve for u in the second equation and substitute it into the first equation to solve for d.

    From the second equation, 32u + 40d = 83, we get:
    32u = 83 - 40d
    u = (83 - 40d) / 32
    u = (83/32) - (40/32)d
    u = 2.59375 - 1.25d

    Substituting this expression for u into the first equation:
    33d + 62(2.59375 - 1.25d) = 114
    33d + 160.8125 - 77.5d = 114
    -44.5d + 160.8125 = 114
    -44.5d = -46.8125
    d = (-46.8125) / (-44.5)
    d = 1.052

    Now that we have the value of d, we can substitute it back into the expression for u:
    u = 2.59375 - 1.25d
    u = 2.59375 - 1.25(1.052)
    u = 2.59375 - 1.315
    u = 1.27875

    Therefore, the values of d and u are:
    d = 1.052
    u = 1.27875

    ### Answer: d = 1.052, u = 1.27875

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

Correct? No

🔗 Problem

Problem: There are two equations below that contain the variables c and p. Can you solve this system of equations to find the values of c and p? 5 * c + 28 * p = 102 20 * c - 67 * p = 13

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, let's solve for c:
    5c + 28p = 102
    5c = 102 - 28p
    c = (102 - 28p) / 5

    Now, substitute this expression for c into the second equation:
    20 * ((102 - 28p) / 5) - 67p = 13

    Simplify:
    (2040 - 560p) / 5 - 67p = 13
    2040/5 - 560p/5 - 67p = 13
    408 - 112p - 67p = 13
    408 - 179p = 13

    Solve for p:
    -179p = 13 - 408
    -179p = -395
    p = -395 / -179
    p = 395/179

    Now that we have the value of p, we can substitute it back into the equation for c:
    c = (102 - 28p) / 5
    c = (102 - 28 * (395/179)) / 5
    c = (102 - 61.8435754) / 5
    c = 40.1564246 / 5
    c = 8.0312849

    Therefore, the solutions are:
    p = 395/179 ≈ 2.2067039
    c = 40.1564246 / 5 ≈ 8.0312849

    ### Answer: p ≈ 2.2067039, c ≈ 8.0312849

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

Correct? No

🔗 Problem

Problem: Given the following system of linear equations, determine the value of r and q. The system of equations is: 58 * r + 12 * q = 147 97 * q - 76 * r = 76.

Gemma 2B:

    To solve this system of linear equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, let's solve for r:
    58 * r + 12 * q = 147
    58 * r = 147 - 12 * q
    r = (147 - 12 * q) / 58

    Now, substitute the expression for r into the second equation:
    97 * q - 76 * r = 76
    97 * q - 76 * ((147 - 12 * q) / 58) = 76

    Simplify the equation:
    97 * q - 76 * (147 - 12 * q) / 58 = 76
    97 * q - (76 * 147 - 76 * 12 * q) / 58 = 76
    97 * q - (11172 - 912 * q) / 58 = 76

    Multiply both sides by 58:
    97 * 58 * q - 11172 + 912 * q = 76 * 58
    5626 * q - 11172 + 912 * q = 4408
    6538 * q - 11172 = 4408
    6538 * q = 15580
    q = 15580 / 6538
    q = 2.38

    Now that we have the value of q, we can substitute it back into the expression for r:
    r = (147 - 12 * q) / 58
    r = (147 - 12 * 2.38) / 58
    r = (147 - 28.56) / 58
    r = 118.44 / 58
    r = 2.04

    Therefore, the values of r and q are:
    r = 2.04
    q = 2.38

    ### Answer: r = 2.04, q = 2.38

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

Correct? No

🔗 Problem

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

Gemma 2B:

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

    Given equations:
    47 * v - 3 * k = 10 ... (1)
    73 * v + 2 * k = 128 ... (2)

    Step 1: Solve equation (1) for k in terms of v.
    47 * v - 3 * k = 10
    -3 * k = -47 * v + 10
    k = (47 * v - 10) / 3 ... (3)

    Step 2: Substitute the expression for k from equation (3) into equation (2).
    73 * v + 2 * ((47 * v - 10) / 3) = 128
    73 * v + (94 * v - 20) / 3 = 128
    219 * v + 94 * v - 20 = 384
    313 * v - 20 = 384
    313 * v = 404
    v = 404 / 313 ... (4)

