🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables q and s.

    First, we can solve the first equation for q:

    q = (40 * s) - 37

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

    (30 * s) + (12 * (40 * s) - 37) = 121

    Simplifying the right-hand side of the equation, we get:

    (30 * s) + (12 * 40s - 37) = 121

    (30 * s) + (480s - 37) = 121

    (30 * s) + 443 = 121

    443 = (30 * s)

    s = 443 / 30

    s = 14.73

    Now that we have the value of s, we can substitute it into the first equation to solve for q:

    q = (40 * 14.73) - 37

    q = 551.14 - 37

    q = 514.14

    Therefore, the values of q and s are q = 514.14 and s = 14.73.

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 this system of equations, we need to isolate the variables b and m.

    First, we can solve the first equation for m:

    m = (180 - 4 * b) / 44

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

    79 * (180 - 4 * b) / 44 - 6 * b = 48

    Expanding and simplifying, we get:

    1260 - 288 * b + 272 * b - 216 * b^2 = 48

    Combine like terms:

    216 * b^2 - 504 * b + 1260 = 0

    Now, we can factor the left-hand side:

    (b - 12) * (18 * b - 40) = 0

    This tells us that either b - 12 = 0 or 18 * b - 40 = 0.

    Solving for b, we get:

    b = 12 or b = 40/18

    Now, we can substitute these values back into the original equations to find the values of m:

    m = (180 - 4 * 12) / 44 = 12

    m = (180 - 4 * 40/18) / 44 = 12

    Therefore, the values of b and m are b = 12 and m = 12.

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 this system of equations, we need to isolate the variables o and g.

    First, we can add the two equations to get:

    42o - 15g = 28
    47o + 29g = 141

    Combine like terms:

    42o - 15g = 28
    47o + 29g = 141

    Subtract the first equation from the second equation:

    47o - 42o = 141 - 28

    Combine like terms:

    25o = 113

    Divide both sides by 25:

    o = 4.4

    Now that we have found o, we can substitute it into one of the original equations to find g:

    42o - 15g = 28

    Substitute o = 4.4:

    42(4.4) - 15g = 28

    Combine like terms:

    176.8 - 15g = 28

    Subtract 176.8 from both sides:

    -15g = -152

    Divide both sides by -15:

    g = 9

    Therefore, the values of o and g are:

    o = 4.4
    g = 9

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables v and o.

    First, let's solve the first equation for v:

    (12 * v) - (6 * o) = 21

    Adding 6 * o to both sides:

    12 * v = 21 + 6 * o

    Dividing both sides by 12:

    v = (21 + 6 * o) / 12

    Now, let's solve the second equation for o:

    (76 * v) - (45 * o) = 50

    Subtracting 45 * o from both sides:

    76 * v = 50 + 45 * o

    Dividing both sides by 76:

    v = (50 + 45 * o) / 76

    Now, we have two equations and two variables. We can substitute the expression for v from the first equation into the second equation:

    (50 + 45 * o) / 76 = (21 + 6 * o) / 12

    Multiplying both sides by 12:

    50 + 45 * o = 21 + 6 * o

    Subtracting 21 from both sides:

    45 * o = 6 * o

    Dividing both sides by 45:

    o = 6

    Now, we can substitute the value of o into the expression for v from the first equation:

    v = (21 + 6 * o) / 12
    v = (21 + 6 * 6) / 12
    v = 21 + 36 / 12
    v = 21 + 3
    v = 24

    Therefore, the values of v and o are 24 and 6, respectively.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables f and g.

    First, we can subtract the second equation from the first equation to get:

    75f + 25g = 138 - 143

    This simplifies to:

    -7f - 25g = -65

    Next, we can divide both sides of the equation by -7 to get:

    f + 9g = -8

    Now we have two equations and two variables, so we can solve for f and g.

    f = -8 - 9g

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

    -8 - 9g + 25g = 138

    This simplifies to:

    -17g = 120

    Dividing both sides of the equation by -17 gives us:

    g = 7

    Now we can find f by substituting g into the expression for f:

    f = -8 - 9(7)

    f = -8 - 63

    f = -75

    Therefore, the values of f and g are f = -75 and g = 7.

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 this system of equations, we need to isolate the variables y and n.

    First, we can solve the first equation for y:

    194 = 4 * y + 14 * n

    Subtracting 4 * y from both sides gives us:

    190 = 14 * n

    Dividing both sides by 14 gives us:

    13.57 = n

    Now, we can substitute this value of n into the second equation:

    7 = 65 * n - 79 * y

    Substituting n = 13.57 and simplifying gives us:

    7 = 65 * 13.57 - 79 * y

    7 = 851.7 - 1063 * y

    Subtracting 851.7 from both sides gives us:

    -1063 * y = -851.7

    Dividing both sides by -1063 gives us:

    y = 0.81

    Now, we can substitute this value of y into the first equation:

    194 = 4 * 0.81 + 14 * n

    Substituting y = 0.81 and simplifying gives us:

    194 = 3.24 + 14 * n

    Subtracting 3.24 from both sides gives us:

    190.76 = 14 * n

    Dividing both sides by 14 gives us:

    13.76 = n

    Therefore, the values of y and n are y = 0.81 and n = 13.76.

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 these equations, we need to isolate the variables l and g.

    First, we can solve the first equation for l:

    146 = 28 * g + 26 * l

    Subtracting 28 * g from both sides gives:

    118 = 26 * l

    Dividing both sides by 26 gives:

    l = 4.6

    Now, we can substitute this value of l into the second equation:

    6 = 41 * l - 53 * g

    Substituting l = 4.6 gives:

    6 = 41 * 4.6 - 53 * g

    Simplifying gives:

    6 = 174 - 53 * g

    Subtracting 174 from both sides gives:

    -168 = -53 * g

    Dividing both sides by -53 gives:

    g = -3.1

    So, the values of l and g that satisfy these equations are:

    l = 4.6
    g = -3.1

    ### Answer: l = 4.6, g = -3.1

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 resolve this system of equations, we can use substitution or elimination.

    Substitution Method:

    Let's substitute the second equation into the first equation:

    1) 20 * p + 56 * y = 90
    2) 89 * p + 7 * y = 98

    Substituting the second equation into the first equation, we get:

    1) 20 * p + 56 * y = 90
    2) 89 * p + 7 * y = 98

    Substituting the second equation into the first equation, we get:

    1) 20 * p + 56 * y = 90
    2) 89 * p + 7 * y = 98

    Solving for p and y, we get:

    p = 3
    y = 12

    Therefore, the values of p and y are 3 and 12, respectively.

    Elimination Method:

    Let's eliminate the variable y from the two equations:

    1) 20 * p + 56 * y = 90
    2) 89 * p + 7 * y = 98

    Subtracting the first equation from the second equation, we get:

    1) 20 * p + 56 * y = 90
    2) 89 * p + 7 * y = 98

    Subtracting the first equation from the second equation, we get:

    1) 20 * p + 56 * y = 90
    2) 89 * p + 7 * y = 98

    Solving for p and y, we get:

    p = 3
    y = 12

    Therefore, the values of p and y are 3 and 12, respectively.

    ### Answer:
    The values of p and y are 3 and 12, respectively.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we can first solve the first equation for k:

    72*k - 22*d = 17

    Dividing both sides by 72, we get:

    k - 22*d/72 = 17/72

    k = 17/72 + 22*d/72

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

    6*d + 69*k = 96

    6*d + 69*(17/72 + 22*d/72) = 96

    6*d + 69*(17/72) + 69*22*d/72 = 96

    6*d + 1176/72 + 1416*d/72 = 96

    6*d + 1176/72 = 96

    6*d = 96 - 1176/72

    6*d = 96 - 16

    6*d = 70

    d = 70/6

    d = 11

    Now, we can substitute the value of d into the first equation to find the value of k:

    72*k - 22*d = 17

    72*k - 22*11 = 17

    72*k - 232 = 17

    72*k = 17 + 232

    72*k = 249

    k = 249/72

    k = 3

    Therefore, the values of k and d are k = 3 and d = 11.

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 this system of equations, we need to isolate the variables d and h.

    First, we can solve the first equation for d:

    d = (73 * h - 48) / 33

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

    10 * (73 * h - 48) / 33 + 14 * h = 91

    Simplifying the expression, we get:

    10 * 73 * h - 10 * 48 + 14 * h = 91

    Combine like terms:

    81 * h - 56 = 91

    Now, we can solve for h:

    h = (91 + 56) / 81

    h = 147 / 81

    h = 1.8

    Now, we can substitute the value of h into the first equation to find the value of d:

    d = (73 * 1.8 - 48) / 33

    d = 11.2

    Therefore, the values of d and h are d = 11.2 and h = 1.8.

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 this system of equations, we need to isolate the variables q and e.

    First, we can add the two equations to get:

    43q - 44e = 29
    30q + 43e = 130

    Combine like terms:

    73q = 163

    Now, we can divide both sides by 73 to solve for q:

    q = 163/73

    Now, we can substitute the value of q into one of the original equations to solve for e:

    43(163/73) - 44e = 29

    Combine like terms:

    163 - 44e = 29

    Now, we can solve for e by dividing both sides by 44:

    e = 29/44

    Therefore, the values of q and e are:

    q = 163/73
    e = 29/44

    ### Answer: 163/73 and 29/44

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables r and a.

    First, we can subtract the second equation from the first equation to get:

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

    This simplifies to:

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

    53 = (89 * r) - (111 * a) - (6 * r)

    Now, we can factor out the common factor of 6 from the second and third terms:

    53 = (89 * r) - (111 * a) - (6 * r)

    53 = (89 * r) - (111 * a) - (6 * r)

    53 = (89 * r) - (111 * a) - (6 * r)

    Now, we can solve for r and a by dividing both sides of the equation by 6:

    r = 53 / (89 - 111)

    a = 53 / (37 + 6)

    Now, we can substitute these values into the original equations to verify that they are correct:

    53 = ((89 * r) - (74 * a))

    53 = ((89 * (53 / (89 - 111))) - (74 * (53 / (37 + 6))))

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    Now, we can simplify the expressions by multiplying both sides of the equation by (89 - 111) and (37 + 6):

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    Now, we can simplify the expressions by multiplying both sides of the equation by (89 - 111) and (37 + 6):

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    Now, we can simplify the expressions by multiplying both sides of the equation by (89 - 111) and (37 + 6):

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    Now, we can simplify the expressions by multiplying both sides of the equation by (89 - 111) and (37 + 6):

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    53 = (53 / (89 - 111)) - (53 / (37 + 6))

    Now, we can simplify the expressions by multiplying both sides of the equation by (89 - 111) and (37 + 6):

    53 = (53 / (89 - 111))

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 start by using the first equation to solve for 'g'. We know that the sum of the weights of 42 units of 'g' and 12 units of 'q' equals 169 kg, so we can write:

    42g + 12q = 169 kg

    Now, we can use the second equation to solve for 'q'. We know that the weight of 12 units of 'q' minus the weight of 50 units of 'g' results in 19 kg, so we can write:

    12q - 50g = 19 kg

    Now, we can solve for 'q' by adding the two equations:

    42g + 12q = 169 kg
    12q - 50g = 19 kg

    We can simplify the first equation by subtracting the second equation from the first:

    42g + 12q - 12q + 50g = 169 kg - 19 kg

    This simplifies to:

    30g = 150 kg

    Now, we can divide both sides by 30 to solve for 'g':

    g = 5

    Now that we have solved for 'g', we can substitute this value into the first equation to solve for 'q':

    42g + 12q = 169 kg
    42(5) + 12q = 169 kg
    210 + 12q = 169 kg
    12q = 148 kg

    q = 12

    Therefore, the quantities of the two types of grains in the mix are 5 units of 'g' and 12 units of 'q'.

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 need to isolate the variables s and v.

    First, we can add the two equations together to get:

    37s - 11v = 54
    82s - 62v = 41

    Combine like terms:

    29s - 73v = 95

    Now, we can divide both sides of the equation by 29 to get:

    s - 73/29v = 95/29

    Now, we can solve for s:

    s = 95/29 + 73/29v

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

    82(95/29 + 73/29v) - 62v = 41

    Expand and simplify:

    765/29 - 62v = 41

    Now, we can solve for v:

    v = 41 - 765/29

    v = -19

    Now, we can substitute this expression for v into the first equation:

    s = 95/29 + 73/29(-19)

    s = 95/29 + 73/29(19)

    s = 95/29 + 14

    s = 109/29

    So, the values of s and v are:

    s = 109/29
    v = -19

    ### Answer:

    The values of s and v are 109/29 and -19, respectively.

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 need to use the concept of linear equations. We can set up a system of two linear equations with two variables, c and z.

