Finite Math Examples

$[\begin{array}{c} 18 & 4 & 5 \\ 20 & 4 & 4 \\ 24 & 3 & 6 \end{array}] [\begin{array}{c} a \\ b \\ c \end{array}] = [\begin{array}{c} 40600 \\ 7500 \\ 9700 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 18 & 4 & 5 \\ 20 & 4 & 4 \\ 24 & 3 & 6 \end{array}] [\begin{array}{c} a \\ b \\ c \end{array}]$ .

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Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $3 \times 3$ and the second matrix is $3 \times 1$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 18 a + 4 b + 5 c \\ 20 a + 4 b + 4 c \\ 24 a + 3 b + 6 c \end{array}] = [\begin{array}{c} 40600 \\ 7500 \\ 9700 \end{array}]$

Step 2

Write as a linear system of equations.

$18 a + 4 b + 5 c = 40600$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $a$ in $18 a + 4 b + 5 c = 40600$ .

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Step 3.1.1

Move all terms not containing $a$ to the right side of the equation.

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Step 3.1.1.1

Subtract $4 b$ from both sides of the equation.

$18 a + 5 c = 40600 - 4 b$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.1.2

Subtract $5 c$ from both sides of the equation.

$18 a = 40600 - 4 b - 5 c$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

$18 a = 40600 - 4 b - 5 c$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2

Divide each term in $18 a = 40600 - 4 b - 5 c$ by $18$ and simplify.

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Step 3.1.2.1

Divide each term in $18 a = 40600 - 4 b - 5 c$ by $18$ .

$\frac{18 a}{18} = \frac{40600}{18} + \frac{- 4 b}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $18$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{18 a}{18} = \frac{40600}{18} + \frac{- 4 b}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{40600}{18} + \frac{- 4 b}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

$a = \frac{40600}{18} + \frac{- 4 b}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

$a = \frac{40600}{18} + \frac{- 4 b}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Simplify each term.

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Step 3.1.2.3.1.1

Cancel the common factor of $40600$ and $18$ .

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Step 3.1.2.3.1.1.1

Factor $2$ out of $40600$ .

$a = \frac{2 (20300)}{18} + \frac{- 4 b}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.3.1.1.2

Cancel the common factors.

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Step 3.1.2.3.1.1.2.1

Factor $2$ out of $18$ .

$a = \frac{2 \cdot 20300}{2 \cdot 9} + \frac{- 4 b}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.3.1.1.2.2

Cancel the common factor.

$a = \frac{2 \cdot 20300}{2 \cdot 9} + \frac{- 4 b}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.3.1.1.2.3

Rewrite the expression.

$a = \frac{20300}{9} + \frac{- 4 b}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

$a = \frac{20300}{9} + \frac{- 4 b}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

$a = \frac{20300}{9} + \frac{- 4 b}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.3.1.2

Cancel the common factor of $- 4$ and $18$ .

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Step 3.1.2.3.1.2.1

Factor $2$ out of $- 4 b$ .

$a = \frac{20300}{9} + \frac{2 (- 2 b)}{18} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.3.1.2.2

Cancel the common factors.

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Step 3.1.2.3.1.2.2.1

Factor $2$ out of $18$ .

$a = \frac{20300}{9} + \frac{2 (- 2 b)}{2 (9)} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.3.1.2.2.2

Cancel the common factor.

$a = \frac{20300}{9} + \frac{2 (- 2 b)}{2 \cdot 9} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.3.1.2.2.3

Rewrite the expression.

$a = \frac{20300}{9} + \frac{- 2 b}{9} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

$a = \frac{20300}{9} + \frac{- 2 b}{9} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

$a = \frac{20300}{9} + \frac{- 2 b}{9} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.3.1.3

Move the negative in front of the fraction.

$a = \frac{20300}{9} - \frac{2 b}{9} + \frac{- 5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.1.2.3.1.4

Move the negative in front of the fraction.

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$20 a + 4 b + 4 c = 7500$

$24 a + 3 b + 6 c = 9700$

Step 3.2

Replace all occurrences of $a$ with $\frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$ in each equation.

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Step 3.2.1

Replace all occurrences of $a$ in $20 a + 4 b + 4 c = 7500$ with $\frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$ .

$20 (\frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify $20 (\frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}) + 4 b + 4 c$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$20 (\frac{20300}{9}) + 20 (- \frac{2 b}{9}) + 20 (- \frac{5 c}{18}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2

Simplify.

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Step 3.2.2.1.1.2.1

Multiply $20 (\frac{20300}{9})$ .

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Step 3.2.2.1.1.2.1.1

Combine $20$ and $\frac{20300}{9}$ .

$\frac{20 \cdot 20300}{9} + 20 (- \frac{2 b}{9}) + 20 (- \frac{5 c}{18}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2.1.2

Multiply $20$ by $20300$ .