    Step 3: Substitute the value of v from equation (4) into equation (3) to find k.
    k = (47 * (404 / 313) - 10) / 3
    k = (18988 / 313 - 10) / 3
    k = (18978 / 313) / 3
    k = 6058 / 313

    Step 4: Simplify the values of v and k.
    v = 404 / 313 ≈ 1.29 ... (5)
    k = 6058 / 313 ≈ 19.35 ... (6)

    Therefore, the solution to the given system of simultaneous linear equations is:
    v ≈ 1.29
    k ≈ 19.35

    ### Answer: v ≈ 1.29, k ≈ 19.35

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

Correct? No

🔗 Problem

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:

    Let's solve this system of equations step by step using the substitution method.

    Given equations:
    1. 89p - 41q = 60
    2. 27p + 91q = 124

    Step 1: Solve equation 1 for p in terms of q.
    89p - 41q = 60
    89p = 60 + 41q
    p = (60 + 41q) / 89

    Step 2: Substitute the expression for p into equation 2.
    27p + 91q = 124
    27((60 + 41q) / 89) + 91q = 124

    Step 3: Simplify the equation and solve for q.
    (1620 + 1107q) / 89 + 91q = 124
    1620 + 1107q + 8099q = 11036
    9206q = 9416
    q = 9416 / 9206
    q = <<9416/9206=1.0228>>1.0228

    Step 4: Substitute the value of q into the expression for p from step 1.
    p = (60 + 41q) / 89
    p = (60 + 41 × 1.0228) / 89
    p = 101.9348 / 89
    p = <<101.9348/89=1.1453>>1.1453

    Therefore, the values of p and q are:
    ### Answer: p = 1.1453, q = 1.0228

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

Correct? No

🔗 Problem

Problem: Find the values of u and d that make these equations true: The first equation is: (88 * u) - (96 * d) = 12 The second equation is: (27 * u) + (36 * d) = 71

Gemma 2B:

    To solve for u and d, we can use the substitution method. Let's solve for u in the first equation and substitute it into the second equation to solve for d.

    First equation: (88 * u) - (96 * d) = 12
    88u - 96d = 12
    88u = 96d + 12
    u = (96d + 12) / 88
    u = (12d + 1.5)

    Now, substitute the expression for u into the second equation:
    Second equation: (27 * u) + (36 * d) = 71
    27 * (12d + 1.5) + 36d = 71
    324d + 40.5 + 36d = 71
    360d + 40.5 = 71
    360d = 30.5
    d = 30.5 / 360
    d = 0.0847222...

    Now that we have the value of d, we can substitute it back into the expression for u:
    u = (12d + 1.5)
    u = (12 * 0.0847222... + 1.5)
    u = 2.5166666...

    Therefore, the values of u and d that make both equations true are approximately:
    d ≈ 0.0847
    u ≈ 2.5167

    ### Answer: u ≈ 2.5167, d ≈ 0.0847

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of f and n: 20 * n - 18 * f = 9 and 3 * n - 2 * f = 70.

Gemma 2B:

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

    Given equations:
    1. 20 * n - 18 * f = 9
    2. 3 * n - 2 * f = 70

    Step 1: Solve equation 2 for n in terms of f.
    3 * n - 2 * f = 70
    3 * n = 70 + 2 * f
    n = (70 + 2 * f) / 3

    Step 2: Substitute the expression for n into equation 1.
    20 * ((70 + 2 * f) / 3) - 18 * f = 9
    (1400 + 40 * f) / 3 - 18 * f = 9
    1400 / 3 + 40 * f / 3 - 18 * f = 9
    1400 / 3 + 40 * f / 3 - 54 * f / 3 = 9
    1400 / 3 - 14 * f = 9
    1400 / 3 - 9 = 14 * f
    1400 / 3 - 9 = 14 * f
    (1400 - 27) / 3 = 14 * f
    1373 / 3 = 14 * f
    1373 / 42 = f
    f ≈ 32.69

    Step 3: Substitute the value of f into the expression for n.
    n = (70 + 2 * 32.69) / 3
    n = (70 + 65.38) / 3
    n = 135.38 / 3
    n ≈ 45.13