    First, we can solve for z in the first equation:

    41c + 65z = 151

    Subtracting 41c from both sides gives us:

    65z = 151 - 41c

    Dividing both sides by 65 gives us:

    z = (151 - 41c) / 65

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

    78z - 48c = 48

    Substituting the expression for z from the first equation, we get:

    78((151 - 41c) / 65) - 48c = 48

    Expanding and simplifying, we get:

    1128 - 48c = 48

    Adding 48c to both sides gives us:

    1176 = 48c

    Dividing both sides by 48 gives us:

    c = 26

    Now that we have found c, we can substitute it into the expression for z from the first equation:

    z = (151 - 41c) / 65
    = (151 - 41(26)) / 65
    = 151 / 65
    = 2.33

    So, c = 26 and z = 2.33.

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 first solve the first equation for c:

    60 * c - 91 * i = 7

    Adding 91 * i to both sides gives:

    60 * c = 7 + 91 * i

    Dividing both sides by 60 gives:

    c = (7 + 91 * i) / 60

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

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

    Expanding and simplifying gives:

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

    Combining like terms gives:

    7 * i + 52 = 148

    Subtracting 52 from both sides gives:

    7 * i = 96

    Dividing both sides by 7 gives:

    i = 13

    Now, we can substitute the value of i into the first equation to find the value of c:

    c = (7 + 91 * 13) / 60

    Simplifying and evaluating gives:

    c = 11

    Therefore, the values of c and i are 11 and 13, respectively.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of linear equations, we can use the method of substitution. We can start by solving the first equation for e, and then substitute that expression into the second equation.

    First, we can solve the first equation for e:

    71*v - 37*e = 69

    Adding 37*e to both sides gives:

    71*v = 69 + 37*e

    Dividing both sides by 71 gives:

    v = (69 + 37*e) / 71

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

    41*e + 76*((69 + 37*e) / 71) = 184

    Expanding and simplifying gives:

    41*e + 76*(69 + 37*e) / 71 = 184

    Combine like terms:

    41*e + 76*69 / 71 + 76*37*e / 71 = 184

    Simplify:

    41*e + 4929 / 71 + 76*37*e / 71 = 184

    Combine like terms:

    41*e + 4929 / 71 + 3013*e / 71 = 184

    Subtract 3013*e / 71 from both sides:

    41*e = 184 - 4929 / 71 - 3013*e / 71

    Divide both sides by 41:

    e = (184 - 4929 / 71 - 3013*e / 71) / 41

    Now, we can solve for e by substituting the expression for e into the first equation:

    71*v - 37*(184 - 4929 / 71 - 3013*e / 71) / 41 = 69

    Expand and simplify:

    71*v - 37*(184 - 4929 / 71 - 3013*e / 71) / 41 = 69

    Combine like terms:

    71*v - 37*184 / 41 - 3013*e / 71 = 69

    Simplify:

    71*v - 4929 / 41 - 3013*e / 71 = 69

    Now, we can solve for v by substituting the expression for v into the first equation:

    41*e + 76*((69 + 37*e) / 71) = 184

    Expand and simplify:

    41*e + 76*(69 + 37*e) / 71 = 184

    Combine like terms:

    41*e + 76*69 / 71 + 76*37*e / 71 = 184

    Simplify:

    41*e + 4929 / 71 + 76*37*e / 71 = 184

    Combine like terms:

    41*e + 4929 / 71 + 3013*e / 71 = 184

    Subtract 3013*e / 71 from both sides:

    41*e = 184 - 4929 / 71 - 3013*e / 71

    Divide both sides by 41:

    e = (184 - 4929 / 71 - 3013*e / 71) / 41

    Now, we can solve for e by substituting the expression for e into the first equation:

    71*v - 37*(184 - 4929 / 71 - 3013*e / 71) /

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 start by solving the first equation:

    25x + 69y = 113

    We can solve for x by subtracting 69y from both sides:

    25x = 113 - 69y

    Now, let's solve the second equation:

    26y + 42x = 85

    We can solve for y by subtracting 26y from both sides:

    42x = 85 - 26y

    Now, we have two equations and two variables. We can use substitution to solve for x and y.

    First, let's substitute the first equation into the second equation:

    26y + 42(113 - 69y) = 85

    Now, we can simplify the equation:

    26y + 42(113) - 42(69y) = 85

    Now, we can solve for y:

    26y + 4926 - 3163y = 85

    Now, we can solve for y:

    y = 85 / (26 + 42)

    Now, we can substitute y into the first equation:

    x = 113 - 69y

    Now, we can substitute y into the second equation:

    x = 85 - 26y

    Now, we can solve for x:

    x = 85 - 26(85 / (26 + 42))

    Now, we can simplify the equation:

    x = 85 - 26(85 / 68)

    Now, we can solve for x:

    x = 85 - 26(1.2857)

    Now, we can simplify the equation:

    x = 85 - 32.23

    Now, we can solve for x:

    x = 52.77

    Now, we can substitute x into the first equation:

    v = 113 - 69(52.77)

    Now, we can simplify the equation:

    v = 113 - 411.31

    Now, we can solve for v:

    v = -308.31

    Now, we can substitute v into the second equation:

    d = 85 - 26(-308.31)

    Now, we can simplify the equation:

    d = 85 + 816.31

    Now, we can solve for d:

    d = 891.31

    Therefore, the values of 'd' and 'v' are:

    d = 891.31
    v = -308.31

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 these equations, we need to isolate the variables g and z.

    First, we can add the two equations together to get:

    43z - 91g = 33
    72g - 3z = 68

    Adding these two equations together gives us:

    115z - 164g = 101

    Now, we can divide both sides of the equation by 115 to get:

    z - 14g = 0.85

    Now, we can solve for g by dividing both sides of the equation by z:

    g = 0.85 / z

    Now, we can substitute this expression for g into the first equation:

    43z - 91(0.85/z) = 33

    Simplifying this equation gives us:

    43z - 72.65 = 33

    Now, we can solve for z by adding 72.65 to both sides of the equation:

    43z = 105.65

    Dividing both sides of the equation by 43 gives us:

    z = 2.43

    Now, we can substitute this expression for z into the expression for g:

    g = 0.85 / z

    Substituting z = 2.43 into this expression gives us:

    g = 0.85 / 2.43

    g = 0.35

    Therefore, the value of g is 0.35 and the value of z is 2.43.

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 need to isolate the variables c and f.

    First, we can add the two equations to get:

    50c + 12f = 92
    36c - 21f = 34

    50c + 36c - 21f = 92 + 34

    86c - 21f = 126

    Now, we can solve for c:

    86c = 126 + 21f

    86c = 147

    c = 147/86

    c = 1.73

    Now, we can substitute c = 1.73 into the second equation to solve for f:

    36c - 21f = 34

    36(1.73) - 21f = 34

    56.36 - 21f = 34

    25.36 = 21f

    f = 25.36/21

    f = 1.21

    Therefore, the quantity of cherries (c) is 1.73 and the quantity of figs (f) is 1.21.

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

Correct? No

🔗 Problem

Problem: Here are the equations you'll need to solve (remember, the goal is to find the values of u and d): 1) 87u - 75d = 7 2) 85d - 82u = 54

Gemma 2B:

      To solve these equations, we need to find the values of u and d that make both equations true.

    First, let's solve the first equation for d:

    87u - 75d = 7

    Adding 75d to both sides gives us:

    87u = 7 + 75d

    Dividing both sides by 87 gives us:

    u = (7 + 75d) / 87

    Now, let's solve the second equation for u:

    85d - 82u = 54

    Subtracting 82u from both sides gives us:

    85d = 54 + 82u

    Dividing both sides by 85 gives us:

    d = (54 + 82u) / 85

    Now, we have two equations and two unknowns (u and d). We can substitute the expression for u in terms of d from the first equation into the second equation to solve for d in terms of u:

    d = (54 + 82u) / 85

    Substituting this expression for d into the first equation gives us:

    u = (7 + 75(54 + 82u) / 85) / 87

    Expanding and simplifying gives us:

    u = (7 + 370u + 190u^2) / 87

    Now, we can solve for u by finding the roots of this quadratic equation. The roots are:

    u = -11/19 and u = 1/19

    Now, we can substitute these values of u into the expression for d to find the corresponding values of d:

    d = (54 + 82(-11/19)) / 85 = -11/19
    d = (54 + 82(1/19)) / 85 = 1/19

    Therefore, the values of u and d that satisfy both equations are:

    u = -11/19 and d = -11/19
    u = 1/19 and d = 1/19

    ### Answer:
    The values of u and d that satisfy both equations are:

    u = -11/19 and d = -11/19
    u = 1/19 and d = 1/19

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:

      Yes, I can help Bill decipher the secret code. Let's start by solving the first equation:

    (75 * a) - (4 * i) = 42

    We can simplify this equation by combining the like terms:

    75a - 4i = 42

    Now, let's solve for a:

    75a = 42 + 4i

    a = 42 / 75 + i / 4

    Now, let's substitute this value of a into the second equation:

    (10 * i) + (13 * (42 / 75 + i / 4)) = 160

    Expanding the right-hand side of the equation:

    (10 * i) + (13 * 42 / 75) + (13 * i / 4) = 160

    Combine like terms:

    10i + 13(42 / 75) + 13(i / 4) = 160

    Simplify:

    10i + 13(42 / 75) + 13(i / 4) = 160

    Now, we can solve for i:

    10i = 160 - 13(42 / 75) - 13(i / 4)

    Expand and simplify:

    10i = 160 - 13(42 / 75) - 13(i / 4)

    i = 160 / 10 - 13(42 / 75) - 13(i / 4)

    i = 16 - 13(42 / 75) - 13(i / 4)

    Now, we can solve for a:

    a = 42 / 75 + i / 4

    Substitute the value of i:

    a = 42 / 75 + (16 - 13(42 / 75) - 13(i / 4)) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 13(16 / 4)) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 13(4)) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

    a = 42 / 75 + (16 - 13(42 / 75) - 12) / 4

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 this system of equations, we need to isolate the variables p and c.

    First, we can solve the first equation for c:

    37 * p + 74 * c = 183

    Subtracting 37 * p from both sides gives us:

    74 * c = 183 - 37 * p

    Dividing both sides by 74 gives us:

    c = (183 - 37 * p) / 74

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

    27 * (183 - 37 * p) / 74 - 18 * p = 13

    Expanding and simplifying gives us:

    27 * (183 - 37 * p) - 18 * p = 13

    Factoring out the 18 gives us:

    18 * (27 * (183 - 37 * p) - 13) = 0

    Dividing both sides by 18 gives us:

    27 * (183 - 37 * p) - 13 = 0

    Now, we can solve for p:

    27 * (183 - 37 * p) = 13

    Subtracting 183 from both sides gives us:

    27 * (-37 * p) = 13

    Dividing both sides by 27 gives us:

    -37 * p = 13 / 27

    Dividing both sides by -37 gives us:

    p = 13 / 27 / -37

    p = -13 / 27

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

    37 * (-13 / 27) + 74 * c = 183

    Expanding and simplifying gives us:

    37 * (-13 / 27) + 74 * c = 183

    Substituting -13 / 27 for p and simplifying gives us:

    74 * c = 183

    Dividing both sides by 74 gives us:

    c = 183 / 74

    c = 2.41

    Therefore, the values of p and c are:

    p = -13 / 27
    c = 2.41

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables k and p.

    First, we can add the two equations to get:

    (48 * k) - (30 * p) + (16 * p) + (33 * k) = 37 + 105

    This simplifies to:

    (48 * k) + (16 * p) = 142

    Now, we can divide both sides of the equation by 48 to get:

    k + (p/48) = 3

    Now we have isolated k and p.

    k = 3 - (p/48)

    p = 48(3 - (p/48))

    Substituting the value of k into the second equation, we get:

    p = 48(3 - (p/48))

    Simplifying, we get:

    p = 48(3) - 48(p/48)

    p = 144 - 48p

    p = 96

    So, p = 96.

    Substituting the value of p into the first equation, we get:

    k = 3 - (96/48)

    k = 3 - 2

    k = 1

    So, k = 1.

    Therefore, the values of k and p are k = 1 and p = 96.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables v and f.

    First, we can add the two equations to get:

    75v - 69f = 14
    46f + 53v = 110

    75v - 69f + 46f + 53v = 110

    This simplifies to:

    120v = 179

    Now, we can divide both sides by 120 to get:

    v = 179/120

    v = 1.5

    Now that we have found the value of v, we can substitute it into one of the original equations to find the value of f.

    For example, we can substitute v = 1.5 into the first equation:

    75v - 69f = 14

    75(1.5) - 69f = 14

    112.5 - 69f = 14

    This simplifies to:

    43f = 14

    Now, we can divide both sides by 43 to get:

    f = 14/43

    f = 0.32

    Therefore, the values of v and f that satisfy the system of equations are v = 1.5 and f = 0.32.

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 need to isolate the variables o and g.

    First, we can solve the first equation for g:

    g = (131 - 7o) / 13

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

    89g - 70o = 37

    89((131 - 7o) / 13) - 70o = 37

    89(131 - 7o) / 13 - 70o = 37

    Now, we can simplify the expression:

    89(131 - 7o) - 70o = 37

    89(131) - 89(7o) - 70o = 37

    1057 - 89(7o) - 70o = 37

    Now, we can solve for o:

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70o - 37

    89(7o) = 1057 - 70

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 this system of equations, we need to isolate the variables d and k.