$\frac{406000}{9} + 20 (- \frac{2 b}{9}) + 20 (- \frac{5 c}{18}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$\frac{406000}{9} + 20 (- \frac{2 b}{9}) + 20 (- \frac{5 c}{18}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2.2

Multiply $20 (- \frac{2 b}{9})$ .

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Step 3.2.2.1.1.2.2.1

Multiply $- 1$ by $20$ .

$\frac{406000}{9} - 20 \frac{2 b}{9} + 20 (- \frac{5 c}{18}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2.2.2

Combine $- 20$ and $\frac{2 b}{9}$ .

$\frac{406000}{9} + \frac{- 20 (2 b)}{9} + 20 (- \frac{5 c}{18}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2.2.3

Multiply $2$ by $- 20$ .

$\frac{406000}{9} + \frac{- 40 b}{9} + 20 (- \frac{5 c}{18}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$\frac{406000}{9} + \frac{- 40 b}{9} + 20 (- \frac{5 c}{18}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.2.3.1

Move the leading negative in $- \frac{5 c}{18}$ into the numerator.

$\frac{406000}{9} + \frac{- 40 b}{9} + 20 (\frac{- 5 c}{18}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2.3.2

Factor $2$ out of $20$ .

$\frac{406000}{9} + \frac{- 40 b}{9} + 2 (10) (\frac{- 5 c}{18}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2.3.3

Factor $2$ out of $18$ .

$\frac{406000}{9} + \frac{- 40 b}{9} + 2 \cdot (10 (\frac{- 5 c}{2 \cdot 9})) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2.3.4

Cancel the common factor.

$\frac{406000}{9} + \frac{- 40 b}{9} + 2 \cdot (10 (\frac{- 5 c}{2 \cdot 9})) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2.3.5

Rewrite the expression.

$\frac{406000}{9} + \frac{- 40 b}{9} + 10 (\frac{- 5 c}{9}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$\frac{406000}{9} + \frac{- 40 b}{9} + 10 (\frac{- 5 c}{9}) + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2.4

Combine $10$ and $\frac{- 5 c}{9}$ .

$\frac{406000}{9} + \frac{- 40 b}{9} + \frac{10 (- 5 c)}{9} + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.2.5

Multiply $- 5$ by $10$ .

$\frac{406000}{9} + \frac{- 40 b}{9} + \frac{- 50 c}{9} + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$\frac{406000}{9} + \frac{- 40 b}{9} + \frac{- 50 c}{9} + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.3

Simplify each term.

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Step 3.2.2.1.1.3.1

Move the negative in front of the fraction.

$\frac{406000}{9} - \frac{40 b}{9} + \frac{- 50 c}{9} + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.1.3.2

Move the negative in front of the fraction.

$\frac{406000}{9} - \frac{40 b}{9} - \frac{50 c}{9} + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$\frac{406000}{9} - \frac{40 b}{9} - \frac{50 c}{9} + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$\frac{406000}{9} - \frac{40 b}{9} - \frac{50 c}{9} + 4 b + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.2

To write $4 b$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{406000}{9} - \frac{50 c}{9} - \frac{40 b}{9} + 4 b \cdot \frac{9}{9} + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.3

Combine $4 b$ and $\frac{9}{9}$ .

$\frac{406000}{9} - \frac{50 c}{9} - \frac{40 b}{9} + \frac{4 b \cdot 9}{9} + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.4

Combine the numerators over the common denominator.

$\frac{406000}{9} - \frac{50 c}{9} + \frac{- 40 b + 4 b \cdot 9}{9} + 4 c = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.5

Combine the numerators over the common denominator.

$4 c + \frac{406000 - 50 c - 40 b + 4 b \cdot 9}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.6

Multiply $9$ by $4$ .

$4 c + \frac{406000 - 50 c - 40 b + 36 b}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.7

Add $- 40 b$ and $36 b$ .

$4 c + \frac{406000 - 50 c - 4 b}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.8

Factor $2$ out of $406000 - 50 c - 4 b$ .

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Step 3.2.2.1.8.1

Factor $2$ out of $406000$ .

$4 c + \frac{2 (203000) - 50 c - 4 b}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.8.2

Factor $2$ out of $- 50 c$ .

$4 c + \frac{2 (203000) + 2 (- 25 c) - 4 b}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.8.3

Factor $2$ out of $- 4 b$ .

$4 c + \frac{2 (203000) + 2 (- 25 c) + 2 (- 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.8.4

Factor $2$ out of $2 (203000) + 2 (- 25 c)$ .

$4 c + \frac{2 (203000 - 25 c) + 2 (- 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.8.5

Factor $2$ out of $2 (203000 - 25 c) + 2 (- 2 b)$ .

$4 c + \frac{2 (203000 - 25 c - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$4 c + \frac{2 (203000 - 25 c - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.9

To write $4 c$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$4 c \cdot \frac{9}{9} + \frac{2 (203000 - 25 c - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.10

Simplify terms.