    ### Answer: f ≈ 32.69, n ≈ 45.13

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, let's solve for l:
    7 * l + 97 * j = 133
    7 * l = 133 - 97 * j
    l = (133 - 97 * j) / 7

    Now, substitute the expression for l into the second equation:
    22 * j + 37 * ((133 - 97 * j) / 7) = 76

    Simplify:
    22 * j + (4921 - 3589 * j) / 7 = 76
    154 * j + 4921 - 3589 * j = 532
    4921 - 3435 * j = 532
    -3435 * j = -4389
    j = -4389 / -3435
    j = 1.278

    Now that we have the value of j, we can substitute it back into the equation for l:
    l = (133 - 97 * 1.278) / 7
    l = (133 - 124) / 7
    l = 9 / 7
    l = 1.286

    Therefore, the values of j and l are:
    j ≈ 1.278
    l ≈ 1.286

    ### Answer: j ≈ 1.278, l ≈ 1.286

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for d in the first equation and substitute it into the second equation to solve for f.

    1. Solve for d in the first equation:
       21d - 46f = 53
       21d = 53 + 46f
       d = (53 + 46f) / 21
       d = (53 + 46f) ÷ 21

    2. Substitute the expression for d into the second equation:
       8d + 84f = 133
       8((53 + 46f) ÷ 21) + 84f = 133
       (8(53 + 46f) ÷ 21) + 84f = 133
       (424 + 368f) ÷ 21 + 84f = 133
       (424 ÷ 21) + (368f ÷ 21) + 84f = 133
       20.19 + 17.52f + 84f = 133
       20.19 + 101.52f = 133
       101.52f = 112.81
       f = 112.81 ÷ 101.52
       f ≈ 1.11

    3. Substitute the value of f into the expression for d:
       d = (53 + 46f) ÷ 21
       d = (53 + 46(1.11)) ÷ 21
       d = (53 + 51.06) ÷ 21
       d = 104.06 ÷ 21
       d ≈ 4.96

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

    ### Answer: d ≈ 4.96, f ≈ 1.11

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

Correct? No

🔗 Problem

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

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, 74u + 11w = 116, let's solve for u:
    74u = 116 - 11w
    u = (116 - 11w) / 74

    Now, substitute this expression for u into the second equation, 66u - 68w = 21:
    66 * ((116 - 11w) / 74) - 68w = 21

    Simplify:
    (66 * (116 - 11w)) / 74 - 68w = 21
    (7656 - 726w) / 74 - 68w = 21
    103.4594594594595 - 9.810810810810813w - 68w = 21
    103.4594594594595 - 77.81081081081081w = 21

    Subtract 103.4594594594595 from both sides:
    -77.81081081081081w = -82.4594594594595

    Divide both sides by -77.81081081081081:
    w = 1.06

    Now that we have the value of w, we can substitute it back into the equation for u:
    u = (116 - 11w) / 74
    u = (116 - 11 * 1.06) / 74
    u = 104.34 / 74
    u = 1.41

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

    ### Answer: u = 1.41, w = 1.06

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

Correct? No

🔗 Problem

Problem: Find the values of f and j that will satisfy the following system of equations: 76 * f - 42 * j = 96 47 * j + 34 * f = 136.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    Given equations:
    76 * f - 42 * j = 96 ... (1)
    47 * j + 34 * f = 136 ... (2)

    Step 1: Solve equation (1) for f in terms of j.
    76 * f - 42 * j = 96
    76 * f = 96 + 42 * j
    f = (96 + 42 * j) / 76
    f = (4 + 21 * j) / 38 ... (3)

    Step 2: Substitute the expression for f from equation (3) into equation (2).
    47 * j + 34 * ((4 + 21 * j) / 38) = 136

    Step 3: Solve the resulting equation for j.
    47 * j + (136 + 714 * j) / 38 = 136
    1786 * j + 136 + 714 * j = 5168
    2500 * j = 5032
    j = 5032 / 2500
    j = <<5032/2500=2.0128>>2.0128