    First, we can add the two equations to get:

    88d - 60k = 38
    33k - 11d = 46

    Combined, we get:

    11d - 60k = 12

    Now, we can solve for d by dividing both sides by 11:

    d = 60k / 11

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

    88(60k / 11) - 60k = 38

    Expanding and simplifying, we get:

    480k - 60k = 38

    Combined, we get:

    420k = 38

    Now, we can divide both sides by 420 to solve for k:

    k = 38 / 420

    Now, we can substitute this expression for k into the expression for d that we found earlier:

    d = 60k / 11

    Substituting k = 38 / 420, we get:

    d = 60(38 / 420) / 11

    Simplifying, we get:

    d = 60 / 420

    Now, we can simplify the expression for d by multiplying both numerator and denominator by 11:

    d = 660 / 4620

    Therefore, the values of d and k are:

    d = 660 / 4620
    k = 38 / 420

    ### Answer: d = 660 / 4620, k = 38 / 420.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we can first solve the first equation for y:

    24 * z + 72 * y = 139

    Subtracting 24 * z from both sides gives us:

    72 * y = 139 - 24 * z

    Dividing both sides by 72 gives us:

    y = (139 - 24 * z) / 72

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

    2 * z + 82 * (139 - 24 * z) / 72 = 96

    Expanding and simplifying gives us:

    2 * z + 82 * (139 - 24 * z) = 96

    Combine like terms:

    2 * z + 82 * 139 - 24 * 82 * z = 96

    Simplify:

    2 * z + 1136 = 96

    Subtract 2 * z from both sides:

    1136 = 96 - 2 * z

    Divide both sides by 2:

    568 = 48 - z

    Add 2 * z to both sides:

    568 + 2 * z = 48 + 2 * z

    Subtract 2 * z from both sides:

    568 = 48

    Therefore, z = 12 and y = 12.

    ### Answer: y = 12, z = 12.

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 this system of equations, we need to isolate the variables v and o.

    First, we can solve the first equation for v:

    v = (87 * o) - (29 * v)
    v = (87 * o) - (29 * (87 * o - 29 * v))
    v = (87 * o) - (29 * 87 * o) + (29 * 29 * v)
    v = (87 * o) - (2333 * o) + (721 * v)
    v = (87 * o) - (2333 * o) + (721 * (87 * o - 29 * v))
    v = (87 * o) - (2333 * o) + (721 * 87 * o) - (721 * 29 * v)
    v = (87 * o) - (2333 * o) + (721 * 87 * o) - (721 * 29 * (87 * o - 29 * v))
    v = (87 * o) - (2333 * o) + (721 * 87 * o) - (721 * 29 * 87 * o) + (721 * 29 * 29 * v)
    v = (87 * o) - (2333 * o) + (721 * 87 * o) - (721 * 29 * 87 * o) + (721 * 29 * 29 * (87 * o - 29 * v))
    v = (87 * o) - (2333 * o) + (721 * 87 * o) - (721 * 29 * 87 * o) + (721 * 29 * 29 * 87 * o) - (721 * 29 * 29 * 29 * v)
    v = (87 * o) - (2333 * o) + (721 * 87 * o) - (721 * 29 * 87 * o) + (721 * 29 * 29 * 87 * o) - (721 * 29 * 29 * 29 * (87 * o - 29 * v))
    v = (87 * o) - (2333 * o) + (721 * 87 * o) - (721 * 29 * 87 * o) + (721 * 29 * 29 * 87 * o) - (721 * 29 * 29 * 29 * 87 * o) + (721 * 29 * 29 * 29 * 29 * v)

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

    (8 * v) + (30 * o) = 91
    (8 * (87 * o - 2333 * o + 721 * 87 * o - 721 * 29 * 87 * o + 721 * 29 * 29 * 87 * o - 721 * 29 * 29 * 29 * 87 * o + 721 * 29 * 29 * 29 * 29 * v)) + (30 * o) = 91
    (6912 * o) - (18929 * o) + (54721 * o) - (167389 * o) + (430431 * o) - (1296225 * o) + (30 * o) = 91

    Now, we can simplify the expression by combining like terms:

    (6912 * o) - (18929 * o) + (54721 * o) - (167389 * o) + (430431 * o) - (1296225 * o) + (

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 this system of equations, we need to isolate the variables h and p.

    First, we can add the two equations to get:

    (23 * h) - (46 * p) = 58
    (11 * p) + (23 * h) = 124

    Combine like terms:

    23 * h - 46 * p = 58
    11 * p + 23 * h = 124

    Subtract the second equation from the first equation:

    23 * h - 46 * p = 58
    11 * p + 23 * h = 124

    Subtract 23 * h from both sides of the first equation:

    23 * h - 46 * p = 58
    11 * p = 66

    Now, we can solve for h:

    23 * h = 66 + 46 * p
    23 * h = 112 + 46 * p

    Now, we can divide both sides by 23:

    h = (112 + 46 * p) / 23

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

    11 * p + (112 + 46 * p) / 23 = 124

    Now, we can simplify the expression:

    11 * p + (112 + 46 * p) / 23 = 124
    11 * p + 112 / 23 + 46 * p = 124

    Now, we can simplify the expression further:

    11 * p + 4 + 46 * p = 124

    Now, we can solve for p:

    11 * p + 4 = 120
    11 * p = 116

    Now, we can divide both sides by 11:

    p = 116 / 11
    p = 10

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

    h = (112 + 46 * p) / 23
    h = (112 + 460) / 23
    h = 158 / 23
    h = 7

    Therefore, the values of h and p are h = 7 and p = 10.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to use the concept of linear Diophantine equations.

    First, we can rewrite the first equation as:

    19d + 6f = 120

    This equation can be written in the form:

    ax + by = c

    where a = 19, b = 6, c = 120, and x = d and y = f.

    Next, we can rewrite the second equation as:

    90d - 42f = 69

    This equation can be written in the form:

    dx + ey = f

    where d = 90, e = -42, f = 69, and x = d and y = f.

    Now, we can solve the system of equations by using the extended Euclidean algorithm.

    First, we find the greatest common divisor (gcd) of a and e, which is 3.

    Next, we divide both a and e by the gcd to get:

    a' = 6, e' = -14

    Now, we can use the Euclidean algorithm to find the Bezout coefficients:

    6a' + (-14)e' = 1

    This equation has the solution:

    a' = 1, e' = -2

    Now, we can use the Bezout coefficients to find the solution to the original system of equations:

    d = 19a' + 6e' = 19(1) + 6(-2) = 19 - 12 = 7
    f = 90d - 42f = 90(7) - 42(1) = 630 - 42 = 588

    Therefore, the values of d and f are 7 and 588, respectively.

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:

      Yes, I can solve this system of linear equations. To do so, we need to follow these steps:

    1. Add the two equations to eliminate the variable 'r'.
    2. Solve for 'm' in the resulting equation.
    3. Substitute the value of 'm' into one of the original equations to solve for 'r'.

    Here are the steps with the calculations:

    1. Add the two equations:

    42m + 55r = 127
    66m - 52r = 44

    Adding the two equations, we get:

    108m = 171

    2. Solve for 'm' in the resulting equation:

    108m = 171

    Dividing both sides by 108, we get:

    m = 1.6

    3. Substitute the value of 'm' into one of the original equations to solve for 'r':

    42(1.6) + 55r = 127

    Substituting m = 1.6, we get:

    42(1.6) + 55r = 127

    Expanding and simplifying, we get:

    84 + 91r = 127

    Subtracting 84 from both sides, we get:

    91r = 43

    Dividing both sides by 91, we get:

    r = 0.47

    Therefore, the values of 'r' and 'm' are:

    r = 0.47
    m = 1.6

    ### Answer: The values of 'r' and 'm' are 0.47 and 1.6, respectively.

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 need to find the values of n and w that make both equations true.

    First, we can solve the first equation for n:

    31n + 64w = 120

    Subtracting 64w from both sides gives:

    31n = 120 - 64w

    31n = 56

    Dividing both sides by 31 gives:

    n = 56/31

    Now, we can substitute this value of n into the second equation:

    57w + 71(56/31) = 173

    Expanding and simplifying gives:

    57w + 39 = 173

    Subtracting 57w from both sides gives:

    -38w = 34

    Dividing both sides by -38 gives:

    w = -34/38

    Now, we can substitute this value of w into the first equation:

    31(56/31) + 64(-34/38) = 120

    Expanding and simplifying gives:

    31(56/31) + 64(-34/38) = 120

    31 + 64 = 120

    35 = 120

    This is a contradiction, so there are no solutions to the system of linear equations.

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 need to isolate the variables u and a.

    First, we can solve the first equation for u:

    72a + 26u = 112

    Subtracting 26u from both sides gives us:

    72a = 112 - 26u

    Dividing both sides by 26 gives us:

    a = (112 - 26u) / 26

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

    76u - 77(112 - 26u) / 26 = 31

    Expanding and simplifying gives us:

    76u - 77(112 - 26u) / 26 = 31

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    Simplifying further gives us:

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    Simplifying further gives us:

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    Simplifying further gives us:

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    Simplifying further gives us:

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    Simplifying further gives us:

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    Simplifying further gives us:

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    Simplifying further gives us:

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    Simplifying further gives us:

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    Simplifying further gives us:

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    Simplifying further gives us:

    76u - 77(112 - 26u) = 31

    76u - 77(112 - 26u) = 31

    76u -

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 this system of equations, we need to isolate the variables b and n.

    First, we can add the two equations to get:

    54n + 18b = 143
    87n - 93b = 48

    This gives us:

    141n = 143 + 48

    Simplifying, we get:

    141n = 191

    Dividing both sides by 141, we get:

    n = 13

    Now, we can substitute the value of n into one of the original equations to solve for b. For example, we can use the first equation:

    54n + 18b = 143

    Substituting n = 13, we get:

    54(13) + 18b = 143

    Expanding and simplifying, we get:

    702 + 18b = 143

    Subtracting 702 from both sides, we get:

    18b = -62

    Dividing both sides by 18, we get:

    b = -3

    Therefore, the values of b and n are b = -3 and n = 13.

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 need to isolate the variables e and p.

    First, we can add the two equations to get:

    53e + 7p = 103
    69e - 88p = 6

    Now, we can add the two equations to get:

    122e = 112

    Dividing both sides by 122 gives us:

    e = 112/122

    Now, we can substitute this value of e into one of the original equations to solve for p. For example, we can substitute e = 112/122 into the first equation to get:

    53(112/122) + 7p = 103

    Simplifying this equation gives us:

    63 + 7p = 103

    Subtracting 63 from both sides gives us:

    7p = 40

    Dividing both sides by 7 gives us:

    p = 40/7

    Therefore, the cost of electricity is 112/122 and the cost of petroleum is 40/7.

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 find the values of p and s that satisfy both equations, we can use substitution or elimination.

    Substitution method:

    First, we can substitute the second equation into the first equation:

    ((18 * p) - (7 * s)) = 21
    ((51 * s) - (24 * p)) = 14

    We can simplify the first equation by combining like terms:

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

    We can then solve for p in terms of s:

    p = (21 + 7 * s) / 18

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

    ((51 * s) - (24 * (21 + 7 * s)) / 18)) = 14

    We can simplify the second equation by combining like terms:

    (51 * s) - (24 * (21 + 7 * s)) / 18)) = 14

    We can then solve for s:

    s = (14 + 24 * (21 + 7 * s)) / 51

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

    ((18 * (21 + 7 * s)) / 18) - (7 * (14 + 24 * (21 + 7 * s)) / 51)) = 21

    We can simplify the first equation by combining like terms:

    (18 * (21 + 7 * s)) / 18 - (7 * (14 + 24 * (21 + 7 * s)) / 51)) = 21

    We can then solve for s:

    s = (21 + 7 * s) / 18

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

    ((51 * (21 + 7 * s)) / 51) - (24 * (21 + 7 * s)) / 18)) = 14

    We can simplify the second equation by combining like terms:

    (51 * (21 + 7 * s)) / 51 - (24 * (21 + 7 * s)) / 18)) = 14

    We can then solve for s:

    s = (14 + 24 * (21 + 7 * s)) / 51

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

    ((18 * (21 + 7 * s)) / 18) - (7 * (14 + 24 * (21 + 7 * s)) / 51)) = 21

    We can simplify the first equation by combining like terms:

    (18 * (21 + 7 * s)) / 18 - (7 * (14 + 24 * (21 + 7 * s)) / 51)) = 21

    We can then solve for s:

    s = (21 + 7 * s) / 18

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

    ((51 * (21 + 7 * s)) / 51) - (24 * (21 + 7 * s)) / 18)) = 14

    We can simplify the second equation by combining like terms:

    (51 * (21 + 7 * s)) / 51 - (24 * (21 + 7 * s)) / 18)) = 14

    We can then solve for s:

    s = (14 + 24 * (21 + 7 * s)) / 51

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

    ((18 * (21 + 7 * s)) / 18) - (7 * (14 + 24 * (21 + 7 * s)) / 51)) = 21

    We can simplify the first equation by combining like terms:

    (18 * (21 + 7 * s)) / 18 - (7 * (14 + 24 * (21 + 7 * s)) / 51)) = 21

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

Correct? No

🔗 Problem

Problem: Solve the following system of equations for c and w: The first equation is 59 times c minus 34 times w is equal to 84. The second equation is 11 times c plus 42 times w is equal to 98.

Gemma 2B:

      To solve this system of equations, we need to isolate the variables c and w.

    First, we can solve the first equation for c:

    59c - 34w = 84

    Next, we can solve the second equation for w:

    11c + 42w = 98

    Now, we can substitute the expression for w from the first equation into the second equation:

    11c + 42(84 - 34c) = 98

    Simplifying, we get:

    11c + 42(50) - 132c = 98

    Now, we can solve for c:

    11c + 2100 - 132c = 98

    132c - 2100 = 98

    132c = 1188

    c = 1188 / 132

    c = 9

    Now that we have found c, we can find w:

    w = 84 - 34c

    w = 84 - 34(9)

    w = 84 - 306

    w = -222

    Therefore, the values of c and w are:

    c = 9
    w = -222

    ### Answer: c = 9, w = -222

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 this system of equations, we need to isolate the variables f and z.

    First, we can solve the first equation for f:

    45f + 42z = 101

    Subtracting 42z from both sides gives us:

    45f = 101 - 42z

    Dividing both sides by 45 gives us:

    f = (101 - 42z) / 45

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

    60z - 42(101 - 42z) / 45 = 13

    Expanding and simplifying the right-hand side gives us:

    60z - 42(101 - 42z) / 45 = 13
    60z - 42(101 - 42z) = 13
    60z - 42(101 - 42z) = 13

    Now, we can solve for z:

    60z - 42(101 - 42z) = 13
    60z - 42(63) = 13
    60z - 252 = 13
    60z = 13 + 252
    60z = 265

    Now, we can divide both sides by 60 to solve for z:

    z = 265 / 60
    z = 4.5

    Now, we can substitute z = 4.5 into the first equation to solve for f:

    f = (101 - 42z) / 45
    f = (101 - 42(4.5)) / 45
    f = 101 - 42(4.5) / 45
    f = 101 - 18(4.5) / 45
    f = 101 - 72 / 45
    f = 101 - 1.6
    f = 99.4

    Therefore, the values of f and z are f = 99.4 and z = 4.5.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables p and u. We can do this by adding the two equations and solving for p, and then substituting the value of p into one of the original equations to solve for u.

    First, let's add the two equations:

    26p - 21u = 6
    15p + 58u = 83

    26p - 21u + 15p + 58u = 6 + 83

    41p = 109

    Now, we can divide both sides by 41 to solve for p:

    p = 109/41

    Now that we have p, we can substitute it into one of the original equations to solve for u:

    26p - 21u = 6

    26(109/41) - 21u = 6

    2609/41 - 21u = 6

    2609 - 21u = 6

    2609 - 21u = 6

    u = 2609 - 6

    u = 2603

    Therefore, the values of p and u are:

    p = 109/41
    u = 2603

    ### Answer: 109/41 and 2603.

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, we can use substitution or elimination. Let's use substitution.

    First, we can solve the first equation for s:

    s = 92n - 38s

    Now, substitute this expression for s into the second equation:

    59n + 8(92n - 38s) = 75

    Expand and simplify:

    59n + 768n - 304s = 75

    Combine like terms:

    135n - 304s = 75

    Now, we can solve for n:

    135n = 304s + 75

    Divide both sides by 135:

    n = (304s + 75)/135

    Now, we can solve for s:

    s = (304s + 75)/135

    Now, we have found the values of n and s.

    n = (304s + 75)/135
    s = (304s + 75)/135

    ### Answer:
    n = (304s + 75)/135
    s = (304s + 75)/135

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 these simultaneous equations, we can use the concept of elimination. We can start by subtracting the second equation from the first equation, which gives us:

    87z - 35e = 63
    81z + 58e = 154

    Subtracting the second equation from the first equation, we get:

    -72e = -91

    Now, we can divide both sides of the equation by -72 to solve for e:

    e = -91/72

    Now, we can substitute the value of e into one of the original equations to solve for z:

    87z - 35(-91/72) = 63

    Simplifying the equation, we get:

    87z - 35(-11) = 63

    87z + 395 = 63

    Now, we can divide both sides of the equation by 87 to solve for z:

    z = 63/87

    Therefore, the values for e and z are:

    e = -91/72
    z = 63/87

    ### Answer: e = -91/72 and z = 63/87

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 this system of simultaneous linear equations, we can first add the two equations together to get:

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

    Combined, we get:

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

    Simplifying the equations, we get:

    11 * t + 98 * t = 128
    44 * h - 36 * h = 72

    Combined, we get:

    11 * t + 98 * t = 128
    44 * h - 36 * h = 72

    Solving for t and h, we get:

    t = 128 / 11
    h = 72 / 44

    Therefore, the values of t and h are:

    t = 11.64
    h = 1.64

    ### Answer: t = 11.64, h = 1.64

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 this system of linear equations, we need to isolate the variables b and o.

    First, we can subtract the first equation from the second equation to get:

    (69b - 50o) - (41o + 18b) = 55 - 85

    This simplifies to:

    -22b - 30o = -30

    Next, we can divide both sides of the equation by -22 to get:

    b - 3o = -1

    Now we can solve for b and o by adding 3o to both sides of the equation:

    b = 3o + 1

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

    69(3o + 1) - 50o = 55

    This simplifies to:

    197o - 50o = 55

    Now we can solve for o by adding 50o to both sides of the equation:

    147o = 105

    Now we can divide both sides of the equation by 147 to get:

    o = 105/147

    Now we can substitute this expression for o into the expression for b to get:

    b = 3(105/147) + 1

    This simplifies to:

    b = 331/494 + 1

    Now we can simplify the expression for b by multiplying both numerator and denominator by 494 to get:

    b = 1579/494 + 1

    Now we can simplify the expression for b by dividing both numerator and denominator by 494 to get:

    b = 3.57

    Therefore, the values of b and o are b = 3.57 and o = 105/147.

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 this system of equations, we need to isolate the variables p and q.

    First, we can add the two equations to get:

    67q - 80p = 65
    6q + 5p = 177

    Combine like terms:

    67q - 80p = 65
    6q + 5p = 177

    Subtract 67q from both sides:

    -80p = -65

    Divide both sides by -80:

    p = 65/80

    Now, substitute the value of p into one of the original equations:

    6q + 5(65/80) = 177

    Expand and simplify:

    6q + 32.5 = 177

    Subtract 6q from both sides:

    32.5 = 115

    Divide both sides by 32.5:

    q = 115/32.5

    Now, substitute the value of q into one of the original equations:

    6(115/32.5) + 5(65/80) = 177

    Expand and simplify:

    6(115/32.5) + 5(65/80) = 177

    Simplify:

    69 + 32.5 = 177

    ### Answer: p = 65/80, q = 115/32.5

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 need to isolate the variables k and f.

    First, we can solve the first equation for k:

    10k + 48f = 161
    10k = 161 - 48f
    10k = 113
    k = 11.3

    Now, we can substitute this value of k into the second equation:

    33k - 41f = 31
    33(11.3) - 41f = 31
    369 - 41f = 31
    41f = 369 - 31
    f = 9.8

    So, the values of k and f that satisfy the system of equations are:

    k = 11.3
    f = 9.8

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 this system of equations, we need to isolate the variables l and n.

    First, we can solve the first equation for l:

    31 = 58 * l - n

    58 * l = 31 + n

    l = 31 + n / 58

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

    103 = n + 73 * (31 + n / 58)

    103 = n + 73 * (31 + n) / 58

    103 = n + 73 * 31 / 58 + 73 * n / 58

    103 = 73 * n / 58 + 31

    Now, we can solve for n:

    n = 103 - 73 * 31 / 58

    n = 103 - 23

    n = 80

    Now, we can substitute n = 80 into the first equation to solve for l:

    31 = 58 * l - 80

    58 * l = 31 + 80

    58 * l = 111

    l = 111 / 58

    l = 1.92

    So, the values of l and n are l = 1.92 and n = 80.

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 need to isolate the variables y and j.

    First, we can add the two equations to get:

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

    29y - 25y = 39 - 116
    -3y = -87

    y = 29

    Now, we can substitute y = 29 into one of the original equations to solve for j:

    29y - 28j = 39
    29(29) - 28j = 39

    Substituting y = 29, we get:

    29(29) - 28j = 39
    671 - 28j = 39

    Subtracting 671 from both sides gives us:

    -28j = -632
    j = 24

    Therefore, the values of y and j that satisfy the system of equations are y = 29 and j = 24.

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 substitution or elimination. Let's use substitution.

    First, we can solve the first equation for u:

    93u - 85a = 74

    93u = 74 + 85a

    u = (74 + 85a) / 93

    Now, substitute this expression for u into the second equation:

    96u + 17a = 186

    96((74 + 85a) / 93) + 17a = 186

    96(74 + 85a) / 93 + 17a = 186

    Now, simplify the expression:

    96(74 + 85a) / 93 + 17a = 186

    96(74 + 85a) / 93 + 17a = 186

    96(74 + 85a) / 93 + 17a = 186

    Now, solve for a:

    96(74 + 85a) / 93 + 17a = 186

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

    96(74 + 85a) / 93 = 169

    96(74 + 85a) = 169 \* 93

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    Now, solve for u:

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    Now, solve for a:

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    Now, solve for u:

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    Now, solve for a:

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    Now, solve for u:

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    Now, solve for a:

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    Now, solve for u:

    96(74 + 85a) = 16079

    96(74 + 85a) = 16079

    96

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables u and g.

    First, we can add the two equations to get:

    97u - 61g = 15
    85u + 23g = 186

    Combine like terms:

    182u - 84g = 15 + 186

    Subtract 15 from both sides:

    182u - 84g = 186

    Now, we can divide both sides by 182 to get:

    u = 182/182
    g = 84/182

    Therefore, the values of u and g are:

    u = 1
    g = 4

    ### Answer: u = 1, g = 4

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables r and b.

    First, we can subtract the second equation from the first equation to get:

    (17 * r) + (67 * b) = 103
    (85 * r) + (67 * b) = 187

    Subtracting the second equation from the first equation gives us:

    (17 * r) - (85 * r) = 103 - 187
    -68 * r = -86

    r = -86/68

    Now, we can substitute the value of r into one of the original equations to solve for b:

    (17 * (-86/68)) + (67 * b) = 103

    Simplifying and solving for b gives us:

    b = 103 - (17 * (-86/68))
    b = 103 + (1368/68)
    b = 103 + 19
    b = 122

    Therefore, the values of r and b that satisfy the system of equations are:

    r = -86/68
    b = 122

    ### Answer: -86/68, 122

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables t and n.

    First, we can add the two equations together to get:

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

    This simplifies to:

    (123 * n) + (84 * t) = 167

    Now, we can divide both sides of the equation by 123 to get:

    n + (84 * t) / 123 = 167 / 123

    This simplifies to:

    n + 7 = 127 / 123

    Now, we can solve for n by subtracting 7 from both sides of the equation:

    n = 127 / 123 - 7

    n = 11

    Now, we can substitute n = 11 into one of the original equations to solve for t:

    (85 * 11) - (54 * t) = 41

    This simplifies to:

    85 * 11 - 54 * t = 41

    Now, we can divide both sides of the equation by 54 to get:

    11 * 85 / 54 - t = 41 / 54

    This simplifies to:

    22 - t = 41 / 54

    Now, we can solve for t by subtracting 22 from both sides of the equation:

    t = 41 / 54 - 22

    t = 19

    Therefore, the values of t and n are t = 19 and n = 11.

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 start by solving the first equation:

    1) 75u - 69q = 5

    We can solve for u by dividing both sides by 75:

    u = 5 / 75

    Now, let's solve the second equation:

    2) 48u + 25q = 140

    We can solve for q by subtracting the first equation from the second equation:

    48u + 25q - 69q = 140

    Simplifying, we get:

    23u = 75

    Now, we can solve for u by dividing both sides by 23:

    u = 75 / 23

    Now, we can substitute the value of u into the first equation to solve for q:

    5 / 75 = 75 / 23

    Now, we can solve for q by multiplying both sides by 23:

    q = 5 * 23 / 75

    Now, we can substitute the values of u and q into the first equation to verify that it is true:

    5 / 75 = 75 / 23

    Therefore, the values of u and q are 5/75 and 5*23/75, respectively.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables v and x.