    Step 4: Substitute the value of j into equation (3) to find f.
    f = (4 + 21 * 2.0128) / 38
    f = 46.2688 / 38
    f = <<46.2688/38=1.2176>>1.2176

    Therefore, the values of f and j that satisfy the given system of equations are:
    f = 1.2176
    j = 2.0128

    ### Answer: f = 1.2176, j = 2.0128

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

Correct? No

🔗 Problem

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

Gemma 2B:

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

    From the first equation, we can express u in terms of e:
    68u + 32e = 174
    68u = 174 - 32e
    u = (174 - 32e) / 68

    Now, substitute this expression for u into the second equation:
    63e - 69u = 1
    63e - 69((174 - 32e) / 68) = 1

    Simplify the equation:
    63e - (69 × 174 - 69 × 32e) / 68 = 1
    63e - (12006 - 2208e) / 68 = 1
    4284e - 12006 + 2208e = 68
    6492e - 12006 = 68
    6492e = 12074
    e = 12074 / 6492

    Simplify the fraction:
    e = <<12074/6492=1.86>>1.86

    Now that we have the value of e, we can substitute it back into the expression for u:
    u = (174 - 32e) / 68
    u = (174 - 32 × 1.86) / 68
    u = (174 - 59.52) / 68
    u = 114.48 / 68

    Simplify the fraction:
    u = <<114.48/68=1.68>>1.68

    Therefore, the values of u and e are:
    u = 1.68
    e = 1.86

    ### Answer: u = 1.68, e = 1.86

Expected answer: {e: 6037/3246, u: 5465/3246}

Correct? No

🔗 Problem

Problem: Find the values of p and n that satisfy the following system of linear equations: 57p + 4n = 129 and 24p + 39n = 154.

Gemma 2B:

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

    1. Solve the first equation for p in terms of n:
       57p + 4n = 129
       57p = 129 - 4n
       p = (129 - 4n) / 57

    2. Substitute the expression for p into the second equation:
       24p + 39n = 154
       24((129 - 4n) / 57) + 39n = 154

    3. Simplify the equation:
       (24(129 - 4n) / 57) + 39n = 154
       (3096 - 96n) / 57 + 39n = 154
       3096/57 - 96n/57 + 39n = 154
       54.3157895 - 1.6842105n + 39n = 154
       37.3157895n + 54.3157895 = 154
       37.3157895n = 99.6842105
       n = 99.6842105 / 37.3157895
       n = 2.67

    4. Substitute n = 2.67 into the equation for p from step 1:
       p = (129 - 4n) / 57
       p = (129 - 4(2.67)) / 57
       p = (129 - 10.68) / 57
       p = 118.32 / 57
       p = 2.075

    Therefore, the values that satisfy the system of linear equations are:
    p ≈ 2.075
    n ≈ 2.67

    ### Answer: p ≈ 2.075, n ≈ 2.67

Expected answer: {n: 1894/709, p: 4415/2127}

Correct? No

🔗 Problem

Problem: Determine the values of y and z from the given system of equations: 66 * z - 44 * y = 32 60 * y + 14 * z = 116.

Gemma 2B:

    To solve the given system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    Given equations:
    66 * z - 44 * y = 32 ... (1)
    60 * y + 14 * z = 116 ... (2)

    Step 1: Solve equation (1) for y in terms of z.
    66 * z - 44 * y = 32
    -44 * y = -66 * z + 32
    y = (66 * z - 32) / 44
    y = (33 * z - 16) / 22 ... (3)

    Step 2: Substitute equation (3) into equation (2) and solve for z.
    60 * ((33 * z - 16) / 22) + 14 * z = 116
    (1980 * z - 960) / 22 + 14 * z = 116
    1980 * z - 960 + 308 * z = 2552
    2288 * z - 960 = 2552
    2288 * z = 3512
    z = 3512 / 2288
    z = <<3512/2288=1.535>>1.535

    Step 3: Substitute the value of z into equation (3) to solve for y.
    y = (33 * 1.535 - 16) / 22
    y = (50.655 - 16) / 22
    y = 34.655 / 22
    y = <<34.655/22=1.575>>1.575

    Therefore, the values of y and z are:
    z = 1.535
    y = 1.575

    ### Answer: y = 1.575, z = 1.535

Expected answer: {y: 901/572, z: 439/286}

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the variables t and h: The first equation is 83 * t - 58 * h = 45. The second equation is 69 * h + 11 * t = 124.