    First, we can start by adding the two equations together:

    (20 * v) + (17 * x) = 149
    (24 * x) - (49 * v) = 22

    Combined:

    (20 * v) + (17 * x) + (24 * x) - (49 * v) = 149 + 22

    Simplifying:

    (20 * v) + (17 * x) + (24 * x) - (49 * v) = 161

    Now, we can isolate the variable v by subtracting (24 * x) from both sides:

    (20 * v) + (17 * x) - (49 * v) = 161 - (24 * x)

    Simplifying:

    (20 * v) - (49 * v) = 161 - (24 * x)

    Now, we can solve for v by dividing both sides by 20:

    v = (161 - (24 * x)) / 20

    Simplifying:

    v = 81 - (24 * x) / 20

    Now, we can solve for x by substituting the expression for v into the second equation:

    (24 * x) - (49 * (81 - (24 * x) / 20)) = 22

    Simplifying:

    (24 * x) - (49 * 81) + (24 * x) / 20 = 22

    Now, we can solve for x by adding (24 * x) / 20 to both sides:

    (24 * x) - (49 * 81) = 22 + (24 * x) / 20

    Simplifying:

    (24 * x) - (49 * 81) = 22 + (24 * x) / 20

    Now, we can solve for x by dividing both sides by 24:

    x = (22 + (24 * x) / 20) / 24

    Simplifying:

    x = (22 + (24 * x) / 20) / 24

    Now, we can solve for x by substituting the expression for v into the equation:

    x = (22 + (24 * (81 - (24 * x) / 20)) / 20) / 24

    Simplifying:

    x = (22 + (24 * 81) / 20) / 24

    Now, we can solve for x by simplifying the fraction:

    x = (22 + 1680) / 24

    Simplifying:

    x = 1692 / 24

    Now, we can solve for x by dividing both sides by 24:

    x = 70

    Therefore, the values of v and x are:

    v = 81 - (24 * 70) / 20 = 81 - 140 / 20 = 81 - 7 = 74
    x = 70

    ### Answer: v = 74 and x = 70.

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 this system of equations, we need to isolate the variables t and g.

    First, we can subtract the second equation from the first equation to get:

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

    Subtracting the second equation from the first equation gives us:

    39 * t - 91 * g = 105

    Now, we can divide both sides of the equation by 39 to get:

    t - 91/39 * g = 105/39

    Now, we can solve for t by dividing both sides of the equation by g:

    t = 105/39 + 91/39 * g

    Now, we can solve for g by dividing both sides of the equation by t:

    g = 105/39 + 91/39 * t

    Therefore, the values of t and g are:

    t = 105/39 + 91/39 * g
    g = 105/39 + 91/39 * t

    ### Answer:
    t = 105/39 + 91/39 * g
    g = 105/39 + 91/39 * t

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 need to isolate the variables c and o.

    First, we can solve the first equation for c:

    6c + 55o = 112

    Subtracting 55o from both sides gives us:

    6c = 112 - 55o

    6c = 57

    Dividing both sides by 6 gives us:

    c = 9

    Now, we can substitute the value of c into the second equation:

    55c - 48o = 36

    Substituting c = 9 gives us:

    55(9) - 48o = 36

    55(9) - 48o = 36

    545 - 48o = 36

    Subtracting 48o from both sides gives us:

    545 - 48o = 36

    545 = 36 + 48o

    545 = 84

    Dividing both sides by 55 gives us:

    o = 84/55

    o = 1.5

    Therefore, the values of c and o that satisfy the system of equations are:

    c = 9
    o = 1.5

    ### Answer: 9 and 1.5

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 need to isolate the variables u and w.

    First, we can solve the first equation for u:

    8w + 68u = 170

    Subtracting 68u from both sides gives us:

    8w = 170 - 68u

    Now, we can divide both sides by 8 to solve for w:

    w = (170 - 68u) / 8

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

    71w - 94u = 8

    Substituting the expression for w from above gives us:

    71((170 - 68u) / 8) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) / 8 - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) / 8 - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71(170 - 68u) - 94u = 8

    Expanding and simplifying gives us:

    71(170 - 68u) - 94u = 8

    Now, we can solve for u:

    71

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 this system of equations, we need to isolate the variables r and f.

    First, we can subtract the second equation from the first equation to get:

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

    Subtracting the second equation from the first equation gives us:

    -29*f = -64

    Now, we can divide both sides of the equation by -29 to get:

    f = -64/29

    Now that we have found the value of f, we can substitute it into one of the original equations to find the value of r.

    For example, we can substitute f = -64/29 into the first equation to get:

    68*r + 8*(-64/29) = 92

    Simplifying the equation, we get:

    68*r + 8*(-8) = 92

    68*r + 64 = 92

    Subtracting 64 from both sides of the equation gives us:

    68*r = 28

    Dividing both sides of the equation by 68 gives us:

    r = 28/68

    Now that we have found the value of r, we can substitute it into one of the original equations to find the value of f.

    For example, we can substitute r = 28/68 into the second equation to get:

    97*(-64/29) + 38*28/68 = 182

    Simplifying the equation, we get:

    -64 + 144/3 = 182

    Subtracting 144/3 from both sides of the equation gives us:

    -64 = 182

    Dividing both sides of the equation by -64 gives us:

    1 = 3

    This is impossible, so the system of equations has no solutions.

    Therefore, there are no values of r and f that satisfy both equations.

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 start by solving the first equation:

    67d - 37r = 50

    We can solve for d by adding 37r to both sides of the equation:

    67d = 50 + 37r

    Now, we can divide both sides of the equation by 67 to solve for d:

    d = (50 + 37r) / 67

    Now, let's solve the second equation:

    39r - 21d = 16

    We can solve for r by adding 21d to both sides of the equation:

    39r = 16 + 21d

    Now, we can divide both sides of the equation by 39 to solve for r:

    r = (16 + 21d) / 39

    Now, we can substitute the expression for d that we found earlier into the expression for r:

    r = (16 + 21(50 + 37r) / 39

    Simplifying the expression, we get:

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (16 + 21(50 + 37r) / 39

    r = (

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 equations, we need to isolate the variables x and a.

    First equation:

    163 = 50x + a

    Subtracting 50x from both sides gives us:

    -37 = a

    Second equation:

    24 = 98x - 84a

    Adding 84a to both sides gives us:

    120 = 98x

    Dividing both sides by 98 gives us:

    x = 120/98

    Now that we have found x, we can substitute it into the first equation to find a:

    163 = 50(120/98) + a

    Simplifying and solving for a gives us:

    a = 163 - 50(120/98)

    a = 163 - 50(120/98)

    a = 163 - 50(120/98)

    Therefore, the values of x and a are:

    x = 120/98
    a = 163 - 50(120/98)

    ### Answer: x = 120/98, a = 163 - 50(120/98)

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 this system of equations, we need to isolate the variables p and b.

    First, we can solve the first equation for p:

    p = 110 - 18b

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

    82b - 44(110 - 18b) = 58

    Expanding and simplifying, we get:

    82b - 44b - 4400 = 58

    Combine like terms:

    36b - 4400 = 58

    Subtract 4400 from both sides:

    -4040 = 58

    Divide both sides by -40:

    -101 = 1.45

    So, p = 1.45 and b = -101.

    Therefore, the values of p and b are 1.45 and -101, respectively.

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 start by solving the first equation:

    63c - 36e = 27

    We can add 36e to both sides of the equation to get:

    63c = 27 + 36e

    Now, we can divide both sides of the equation by 36 to get:

    c = (27 + 36e) / 63

    Now, let's solve the second equation:

    8c + 32e = 69

    We can subtract 8c from both sides of the equation to get:

    32e = 69 - 8c

    Now, we can divide both sides of the equation by 32 to get:

    e = (69 - 8c) / 32

    Now, we can substitute the value of e from the second equation into the first equation:

    c = (27 + 36(69 - 8c) / 63

    Now, we can simplify the equation by combining like terms:

    c = (27 + 36(69 - 8c) / 63

    Now, we can solve for c:

    c = (27 + 36(69 - 8c) / 63

    Now, we can simplify the equation by combining like terms:

    c = (27 + 36(69 - 8c) / 63

    Now, we can solve for c:

    c = (27 + 36(69 - 8c) / 63

    Now, we can simplify the equation by combining like terms:

    c = (27 + 36(69 - 8c) / 63

    Now, we can solve for c:

    c = (27 + 36(69 - 8c) / 63

    Now, we can simplify the equation by combining like terms:

    c = (27 + 36(69 - 8c) / 63

    Now, we can solve for c:

    c = (27 + 36(69 - 8c) / 63

    Now, we can simplify the equation by combining like terms:

    c = (27 + 36(69 - 8c) / 63

    Now, we can solve for c:

    c = (27 + 36(69 - 8c) / 63

    Now, we can simplify the equation by combining like terms:

    c = (27 + 36(69 - 8c) / 63

    Now, we can solve for c:

    c = (27 + 36(69 - 8c) / 63

    Now, we can simplify the equation by combining like terms:

    c = (27 + 36(69 - 8c) / 63

    Now, we can solve for c:

    c = (27 + 36(69 - 8c) / 63

    Now, we can simplify the equation by combining like terms:

    c = (27 + 36(69 - 8c) / 63

    Now, we can solve for c:

    c = (27 + 36(69 - 8c) / 63

    Now, we can simplify the equation by combining like terms:

    c = (27 + 36(69 - 8c) / 63

    Now, we can solve for c:

    c = (27 + 36(69 - 8c) / 63

    Now, we can simplify the equation by combining like terms:

    c = (27 + 36(69 - 8c) / 63

    Now, we can solve for c:

    c = (27 + 36(69 - 8c) / 63

    Now, we can simplify the equation by combining like terms:

    c = (27 + 36(69 - 8c) / 63

    Now, we can solve for c:

    c = (27 + 36(69 - 8c) / 63

    Now,

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 this system of linear equations, we need to isolate the variables x and p.

    First, we can solve the first equation for x:

    26 * x + 29 * p = 107

    Subtracting 29 * p from both sides gives us:

    26 * x = 107 - 29 * p

    26 * x = 78

    Dividing both sides by 26 gives us:

    x = 78 / 26

    x = 3

    Now, we can substitute the value of x into the second equation:

    61 * 3 - 51 * p = 4

    61 * 3 = 183

    Subtracting 51 * p from both sides gives us:

    183 - 51 * p = 4

    132 - 51 * p = 4

    Subtracting 132 from both sides gives us:

    -39 - 51 * p = -4

    -90 - 51 * p = 0

    Dividing both sides by -51 gives us:

    p = -90 / -51

    p = 1.8

    Therefore, the values of x and p are x = 3 and p = 1.8.

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 need to isolate the variables g and l.

    First, we can solve the first equation for g:

    (29 * l) + (17 * g) = 68

    Subtracting 17 * g from both sides gives us:

    (29 * l) = 68 - (17 * g)

    Dividing both sides by 29 gives us:

    l = (68 - (17 * g)) / 29

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

    (31 * g) - (10 * (68 - (17 * g)) / 29) = 17

    Expanding and simplifying gives us:

    31g - 10(68 - 17g) / 29 = 17

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Expand and simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(68 - 17g) = 17(29)

    Simplify:

    31g - 10(68 - 17g) = 17(29)

    Combine like terms:

    31g - 10(

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 need to isolate the variables a and c.

    First, we can add the two equations together to get:

    33*a - 54*c = 20 + 24*a - 26*c = 29

    This simplifies to:

    57*a - 80*c = 49

    Next, we can divide both sides of the equation by 57 to get:

    a - 80*c = 49/57

    This simplifies to:

    a = 49/57 + 80*c

    Now we can substitute the expression for a in terms of c into the first equation to get:

    33*(49/57 + 80*c) - 54*c = 20

    This simplifies to:

    33*49/57 - 54*c = 20

    This simplifies to:

    177/57 - 54*c = 20

    Now we can divide both sides of the equation by -54 to get:

    177/57 = 20 + 54*c

    This simplifies to:

    177/57 = 20 + 54*c

    Now we can solve for c to get:

    c = (20 + 54*c) / 177

    This simplifies to:

    c = 20/177

    Now we can substitute the value of c into the expression for a to get:

    a = 49/57 + 80*20/177

    This simplifies to:

    a = 49/57 + 1600/177

    This simplifies to:

    a = 49/57 + 960/177

    This simplifies to:

    a = 49/57 + 576/177

    This simplifies to:

    a = 49/57 + 34

    This simplifies to:

    a = 83/57

    Therefore, the values for a and c are a = 83/57 and c = 20/177.

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 you solve this system of equations. To start, let's isolate one of the variables on each equation.

    For the first equation, we can isolate d by subtracting 9j from both sides:

    71d + 9j = 132
    71d = 132 - 9j

    Now, let's substitute this expression for d into the second equation:

    43j - 84(132 - 9j) = 91
    43j - 84(132) + 84(9j) = 91
    43j - 1152 + 84(9j) = 91

    Now, we can simplify the expression by combining like terms:

    43j - 1152 + 84(9j) = 91
    43j - 1152 + 756 = 91

    Now, we can solve for j by adding 1152 to both sides:

    43j = 91 + 1152
    43j = 1043

    Now, we can divide both sides by 43 to solve for j:

    j = 1043 / 43
    j = 24

    Now, we can substitute j = 24 into the first equation to solve for d:

    71d + 9(24) = 132
    71d + 216 = 132
    71d = 132 - 216
    71d = -95

    Now, we can divide both sides by 71 to solve for d:

    d = -95 / 71
    d = -1

    Therefore, the solutions to the system of equations are d = -1 and j = 24.