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    Given equations:
    1. 83t - 58h = 45
    2. 69h + 11t = 124

    Step 1: Solve the first equation for t in terms of h.
    83t - 58h = 45
    83t = 58h + 45
    t = (58h + 45) / 83

    Step 2: Substitute the expression for t into the second equation.
    69h + 11((58h + 45) / 83) = 124

    Step 3: Simplify the equation and solve for h.
    69h + (638h + 495) / 83 = 124
    5727h + 495 = 10292
    5727h = 9797
    h = 9797 / 5727
    h = 1.71

    Step 4: Substitute the value of h into the expression for t from Step 1.
    t = (58(1.71) + 45) / 83
    t = 143.98 / 83
    t = 1.73

    Therefore, the solution to the system of equations is:
    h = 1.71
    t = 1.73

    ### Answer: h = 1.71, t = 1.73

Expected answer: {h: 9797/6365, t: 10297/6365}

Correct? No

🔗 Problem

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:

    Let's solve this system of equations step by step using the substitution method.

    Given equations:
    1. ((79 * h) - (61 * o)) = 9
    2. ((16 * h) - (12 * o)) = 9

    Step 1: Simplify equation 2 to isolate h in terms of o.
    ((16 * h) - (12 * o)) = 9
    16h - 12o = 9
    16h = 12o + 9
    h = (12o + 9) / 16
    h = (3o + 9/4) / 4
    h = (3/4)o + 9/16

    Step 2: Substitute the expression for h into equation 1.
    ((79 * h) - (61 * o)) = 9
    (79 * ((3/4)o + 9/16)) - (61 * o) = 9
    (237/4)o + 711/16 - 61o = 9

    Step 3: Simplify the equation and solve for o.
    (237/4)o - 61o + 711/16 = 9
    (237/4 - 61)o + 711/16 = 9
    (237/4 - 244/4)o + 711/16 = 9
    (-7/4)o + 711/16 = 9
    -7o/4 = 9 - 711/16
    -7o/4 = (144 - 711)/16
    -7o/4 = -567/16
    -7o = -567/4
    o = 81/7

    Step 4: Substitute the value of o into the expression for h to find its value.
    h = (3/4)o + 9/16
    h = (3/4)(81/7) + 9/16
    h = 243/28 + 9/16
    h = (243/28 + 63/112)
    h = (243 + 63)/28
    h = 306/28
    h = 153/14

    Therefore, the values of o and h are:
    o = 81/7
    h = 153/14

    ### Answer: o = 81/7, h = 153/14

Expected answer: {h: 63/4, o: 81/4}

Correct? No

🔗 Problem

Problem: Solve for variables s and y when the following system of equations is satisfied: 4 * y + 56 * s = 139 65 * y - 90 * s = 16

Gemma 2B:

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

    Given equations:
    4y + 56s = 139 ... (1)
    65y - 90s = 16 ... (2)

    Step 1: Solve equation (1) for y in terms of s.
    4y + 56s = 139
    4y = 139 - 56s
    y = (139 - 56s) ÷ 4
    y = 34.75 - 14s ... (3)

    Step 2: Substitute the expression for y from equation (3) into equation (2).
    65(34.75 - 14s) - 90s = 16
    2258.75 - 910s - 90s = 16
    2258.75 - 1000s = 16
    -1000s = 16 - 2258.75
    -1000s = -2242.75
    s = 2.24275

    Step 3: Substitute the value of s into equation (3) to find y.
    y = 34.75 - 14(2.24275)
    y = 34.75 - 31.3985
    y = 3.3515

    Therefore, the solution to the system of equations is:
    s ≈ 2.24275
    y ≈ 3.3515

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

Expected answer: {s: 8971/4000, y: 6703/2000}

Correct? No

🔗 Problem

Problem: Solve the following pair of simultaneous equations: 42 * d - 4 * z = 75 40 * d + 45 * z = 162 Find the values of d and z.

Gemma 2B:

    To solve the pair of simultaneous equations, we can use the substitution method.

    Given equations:
    42 * d - 4 * z = 75 ... (1)
    40 * d + 45 * z = 162 ... (2)

    Step 1: Solve equation (1) for d in terms of z.
    42 * d - 4 * z = 75
    42 * d = 75 + 4 * z
    d = (75 + 4 * z) / 42
    d = 75/42 + (4/42) * z
    d = 25/14 + (1/21) * z ... (3)

    Step 2: Substitute the expression for d from equation (3) into equation (2).
    40 * (25/14 + (1/21) * z) + 45 * z = 162
    1000/14 + 40/21 * z + 45 * z = 162
    1000/14 + (40/21 + 45) * z = 162
    1000/14 + (200/21 + 945/21) * z = 162
    1000/14 + 1145/21 * z = 162

    Step 3: Solve the resulting equation for z.
    1000/14 + 1145/21 * z = 162
    1145/21 * z = 162 - 1000/14
    1145/21 * z = (3402 - 1500)/42
    1145/21 * z = 1902/42
    z = (1902/42) / (1145/21)
    z = (1902/42) * (21/1145)
    z = 1902 / (42 * 1145)
    z = 1902 / 48090
    z = 1/253

    Step 4: Substitute the value of z into equation (3) to find the value of d.
    d = 25/14 + (1/21) * (1/253)
    d = 25/14 + 1/5313
    d = (132825 + 14) / (14 * 5313)
    d = 132839 / 74382
    d = 33/18579

    Therefore, the values of d and z are:
    z = 1/253
    d = 33/18579

    ### Answer: z = 1/253, d = 33/18579

Expected answer: {d: 4023/2050, z: 1902/1025}

Correct? No

🔗 Problem

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:

    Let's solve this step by step using the substitution method.

    1. From the first equation, express 'o' in terms of 'a':
       (42 * o) + (24 * a) = 142
       42o = 142 - 24a
       o = (142 - 24a) / 42
       o = 142/42 - 24a/42
       o = 3.380952381 - 0.5714285714a

    2. Substitute the expression for 'o' into the second equation:
       (42 * a) + (6 * (3.380952381 - 0.5714285714a)) = 75
       42a + 20.28571429 - 3.428571429a = 75
       38.57142857a + 20.28571429 = 75
       38.57142857a = 54.71428571
       a = 54.71428571 / 38.57142857
       a = 1.419354839

    3. Substitute the value of 'a' back into the expression for 'o':
       o = 3.380952381 - 0.5714285714a
       o = 3.380952381 - 0.5714285714 * 1.419354839
       o = 3.380952381 - 0.8110749185
       o = 2.569877463

    Therefore, the values of 'a' and 'o' are:
    a ≈ 1.419354839
    o ≈ 2.569877463

    ### Answer: a ≈ 1.419354839, o ≈ 2.569877463

Expected answer: {a: 383/270, o: 347/135}

Correct? No

🔗 Problem

Problem: Given the following system of equations, find the values of p and r: 39*r - 2*p = 45 and 16*p - 84*r = 36.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation.

    From the first equation, 39*r - 2*p = 45, let's solve for p:
    39*r - 2*p = 45
    -2*p = -39*r + 45
    p = (39*r - 45) / 2

    Now, substitute this expression for p into the second equation:
    16*p - 84*r = 36
    16*((39*r - 45) / 2) - 84*r = 36
    (16*(39*r - 45)) / 2 - 84*r = 36
    (624*r - 720) / 2 - 84*r = 36
    312*r - 360 - 84*r = 36
    228*r - 360 = 36
    228*r = 396
    r = 396 / 228
    r = <<396/228=1.7368421052631578>>1.7368421052631578

    Now that we have the value of r, we can substitute it back into the expression for p:
    p = (39*r - 45) / 2
    p = (39*1.7368421052631578 - 45) / 2
    p = (67.73684210526316 - 45) / 2
    p = 22.73684210526316 / 2
    p = <<22.73684210526316/2=11.36842105263158>>11.36842105263158

    ### Answer: p = 11.36842105263158, r = 1.7368421052631578

Expected answer: {p: 216/19, r: 33/19}

Correct? No

🔗 Problem

Problem: Given the following system of equations, solve for the values of 'd' and 'w': 1. (23 * d) - (52 * w) = 7 2. (64 * w) - (8 * d) = 58

Gemma 2B:

    To solve for the values of 'd' and 'w', we can use the substitution method. Let's solve for 'd' in terms of 'w' using the first equation, then substitute this expression into the second equation to solve for 'w'.