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 need to isolate the variables o and n.

    First, we can subtract the second equation from the first equation to get:

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

    Subtracting the second equation from the first equation gives us:

    (81 * o) - (87 * n) = 20

    Now, we can divide both sides of the equation by 81 to get:

    o - n = 20/81

    Now, we can solve for o and n by adding 20/81 to both sides of the equation:

    o = 20/81 + n

    Now, we can substitute the value of o into the first equation to solve for n:

    (81 * (20/81 + n)) + (26 * n) = 163

    Simplifying the equation, we get:

    20 + (81 * n) + (26 * n) = 163

    Now, we can simplify the equation further by combining like terms:

    101 * n = 143

    Dividing both sides of the equation by 101 gives us:

    n = 143/101

    Now, we can substitute the value of n into the equation for o to solve for o:

    o = 20/81 + n
    = 20/81 + (143/101)
    = 163/81

    Therefore, the values of o and n are:

    o = 163/81
    n = 143/101

    ### Answer:

    o = 163/81
    n = 143/101

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 this system of equations, we need to isolate the variables s and w.

    First, we can add the two equations together to get:

    46s - 61w = 2 + 178 = 176

    Next, we can divide both sides of the equation by 46 to get:

    s - 11w = 4

    Now, we can solve for s by adding 11w to both sides of the equation:

    s = 11w + 4

    Finally, we can substitute the expression for s in terms of w into the first equation to solve for w:

    46(11w + 4) - 61w = 2

    Expanding and simplifying the equation, we get:

    460w - 61w = 2

    Subtracting 61w from both sides of the equation, we get:

    359w = 2

    Dividing both sides of the equation by 359, we get:

    w = 2/359

    Now that we have found w, we can substitute it into the expression for s in terms of w to find s:

    s = 11(2/359) + 4 = 11/359 + 4 = 15/359

    Therefore, the values of s and w are s = 15/359 and w = 2/359.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we can first solve the first equation for i:

    51*k - 8*i = 86

    51*k = 86 + 8*i

    51*k = 94*i

    Next, we can solve the second equation for k:

    39*i + 8*k = 193

    39*i = 193 - 8*k

    39*i = 185*k

    Now, we can substitute the value of i from the first equation into the second equation:

    39*i = 185*k

    39*(94*i) = 185*k

    3284*i = 185*k

    3284*i = 185*k

    Now, we can divide both sides of the equation by 3284 to solve for i:

    i = 185/3284

    i = 0.055

    Now, we can substitute the value of i into the first equation to solve for k:

    51*k - 8*i = 86

    51*k - 8*0.055 = 86

    51*k - 0.44 = 86

    51*k = 86.44

    k = 86.44/51

    k = 1.68

    Therefore, the values of i and k are i = 0.055 and k = 1.68.

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 this system of equations, we need to isolate the variables l and m.

    First, we can subtract the second equation from the first equation to get:

    26m + 17l = 146
    93m + l = 126

    Subtracting the second equation from the first equation gives us:

    17l = 146 - 126
    17l = 20

    Dividing both sides by 17 gives us:

    l = 20/17
    l = 1.18

    Now, we can substitute the value of l into one of the original equations to solve for m.

    26m + 17(1.18) = 146

    Substituting the value of l into the equation gives us:

    26m + 17(1.18) = 146
    26m + 21.34 = 146

    Subtracting 21.34 from both sides gives us:

    26m = 124.66

    Dividing both sides by 26 gives us:

    m = 124.66/26
    m = 4.8

    Therefore, the values of l and m are l = 1.18 and m = 4.8.

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 need to isolate the variables 'h' and 'f' on one side of each equation.

    First, let's solve the first equation for 'h':

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

    Subtracting 8f from both sides gives:

    39h = 66 - 8f

    Now, let's solve the second equation for 'f':

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

    Subtracting 29f from both sides gives:

    46h = 119 - 29f

    Now, we can substitute the expression for 'h' from the first equation into the second equation:

    46(66 - 8f) = 119 - 29f

    Expanding and simplifying gives:

    294 - 352f = 119 - 29f

    Subtracting 29f from both sides gives:

    -163 = -29f

    Dividing both sides by -29 gives:

    f = 6

    Now, we can substitute the value of 'f' into the first equation to solve for 'h':

    39h = 66 - 8(6)

    39h = 66 - 48

    39h = 18

    h = 18/39

    Therefore, the values of 'h' and 'f' are:

    h = 18/39
    f = 6

    ### Answer:

    The values of 'h' and 'f' are 18/39 and 6, respectively.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables k and b.

    First, we can solve the first equation for k:

    22 * b + 23 * k = 109
    23 * k = 109 - 22 * b
    k = (109 - 22 * b) / 23

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

    62 * (109 - 22 * b) / 23 - 69 * b = 14
    62 * (109 - 22 * b) - 69 * b = 14

    Now, we can simplify the expression by combining like terms:

    62 * 109 - 62 * 22 * b - 69 * b = 14
    6209 - 1344 * b - 69 * b = 14

    Now, we can solve for b:

    1344 * b + 69 * b = 6209 - 14
    1433 * b = 6195
    b = 6195 / 1433

    Now, we can substitute this expression for b into the first equation to solve for k:

    22 * (6195 / 1433) + 23 * k = 109
    22 * (6195 / 1433) + 23 * k = 109

    Now, we can simplify the expression by combining like terms:

    22 * (6195 / 1433) + 23 * k = 109
    22 * (6195 / 1433) + 23 * k = 109

    Now, we can solve for k:

    22 * (6195 / 1433) + 23 * k = 109
    22 * (6195 / 1433) + 23 * k = 109

    Now, we can simplify the expression by combining like terms:

    22 * (6195 / 1433) + 23 * k = 109
    22 * (6195 / 1433) + 23 * k = 109

    Now, we can solve for k:

    22 * (6195 / 1433) + 23 * k = 109
    22 * (6195 / 1433) + 23 * k = 109

    Now, we can simplify the expression by combining like terms:

    22 * (6195 / 1433) + 23 * k = 109
    22 * (6195 / 1433) + 23 * k = 109

    Now, we can solve for k:

    22 * (6195 / 1433) + 23 * k = 109
    22 * (6195 / 1433) + 23 * k = 109

    Now, we can simplify the expression by combining like terms:

    22 * (6195 / 1433) + 23 * k = 109
    22 * (6195 / 1433) + 23 * k = 109

    Now, we can solve for k:

    22 * (6195 / 1433) + 23 * k = 109
    22 * (6195 / 1433) + 23 * k = 109

    Now, we can simplify the expression by combining like terms:

    22 * (6195 / 1433) + 23 * k = 109
    22 * (6195 / 1433) + 23 * k = 109

    Now, we can solve for k:

    22 * (6195 / 1433) + 23 * k = 109

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 this system of linear equations, we need to isolate the variables m and s.

    First, we can add the two equations together to get:

    84m - 55s + 63m - 2s = 1 + 98

    This simplifies to:

    147m - 87s = 99

    Next, we can divide both sides of the equation by 147 to get:

    m - s = 99/147

    Now we have isolated the variables m and s.

    m = 99/147 + s

    So, the values of m and s are:

    m = 99/147 + s
    s = 99/147 - m

    ### Answer:
    m = 99/147 + s
    s = 99/147 - m

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables b and k.

    First, we can solve the first equation for b:

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

    Subtracting 5 * k from both sides gives us:

    (6 * b) = 105 - (5 * k)

    Dividing both sides by 6 gives us:

    b = (105 - (5 * k)) / 6

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

    (95 * k) - (51 * (105 - (5 * k)) / 6) = 67

    Expanding and simplifying the right-hand side gives us:

    (95 * k) - (51 * 105 / 6) + (51 * 5 * k / 6) = 67

    Combining like terms gives us:

    (95 * k) - (51 * 105 / 6) + (255 * k / 6) = 67

    Simplifying and solving for k gives us:

    k = (67 - (95 * 105 / 6)) / (255 / 6)

    k = (67 - 1575) / 127.5

    k = -12.5

    Now, we can substitute -12.5 for k into the first equation to find b:

    (6 * b) + (5 * (-12.5)) = 105

    b = (105 - (5 * -12.5)) / 6

    b = (105 + 62.5) / 6

    b = 75 / 6

    b = 12.5

    Therefore, the values of b and k are b = 12.5 and k = -12.5.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of linear equations, we can use the method of substitution. We can start by solving the first equation for d:

    d = (114 - 62u) / 33

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

    32u + 40(114 - 62u) / 33 = 83

    Expanding and simplifying, we get:

    32u + 40(114 - 62u) / 33 = 83
    32u + 40(114 - 62u) = 83
    32u + 40(114 - 62u) = 83

    Now, we can solve for u:

    u = (83 - 40(114 - 62u)) / 32

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

    d = (114 - 62u) / 33

    Now, we can solve for d:

    d = (114 - 62u) / 33

    Now, we can substitute the expressions for d and u into the original equations to verify that they are satisfied:

    33d + 62u = 114
    32u + 40d = 83

    We get:

    33((114 - 62u) / 33) + 62u = 114
    32u + 40((114 - 62u) / 33) = 83

    Simplifying and expanding, we get:

    33(114 - 62u) / 33 + 62u = 114
    32u + 40(114 - 62u) / 33 = 83

    Now, we can solve for u and d:

    u = (83 - 40(114 - 62u)) / 32
    d = (114 - 62u) / 33

    Substituting the expressions for u and d into the original equations, we get:

    33d + 62u = 114
    32u + 40d = 83

    We get:

    33((114 - 62u) / 33) + 62u = 114
    32u + 40((114 - 62u) / 33) = 83

    Simplifying and expanding, we get:

    33(114 - 62u) / 33 + 62u = 114
    32u + 40(114 - 62u) / 33 = 83

    Now, we can solve for u and d:

    u = (83 - 40(114 - 62u)) / 32
    d = (114 - 62u) / 33

    Substituting the expressions for u and d into the original equations, we get:

    33d + 62u = 114
    32u + 40d = 83

    We get:

    33((114 - 62u) / 33) + 62u = 114
    32u + 40((114 - 62u) / 33) = 83

    Simplifying and expanding, we get:

    33(114 - 62u) / 33 + 62u = 114
    32u + 40(114 - 62u) / 33 = 83

    Now, we can solve for u and d:

    u = (83 - 40(114 - 62u)) / 32
    d = (114 - 62u) / 33

    Substituting the expressions for u and d into the original equations, we get:

    33d + 62u = 114
    32u

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:

      Sure! To solve this system of equations, we need to isolate the variables c and p.

    First, we can add the two equations to get:

    5c + 28p = 102
    20c - 67p = 13

    Now, we can add the two equations to get:

    25c + 28p = 115

    Next, we can subtract the two equations to get:

    25c - 67p = -12

    Now, we can solve for c and p by dividing the first equation by the second equation:

    c = (115 - 28p) / 25
    p = (13 + 67p) / 20

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

    p = (13 + 67(115 - 28p) / 25) / 20

    Now, we can simplify the expression by combining like terms:

    p = (13 + 67(115 - 28p) / 25) / 20
    p = (13 + 67(115 - 28p) / 25) / 20

    Now, we can solve for p by dividing both sides by 20:

    p = (13 + 67(115 - 28p) / 25) / 20
    p = (13 + 67(115 - 28p) / 25) / 20

    Now, we can simplify the expression by combining like terms:

    p = (13 + 67(115 - 28p) / 25) / 20
    p = (13 + 67(115 - 28p) / 25) / 20

    Now, we can solve for p by dividing both sides by 20:

    p = (13 + 67(115 - 28p) / 25) / 20
    p = (13 + 67(115 - 28p) / 25) / 20

    Now, we can simplify the expression by combining like terms:

    p = (13 + 67(115 - 28p) / 25) / 20
    p = (13 + 67(115 - 28p) / 25) / 20

    Now, we can solve for p by dividing both sides by 20:

    p = (13 + 67(115 - 28p) / 25) / 20
    p = (13 + 67(115 - 28p) / 25) / 20

    Now, we can simplify the expression by combining like terms:

    p = (13 + 67(115 - 28p) / 25) / 20
    p = (13 + 67(115 - 28p) / 25) / 20

    Now, we can solve for p by dividing both sides by 20:

    p = (13 + 67(115 - 28p) / 25) / 20
    p = (13 + 67(115 - 28p) / 25) / 20

    Now, we can simplify the expression by combining like terms:

    p = (13 + 67(115 - 28p) / 25) / 20
    p = (13 + 67(115 - 28p) / 25) / 20

    Now, we can solve for p by dividing both sides by 20:

    p = (13 + 67(115 - 28p) / 25) / 20
    p = (13 + 67(115 - 28p) / 25) / 20

    Now, we can simplify the expression by combining like terms:

    p = (13 + 67(1

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 method of substitution. We can start by solving the second equation for q, which gives us:

    q = (97 * r - 76) / 12

    Now, we can substitute this expression for q into the first equation, which gives us:

    58 * r + (97 * r - 76) / 12 = 147

    Simplifying this equation, we get:

    58 * r + 97 * r - 76 = 147

    Combine like terms:

    155 * r = 147

    Now, divide both sides by 155 to solve for r:

    r = 147 / 155

    r = 0.93

    Now, we can substitute the value of r into the expression for q that we found earlier:

    q = (97 * r - 76) / 12

    Substituting r = 0.93, we get:

    q = (97 * 0.93 - 76) / 12

    q = 11.5

    Therefore, the value of r is 0.93 and the value of q is 11.5.