    From equation 1: (23 * d) - (52 * w) = 7
    23d - 52w = 7
    23d = 52w + 7
    d = (52w + 7) / 23
    d = (52/23)w + (7/23)

    Now, substitute this expression for 'd' into equation 2:
    64w - 8((52/23)w + (7/23)) = 58
    64w - (416/23)w - (56/23) = 58
    (1472/23)w - (416/23)w = 58 + (56/23)
    (1056/23)w = (1398/23)
    w = (1398/1056)
    w = <<1398/1056=1.323863636363636>>1.32

    Now that we have the value for 'w', we can substitute it back into the expression for 'd':
    d = (52/23)w + (7/23)
    d = (52/23)(1.32) + (7/23)
    d = 2.98 + 0.30
    d = <<2.98+0.30=3.28>>3.28

    ### Answer: d = 3.28, w = 1.32

Expected answer: {d: 433/132, w: 695/528}

Correct? No

🔗 Problem

Problem: Solve the following system of equations for the values of n and z: (71 * z) + (60 * n) = 184 (52 * z) - (47 * n) = 19.

Gemma 2B:

    To solve the system of equations, we can use the substitution method. Let's solve for one variable in the first equation and substitute it into the second equation.

    From the first equation, let's solve for n:
    (71 * z) + (60 * n) = 184
    60 * n = 184 - (71 * z)
    n = (184 - (71 * z)) / 60

    Now, substitute the expression for n into the second equation:
    (52 * z) - (47 * ((184 - (71 * z)) / 60)) = 19

    Simplify:
    (52 * z) - (47 * (184 - (71 * z))) / 60 = 19
    (52 * z) - ((8648 - (3337 * z)) / 60) = 19
    (52 * z) - (144.1333... - (55.6166... * z)) = 19
    (52 * z) - 144.1333... + (55.6166... * z) = 19
    (107.6166... * z) - 144.1333... = 19
    (107.6166... * z) = 163.1333...
    z = 163.1333... / 107.6166...
    z = 1.5161290322580645

    Now that we have the value of z, we can substitute it back into the expression for n:
    n = (184 - (71 * 1.5161290322580645)) / 60
    n = (184 - 107.64516129032258) / 60
    n = 76.35483870967742 / 60
    n = 1.2725806451612903

    Therefore, the solution to the system of equations is:
    z ≈ 1.5161290322580645
    n ≈ 1.2725806451612903

    ### Answer: z ≈ 1.5161290322580645, n ≈ 1.2725806451612903

Expected answer: {n: 8219/6457, z: 9788/6457}

Correct? No

🔗 Problem

Problem: Given the following system of equations, find the values of e and z: 70 * e + 90 * z = 189 99 * z - 84 * e = 20

Gemma 2B:

    To solve the given system of equations, we can use the substitution method. Let's solve for one variable in terms of the other using the first equation, then substitute it into the second equation to solve for the remaining variable.

    Given equations:
    70 * e + 90 * z = 189 ... (1)
    99 * z - 84 * e = 20 ... (2)

    Step 1: Solve equation (1) for z in terms of e.
    70 * e + 90 * z = 189
    90 * z = 189 - 70 * e
    z = (189 - 70 * e) / 90
    z = 2.1 - (7/9) * e ... (3)

    Step 2: Substitute the expression for z from equation (3) into equation (2).
    99 * (2.1 - (7/9) * e) - 84 * e = 20
    207.9 - 77 * e - 84 * e = 20
    207.9 - 161 * e = 20
    -161 * e = -187.9
    e = 187.9 / 161
    e ≈ 1.167 ... (4)

    Step 3: Substitute the value of e from equation (4) into equation (3) to find z.
    z = 2.1 - (7/9) * 1.167
    z = 2.1 - 0.907
    z ≈ 1.193 ... (5)

    Therefore, the values of e and z are approximately:
    e ≈ 1.167
    z ≈ 1.193

    ### Answer: e ≈ 1.167, z ≈ 1.193

Expected answer: {e: 1879/1610, z: 1234/1035}

Correct? No

🔗 Problem

Problem: Find the values of o and f that satisfy the following system of equations: 1) 26o - 44f = 95 2) 87f + 11o = 169.