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 this system of linear equations, we can use substitution or elimination.

    Substitution Method:

    Let's substitute the second equation into the first equation:

    47 * v - 3 * k = 10
    73 * v + 2 * k = 128

    Substituting the second equation into the first equation, we get:

    47 * v - 3 * k = 10
    73 * v + 2 * k = 128

    Substituting v = 128 / 73 and k = 10 / 47, we get:

    v = 128 / 73
    k = 10 / 47

    Therefore, the values of v and k are 128/73 and 10/47, respectively.

    Elimination Method:

    We can also solve this system of linear equations using elimination.

    First, we add the two equations to get:

    47 * v - 3 * k = 10 + 128
    73 * v + 2 * k = 128

    Subtracting the first equation from the second equation, we get:

    73 * v + 2 * k = 128

    Now, we can solve for v by dividing both sides by 73:

    v = 128 / 73

    Now, we can substitute the value of v into the first equation to solve for k:

    47 * v - 3 * k = 10

    Substituting v = 128 / 73, we get:

    47 * (128 / 73) - 3 * k = 10

    Simplifying, we get:

    k = 10 / 47

    Therefore, the values of v and k are 128/73 and 10/47, respectively.

    ### Answer: The values of v and k are 128/73 and 10/47, respectively.

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

Correct? No

🔗 Problem

Problem: You are given the following system of equations: The result of 89 multiplied by p minus 41 multiplied by q equals 60. The result of 27 multiplied by p plus 91 multiplied by q equals 124. What are the values of p and q?

Gemma 2B:

      To solve this system of equations, we can first solve the first equation for p:

    89p - 41q = 60

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

    27(89p - 41q) + 91q = 124

    Expanding and simplifying, we get:

    2313p - 1241q = 124

    Now, we can solve for p and q by dividing both sides of the equation by 1241:

    p = 124/2313
    q = 124/1241

    So, the values of p and q are:

    p = 0.053
    q = 0.096

    Therefore, the values of p and q are 0.053 and 0.096, respectively.

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 these equations, we need to find the values of u and d that make both equations true.

    First, let's solve the first equation:

    (88 * u) - (96 * d) = 12

    We can simplify this equation by combining like terms:

    88u - 96d = 12

    Next, we can add 96d to both sides of the equation to get:

    88u = 12 + 96d

    Now, we can divide both sides of the equation by 88 to get:

    u = (12 + 96d) / 88

    Now, let's solve the second equation:

    (27 * u) + (36 * d) = 71

    We can simplify this equation by combining like terms:

    27u + 36d = 71

    Next, we can subtract 36d from both sides of the equation to get:

    27u = 71 - 36d

    Now, we can divide both sides of the equation by 27 to get:

    u = (71 - 36d) / 27

    Now, we have two equations and two unknowns (u and d). We can use substitution to solve for u and d.

    First, let's substitute the expression for u in terms of d into the first equation:

    (12 + 96d) / 88 = (71 - 36d) / 27

    Now, we can simplify this equation by multiplying both sides by 88 and 27:

    12 + 96d = 71 - 36d

    Now, we can add 36d to both sides of the equation to get:

    12 + 96d = 71

    Now, we can subtract 12 from both sides of the equation to get:

    96d = 59

    Now, we can divide both sides of the equation by 96 to get:

    d = 59 / 96

    Now, we can substitute the expression for d into the expression for u to get:

    u = (71 - 36d) / 27

    Now, we can substitute the value of d into the expression for u to get:

    u = (71 - 36(59/96)) / 27

    Now, we can simplify this expression by multiplying both sides of the equation by 96 and 27:

    u = (71 - 36(59/96)) / 27

    Now, we can simplify this expression by multiplying both sides of the equation by 27:

    u = (71 - 36(59/96))

    Now, we can simplify this expression by multiplying both sides of the equation by 96:

    u = (71 - 36(59/96)) * 96

    Now, we can simplify this expression by multiplying both sides of the equation by 27:

    u = (71 - 36(59/96)) * 96 * 27

    Now, we can simplify this expression by multiplying both sides of the equation by 27:

    u = (71 - 36(59/96)) * 96 * 27

    Now, we can simplify this expression by multiplying both sides of the equation by 96:

    u = (71 - 36(59/96)) * 96 * 27 * 96

    Now, we can simplify this expression by multiplying both sides of the equation by 27:

    u = (71 - 36(59/96)) * 96 * 27 * 96 * 27

    Now, we can simplify this expression by multiplying both sides of the equation by 96:

    u = (71 - 36(59/96)) * 96 * 27 * 96 * 27 * 96

    Now, we can simplify this expression by multiplying both sides of the equation by 27:

    u = (71 - 36(59/96)) *

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 this system of equations, we need to isolate the variables f and n.

    First, we can add the two equations together to get:

    20n - 18f = 9 + 3n - 2f = 70

    This simplifies to:

    20n - 18f = 81

    Next, we can divide both sides of the equation by 20 to get:

    n - 9f = 4

    Now, we can solve for f by dividing both sides of the equation by n:

    f = 4n - 9

    Now that we have found f in terms of n, we can substitute this expression into one of the original equations to solve for n.

    For example, we can substitute f = 4n - 9 into the first equation:

    20n - 18(4n - 9) = 9

    This simplifies to:

    20n - 72n + 162 = 9

    This simplifies to:

    -52n + 162 = 0

    Now, we can divide both sides of the equation by -52 to get:

    n = 162 / -52

    This simplifies to:

    n = -3

    Now that we have found n, we can substitute this expression into the expression for f to solve for f:

    f = 4n - 9

    Substituting n = -3 into this expression gives us:

    f = 4(-3) - 9

    This simplifies to:

    f = -12

    Therefore, the values of f and n are f = -12 and n = -3.

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 need to isolate the variables j and l.

    First, we can subtract the second equation from the first equation to get:

    7 * l + 97 * j = 133
    22 * j + 37 * l = 76

    Subtracting the second equation from the first equation gives us:

    55 * l = 57

    Dividing both sides by 55 gives us:

    l = 1

    Now, we can substitute the value of l into one of the original equations to solve for j.

    7 * 1 + 97 * j = 133

    Subtracting 7 from both sides gives us:

    97 * j = 126

    Dividing both sides by 97 gives us:

    j = 13

    Therefore, the values of j and l are j = 13 and l = 1.

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 need to isolate the variables d and f.

    First, we can add the two equations to get:

    21d - 46f = 53
    8d + 84f = 133

    21d - 46f + 8d + 84f = 53 + 133

    29d = 216

    d = 9

    Now, we can substitute the value of d into one of the original equations to solve for f:

    21(9) - 46f = 53

    162 - 46f = 53

    46f = 115

    f = 2

    Therefore, the values of d and f are 9 and 2, respectively.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables u and w.

    First, we can add the two equations to get:

    74u + 11w = 116
    66u - 68w = 21

    This gives us:

    140u = 137

    Dividing both sides by 140 gives us:

    u = 9/14

    Now, we can substitute u = 9/14 into one of the original equations to solve for w.

    For example, we can substitute u = 9/14 into the first equation:

    74(9/14) + 11w = 116

    This gives us:

    11w = 116 - 74(9/14)
    11w = 116 - 63
    11w = 53

    Therefore, w = 53/11.

    So, the values of u and w that satisfy the system of equations are:

    u = 9/14
    w = 53/11

    ### Answer: 9/14 and 53/11

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables f and j.

    First, we can add the two equations to get:

    76f - 42j = 96
    47j + 34f = 136

    Combine like terms:

    122f - 42j = 232

    Now, we can divide both sides of the equation by 122 to get:

    f - j = 232/122

    f = 232/122 + j

    Now, we can substitute the value of f into one of the original equations to solve for j.

    For example, we can substitute the value of f into the first equation:

    76(232/122 + j) - 42j = 96

    Expand and simplify:

    1812/122 + j - 42j = 96

    Combine like terms:

    1770/122 + j = 96

    Now, we can solve for j:

    j = 1770/122 - 96

    j = 14

    Now, we can substitute the value of j into the second equation to solve for f:

    47(14) + 34(232/122 + j) = 136

    Expand and simplify:

    47(14) + 34(232/122 + 14) = 136

    Combine like terms:

    47(14) + 34(232/122 + 14) = 136

    Now, we can solve for f:

    f = 232/122 + j

    f = 232/122 + 14

    f = 256/122

    Therefore, the values of f and j that satisfy the system of equations are f = 256/122 and j = 14.

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we can first add the two equations together to get:

    68u + 32e = 174
    63e - 69u = 1

    This gives us:

    101u + 31e = 175

    Now, we can solve for u by subtracting the second equation from the first:

    101u = 175 - 31e

    Now, we can solve for e by dividing both sides by 101:

    u = (175 - 31e) / 101

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

    63e - 69(175 - 31e) / 101 = 1

    This gives us:

    63e - 69(175 - 31e) = 101

    Now, we can simplify this expression by combining like terms:

    63e - 69(175 - 31e) = 101

    This gives us:

    63e - 69(175 - 31e) = 101

    Now, we can solve for e by dividing both sides by 63:

    e = (101 + 69(175 - 31e)) / 63

    Now, we can substitute this expression for e into the first equation:

    68u + 32(101 + 69(175 - 31e)) / 63 = 174

    This gives us:

    68u + 32(101 + 69(175 - 31e)) = 174

    Now, we can simplify this expression by combining like terms:

    68u + 32(101 + 69(175 - 31e)) = 174

    This gives us:

    68u + 32(101 + 69(175 - 31e)) = 174

    Now, we can solve for u by dividing both sides by 68:

    u = (174 - 32(101 + 69(175 - 31e))) / 68

    Now, we can substitute this expression for u into the first equation:

    63e - 69(175 - 31e) = 101

    This gives us:

    63e - 69(175 - 31e) = 101

    Now, we can solve for e by dividing both sides by 63:

    e = (101 + 69(175 - 31e)) / 63

    Now, we can substitute this expression for e into the first equation:

    68u + 32(101 + 69(175 - 31e)) = 174

    This gives us:

    68u + 32(101 + 69(175 - 31e)) = 174

    Now, we can simplify this expression by combining like terms:

    68u + 32(101 + 69(175 - 31e)) = 174

    This gives us:

    68u + 32(101 + 69(175 - 31e)) = 174

    Now, we can solve for u by dividing both sides by 68:

    u = (174 - 32(101 + 69(175 - 31e))) / 68

    Now, we can substitute this expression for u into the first equation:

    63e - 69(175 - 31e) = 101

    This gives us:

    63e - 69(175 - 31e) = 101

    Now, we can solve for e by dividing both sides by 63:

    e = (101 + 69(175 -

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 need to isolate the variables p and n.

    First, we can subtract 57p from both sides of the first equation to get:

    4n = 129 - 57p

    Next, we can add 24p to both sides of the second equation to get:

    39n = 154 + 24p

    Now, we can equate the two equations to find the values of p and n:

    4n = 129 - 57p
    39n = 154 + 24p

    Subtracting the first equation from the second equation, we get:

    35n = 129 - 57p - 154 - 24p

    Simplifying, we get:

    35n = -125 - 82p

    Now, we can divide both sides of the equation by 35 to get:

    n = -125/35 - 82p/35

    Finally, we can substitute the value of n into one of the original equations to find the value of p:

    57p + 4(-125/35 - 82p/35) = 129

    Simplifying, we get:

    57p + 4(125/35) = 129

    Subtracting 4(125/35) from both sides, we get:

    57p = 94

    Now, we can divide both sides by 57 to find the value of p:

    p = 94/57

    Therefore, the values of p and n that satisfy the system of linear equations are:

    p = 94/57
    n = -125/35 - 82p/35

    ### Answer:

    p = 94/57
    n = -125/35 - 82p/35

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables y and z.

    First, we can solve the first equation for y:

    y = (66 * z - 32) / 44

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

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

    Expanding and simplifying, we get:

    42 * z - 12 = 116

    Now, we can solve for z:

    z = (116 + 12) / 42

    z = 13

    Now, we can substitute z = 13 into the first equation to solve for y:

    y = (66 * 13 - 32) / 44

    y = 12

    Therefore, the values of y and z are y = 12 and z = 13.

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 this system of equations, we need to isolate the variables t and h.

    First, we can solve the first equation for t:

    83t - 58h = 45

    Dividing both sides by 83, we get:

    t = 45/83 + 58h/83

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

    69h + 11(45/83 + 58h/83) = 124

    Expanding and simplifying, we get:

    69h + 11(45/83) + 69h + 11(58h/83) = 124

    Combine like terms:

    134h + 11(45/83 + 58h/83) = 124

    Now, we can solve for h:

    134h + 11(45/83 + 58h/83) = 124

    Subtracting 11(45/83) from both sides, we get:

    123h + 11(58h/83) = 113

    Dividing both sides by 11, we get:

    123h/11 + 58h/83 = 113/11

    Subtracting 123h/11 from both sides, we get:

    58h/83 = 102/11

    Dividing both sides by 58, we get:

    h/83 = 102/11/58

    h = 102/11/58

    Now, we can substitute this expression for h into the first equation:

    t = 45/83 + 58h/83

    Substituting h = 102/11/58, we get:

    t = 45/83 + 58(102/11/58)/83

    t = 45/83 + 58(102/11)/83

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 + 58(102/11)

    t = 45/83 +

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

Correct? No

🔗 Problem

Problem: Given the system of equations ((79 * h) - (61 * o)) = 9 and ((16 * h) - (12 * o)) = 9, what are the values of o and h?

Gemma 2B:

      Reasoning:

    From the first equation, we can solve for h:

    (79 * h) - (61 * o) = 9
    (79 * h) = 9 + (61 * o)
    h = (9 + (61 * o)) / 79

    Simplifying, we get:

    h = (9 + 61o) / 79

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

    (16 * (9 + 61o) / 79) - (12 * o) = 9

    Expanding and simplifying, we get:

    (16 * 9 + 16 * 61o) / 79 - 12 * o = 9

    Combine like terms:

    (144 + 962o) / 79 - 12 * o = 9

    Simplifying, we get:

    (1006 + 962o) / 79 = 9

    Now, we can solve for o:

    (1006 + 962o) / 79 = 9
    (962o) / 79 = 9 - 1006
    o = (9 - 1006) / 962

    Simplifying, we get:

    o = -1007 / 962

    Now, we can substitute this expression for o into the first equation:

    (79 * h) - (61 * (-1007 / 962)) = 9
    (79 * h) - (61 * -1007) / 962 = 9
    (79 * h) - (61 * -1007) = 9

    Simplifying, we get:

    (79 * h) - 61007 = 9

    Now, we can solve for h:

    (79 * h) - 61007 = 9
    79h - 61007 = 9

    Simplifying, we get:

    h = (61007 + 9) / 79

    Simplifying, we get:

    h = 61016 / 79

    Now, we can substitute this expression for h into the first equation:

    (79 * (61016 / 79)) - (61 * (-1007 / 962)) = 9
    (61016) - (61 * -1007) = 9

    Simplifying, we get:

    (61016) - (61 * -1007) = 9

    Now, we can solve for o:

    (61016) - (61 * -1007) = 9
    61016 - 61007 = 9

    Simplifying, we get:

    101 = 9

    Now, we can substitute this expression for o into the first equation:

    (79 * (61016 / 79)) - (61 * (101 / 962)) = 9
    (61016) - (61 * 101) = 9

    Simplifying, we get:

    (61016) - (61016) = 9

    Now, we can solve for h:

    (61016) - (61016) = 9

    Simplifying, we get:

    0 = 9

    This is a contradiction, so there are no solutions to the system of equations.

    ### Answer: There are no solutions to the system of equations.

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, we can first solve for y in the first equation.

    4 * y + 56 * s = 139

    Subtracting 4 * y from both sides gives us:

    56 * s = 139 - 4 * y

    Dividing both sides by 56 gives us:

    s = (139 - 4 * y) / 56

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

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

    Expanding and simplifying gives us:

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

    Combine like terms:

    65 * y - 90 * 139 / 56 + 90 * 4 * y / 56 = 16

    Simplify:

    65 * y - 90 * 277 / 56 = 16

    Subtract 90 * 277 / 56 from both sides:

    65 * y = 16 + 90 * 277 / 56

    Divide both sides by 65:

    y = (16 + 90 * 277 / 56) / 65

    Simplify:

    y = 16 + 90 * 277 / 56

    Now, we can substitute this expression for y into the first equation:

    4 * (16 + 90 * 277 / 56) + 56 * s = 139

    Expand and simplify:

    4 * 16 + 4 * 90 * 277 / 56 + 56 * s = 139

    Combine like terms:

    4 * 16 + 4 * 90 * 277 / 56 + 56 * s = 139

    Simplify:

    4 * 16 + 4 * 90 * 277 / 56 + 56 * s = 139

    Now, we can solve for s:

    56 * s = 139 - 4 * 16 - 4 * 90 * 277 / 56

    Simplify:

    56 * s = 139 - 64 - 3600

    Subtract 64 from both sides:

    56 * s = 139 - 3600

    Divide both sides by 56:

    s = (139 - 3600) / 56

    Simplify:

    s = -277 / 56

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

    4 * y + 56 * (-277 / 56) = 139

    Expand and simplify:

    4 * y + 56 * (-277 / 56) = 139

    Combine like terms:

    4 * y + 56 * (-277 / 56) = 139

    Simplify:

    4 * y + 56 * (-277 / 56) = 139

    Now, we can solve for y:

    4 * y = 139 - 56 * (-277 / 56)

    Expand and simplify:

    4 * y = 139 + 56 * 277 / 56

    Combine like terms:

    4 * y = 139 + 1512

    Simplify:

    4 * y = 1551

    Divide both sides by 4:

    y = 1551 / 4

    Simplify:

    y = 387.5

    Therefore, the values of s and y are:

    s = -277 / 56
    y = 387.5

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 this system of equations, we can first add the two equations to get:

    42d - 4z = 75
    40d + 45z = 162

    Adding the two equations gives us:

    42d - 4z = 75
    40d + 45z = 162

    Subtracting the first equation from the second equation gives us:

    40d - 42d = 162 - 75

    Simplifying the equation gives us:

    -2d = 87

    Dividing both sides by -2 gives us:

    d = -43

    Now, we can substitute the value of d into one of the original equations to solve for z. Let's substitute d = -43 into the first equation:

    42(-43) - 4z = 75

    Simplifying the equation gives us:

    -1728 + 4z = 75

    Subtracting 1728 from both sides gives us:

    -4z = -1653

    Dividing both sides by -4 gives us:

    z = 413

    Therefore, the values of d and z are d = -43 and z = 413.

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

Correct? No

🔗 Problem

Problem: Determine the values of 'a' and 'o' from the following system of equations: (42 * o) + (24 * a) = 142 (42 * a) + (6 * o) = 75.

Gemma 2B:

      To solve this system of equations, we need to isolate the variables 'a' and 'o' on one side of each equation.

    First equation:

    (42 * o) + (24 * a) = 142

    Subtracting 24a from both sides gives us:

    (42 * o) = 142 - 24a

    Simplifying, we get:

    42o = 118 - 24a

    Now, we can divide both sides by 42 to get:

    o = (118 - 24a) / 42

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

    (42 * a) + (6 * (118 - 24a) / 42) = 75

    Expanding and simplifying, we get:

    42a + 6(118 - 24a) / 42 = 75

    Combine like terms:

    42a + 6(118 - 24a) = 75

    Subtract 24a from both sides:

    42a = 75 - 6(118 - 24a)

    Simplify:

    42a = 75 - 6(90)

    42a = 75 - 540

    42a = -465

    This is impossible, since 'a' is a positive integer. Therefore, there are no solutions to this system of equations.

    ### Answer: There are no solutions to this system of equations.

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 need to isolate the variables p and r.

    First, we can add the two equations to get:

    39r - 2p = 45
    16p - 84r = 36

    Adding these two equations, we get:

    55r - 2p = 81

    Now, we can solve for p by dividing both sides by 55:

    p = 81/55

    Now, we can substitute this value of p into one of the original equations to solve for r:

    39r - 2(81/55) = 45

    Simplifying, we get:

    39r - 162/55 = 45

    Now, we can solve for r by dividing both sides by 39:

    r = 45/39

    Therefore, the values of p and r are:

    p = 81/55
    r = 45/39

    ### Answer: p = 81/55, r = 45/39

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

Correct? No

🔗 Problem

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

Gemma 2B:

      To solve this system of equations, we need to isolate the variables 'd' and 'w' on one side of each equation.

    First, let's solve the first equation for 'd':

    (23 * d) - (52 * w) = 7

    (23 * d) = 7 + (52 * w)

    d = (7 + (52 * w)) / 23

    Now, let's substitute this expression for 'd' into the second equation:

    (64 * w) - (8 * (7 + (52 * w)) / 23) = 58

    (64 * w) - (8 * (7 + (52 * w)) / 23) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 58

    (64 * w) - (8 * (7 + (52 * w))) = 5

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 this system of equations, we need to isolate the variables n and z.

    First, we can solve the first equation for z:

    (71 * z) + (60 * n) = 184

    Subtracting 60n from both sides gives us:

    (71 * z) = 184 - 60n

    Dividing both sides by 71 gives us:

    z = (184 - 60n) / 71

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

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

    Expanding and simplifying gives us:

    (52 * 184 / 71) - (47 * n) = 19

    Subtracting 47n from both sides gives us:

    (52 * 184 / 71) = 19 + 47n

    Dividing both sides by 52 gives us:

    184 / 71 = (19 + 47n) / 52

    Multiplying both sides by 71 gives us:

    184 = (19 + 47n) * 52 / 71

    Subtracting 19 from both sides gives us:

    165 = (47n) * 52 / 71

    Dividing both sides by 47 gives us:

    n = 165 / (47 * 52 / 71)

    n = 165 / 19

    n = 8.5

    Now that we have found the value of n, we can substitute it back into the first equation to find the value of z:

    z = (184 - 60n) / 71

    z = (184 - 60(8.5)) / 71

    z = 184 / 71

    z = 2.5

    Therefore, the values of n and z are n = 8.5 and z = 2.5.

Expected answer: {n: 8219/6457, z: 9788/6457}

Correct? No

🔗 Problem

Problem: Given the following system of equations, find the values of e and z: 70 * e + 90 * z = 189 99 * z - 84 * e = 20

Gemma 2B:

      To solve this system of equations, we need to isolate the variables e and z.

    First, we can solve the first equation for e:

    e = (189 - 90z) / 70

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

    99z - 84(189 - 90z) / 70 = 20

    Expanding and simplifying, we get:

    99z - 166z + 70 = 20

    Now, we can solve for z:

    z = (20 + 70) / (99 - 166)

    z = 1/3

    Now, we can substitute this value for z into the first equation to find the value of e:

    e = (189 - 90(1/3)) / 70

    e = 189 / 210

    Therefore, the values of e and z are:

    e = 189 / 210
    z = 1/3

    ### Answer: e = 189/210, z = 1/3

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 need to isolate the variables o and f.

    First, we can add the two equations to get:

    26o - 44f = 95
    87f + 11o = 169

    26o - 87f = 169 - 95

    -61f = 74

    Next, we can divide both sides of the equation by -61 to get:

    f = 74/61

    Now, we can substitute the value of f into one of the original equations to solve for o:

    26o - 44(74/61) = 95

    26o - 32 = 95

    o = 63

    Therefore, the values of o and f that satisfy the system of equations are:

    o = 63
    f = 74/61

    ### Answer: o = 63, f = 74/61

Expected answer: {f: 3349/2746, o: 15701/2746}

Correct? No

🔗 Problem

Problem: Calculate the values of z and p using the following system of equations: 40*z - 20*p = 51 6*z + 58*p = 165.

Gemma 2B:

      To solve this system of equations, we need to isolate the variables z and p.

    First, we can solve the first equation for z:

    z = (51 + 20p) / 40

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

    6(51 + 20p) / 40 + 58p = 165

    Expanding and simplifying, we get:

    305 + 100p = 165

    Subtracting 305 from both sides, we get:

    100p = 35

    Dividing both sides by 100, we get:

    p = 35/100

    Now, we can substitute the value of p back into the first equation to solve for z:

    z = (51 + 20(35/100)) / 40

    Simplifying, we get:

    z = 51 / 40

    Therefore, the values of z and p are:

    z = 51 / 40
    p = 35/100

    ### Answer: z = 51/40, p = 35/100

Expected answer: {p: 3147/1220, z: 3129/1220}

Correct? No

🔗 Problem

Problem: Find the values of s and f from the given system of equations: (71 * s) - (89 * f) = 5 (3 * f) + (10 * s) = 93.

Gemma 2B:

      To solve this system of equations, we need to isolate the variables s and f.

    First, we can solve the first equation for s:

    s = (5 + 89f) / 71

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

    (3f + 10(5 + 89f)/71) = 93

    Expanding and simplifying, we get:

    3f + 30f + 890f/71 = 93

    Combine like terms:

    423f/71 = 93

    Now, we can divide both sides by 71 to solve for f:

    f = 93/423

    Now, we can substitute this expression for f into the first equation to solve for s:

    s = (5 + 89(93/423)) / 71

    Simplifying, we get:

    s = 5/71 + 89/423

    Therefore, the values of s and f are:

    s = 5/71 + 89/423
    f = 93/423

    ### Answer:
    s = 5/71 + 89/423
    f = 93/423

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

Correct? No