Gemma 2B:

    To solve this system of equations, we can use the substitution method. Let's solve equation (1) for o in terms of f, then substitute the expression for o into equation (2) to solve for f. Finally, we'll substitute the value of f back into equation (1) to find o.

    From equation (1), we have:
    26o - 44f = 95
    26o = 44f + 95
    o = (44f + 95) / 26
    o = (44/26)f + 95/26

    Now, let's substitute this expression for o into equation (2):
    87f + 11((44/26)f + 95/26) = 169
    87f + (484/26)f + 1045/26 = 169
    (87 + 484/26)f = 169 - 1045/26
    (2262/26 + 484/26)f = 169 - 1045/26
    (2746/26)f = 169 - 1045/26
    f = (169 - 1045/26) / (2746/26)
    f = (4394 - 1045) / 2746
    f = 3349 / 2746
    f = <<3349/2746=1.22>>1.22

    Now that we have the value of f, let's substitute it back into the equation for o:
    o = (44/26)f + 95/26
    o = (44/26)(1.22) + 95/26
    o = <<(44/26)*(1.22)+(95/26)=5.32>>5.32

    Therefore, the values that satisfy the system of equations are:
    f = 1.22
    o = 5.32

    ### Answer: o = 5.32, f = 1.22

Expected answer: {f: 3349/2746, o: 15701/2746}

Correct? No

🔗 Problem

Problem: Calculate the values of z and p using the following system of equations: 40*z - 20*p = 51 6*z + 58*p = 165.

Gemma 2B:

    To solve for z and p, we can use the substitution method. Let's solve for z in the first equation and substitute it into the second equation to solve for p.

    1. Solve for z in the first equation:
       40*z - 20*p = 51
       40*z = 51 + 20*p
       z = (51 + 20*p) / 40
       z = 51/40 + (20/40)*p
       z = 1.275 + 0.5*p

    2. Substitute z into the second equation:
       6*z + 58*p = 165
       6*(1.275 + 0.5*p) + 58*p = 165
       7.65 + 3*p + 58*p = 165
       7.65 + 61*p = 165
       61*p = 165 - 7.65
       61*p = 157.35
       p = 157.35 / 61
       p = 2.58

    3. Substitute p back into the equation for z:
       z = 1.275 + 0.5*p
       z = 1.275 + 0.5*2.58
       z = 1.275 + 1.29
       z = 2.565

    Therefore, the values of z and p are:
    p = 2.58
    z = 2.565

    ### Answer: z = 2.565, p = 2.58

Expected answer: {p: 3147/1220, z: 3129/1220}

Correct? No

🔗 Problem

Problem: Find the values of s and f from the given system of equations: (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93.

Gemma 2B:

    To solve for s and f, we can use the substitution method. Let's solve for one variable in terms of the other using the second equation and then substitute it into the first equation.

    From the second equation, we can solve for f in terms of s:
    (3 * f) + (10 * s) = 93
    3f = 93 - 10s
    f = (93 - 10s) / 3
    f = 31 - (10s / 3)

    Now, let's substitute this expression for f into the first equation:
    (71 * s) - (89 * f) = 5
    (71 * s) - (89 * (31 - (10s / 3))) = 5
    (71 * s) - (2759 - (890s / 3)) = 5
    (71 * s) - 2759 + (890s / 3) = 5
    71s + (890s / 3) = 2764
    213s + 890s = 8292
    1103s = 8292
    s = 8292 / 1103
    s = <<8292/1103=752/103>>752/103

    Now that we have the value of s, we can substitute it back into the expression for f:
    f = 31 - (10s / 3)
    f = 31 - (10 * (752/103) / 3)
    f = 31 - (7520/309)
    f = (9579/309) - (7520/309)
    f = 2059/309

    Therefore, the values of s and f are:
    s = 752/103
    f = 2059/309

    ### Answer: s = 752/103, f = 2059/309

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

Correct? No