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Step 3.2.2.1.10.1

Combine $4 c$ and $\frac{9}{9}$ .

$\frac{4 c \cdot 9}{9} + \frac{2 (203000 - 25 c - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.10.2

Combine the numerators over the common denominator.

$\frac{4 c \cdot 9 + 2 (203000 - 25 c - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$\frac{4 c \cdot 9 + 2 (203000 - 25 c - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.11

Simplify the numerator.

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Step 3.2.2.1.11.1

Factor $2$ out of $4 c \cdot 9 + 2 (203000 - 25 c - 2 b)$ .

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Step 3.2.2.1.11.1.1

Factor $2$ out of $4 c \cdot 9$ .

$\frac{2 (2 c \cdot 9) + 2 (203000 - 25 c - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.11.1.2

Factor $2$ out of $2 (2 c \cdot 9) + 2 (203000 - 25 c - 2 b)$ .

$\frac{2 (2 c \cdot 9 + 203000 - 25 c - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$\frac{2 (2 c \cdot 9 + 203000 - 25 c - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.11.2

Multiply $9$ by $2$ .

$\frac{2 (18 c + 203000 - 25 c - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.11.3

Subtract $25 c$ from $18 c$ .

$\frac{2 (- 7 c + 203000 - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$\frac{2 (- 7 c + 203000 - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.12

Simplify with factoring out.

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Step 3.2.2.1.12.1

Factor $- 1$ out of $- 7 c$ .

$\frac{2 (- (7 c) + 203000 - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.12.2

Rewrite $203000$ as $- 1 (- 203000)$ .

$\frac{2 (- (7 c) - 1 \cdot - 203000 - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.12.3

Factor $- 1$ out of $- (7 c) - 1 (- 203000)$ .

$\frac{2 (- (7 c - 203000) - 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.12.4

Factor $- 1$ out of $- 2 b$ .

$\frac{2 (- (7 c - 203000) - (2 b))}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.12.5

Factor $- 1$ out of $- (7 c - 203000) - (2 b)$ .

$\frac{2 (- (7 c - 203000 + 2 b))}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.12.6

Simplify the expression.

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Step 3.2.2.1.12.6.1

Rewrite $- (7 c - 203000 + 2 b)$ as $- 1 (7 c - 203000 + 2 b)$ .

$\frac{2 (- 1 (7 c - 203000 + 2 b))}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.2.1.12.6.2

Move the negative in front of the fraction.

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$24 a + 3 b + 6 c = 9700$

Step 3.2.3

Replace all occurrences of $a$ in $24 a + 3 b + 6 c = 9700$ with $\frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$ .

$24 (\frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4

Simplify the left side.

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Step 3.2.4.1

Simplify $24 (\frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}) + 3 b + 6 c$ .

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Step 3.2.4.1.1

Simplify each term.

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Step 3.2.4.1.1.1

Apply the distributive property.

$24 (\frac{20300}{9}) + 24 (- \frac{2 b}{9}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2

Simplify.

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Step 3.2.4.1.1.2.1

Cancel the common factor of $3$ .

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Step 3.2.4.1.1.2.1.1

Factor $3$ out of $24$ .

$3 (8) (\frac{20300}{9}) + 24 (- \frac{2 b}{9}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.1.2

Factor $3$ out of $9$ .

$3 \cdot (8 (\frac{20300}{3 \cdot 3})) + 24 (- \frac{2 b}{9}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.1.3

Cancel the common factor.

$3 \cdot (8 (\frac{20300}{3 \cdot 3})) + 24 (- \frac{2 b}{9}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.1.4

Rewrite the expression.

$8 (\frac{20300}{3}) + 24 (- \frac{2 b}{9}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$8 (\frac{20300}{3}) + 24 (- \frac{2 b}{9}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.2

Combine $8$ and $\frac{20300}{3}$ .

$\frac{8 \cdot 20300}{3} + 24 (- \frac{2 b}{9}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.3

Multiply $8$ by $20300$ .

$\frac{162400}{3} + 24 (- \frac{2 b}{9}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.4

Cancel the common factor of $3$ .

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Step 3.2.4.1.1.2.4.1

Move the leading negative in $- \frac{2 b}{9}$ into the numerator.

$\frac{162400}{3} + 24 (\frac{- 2 b}{9}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.4.2

Factor $3$ out of $24$ .

$\frac{162400}{3} + 3 (8) (\frac{- 2 b}{9}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.4.3

Factor $3$ out of $9$ .

$\frac{162400}{3} + 3 \cdot (8 (\frac{- 2 b}{3 \cdot 3})) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.4.4

Cancel the common factor.

$\frac{162400}{3} + 3 \cdot (8 (\frac{- 2 b}{3 \cdot 3})) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.4.5

Rewrite the expression.

$\frac{162400}{3} + 8 (\frac{- 2 b}{3}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$\frac{162400}{3} + 8 (\frac{- 2 b}{3}) + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.5

Combine $8$ and $\frac{- 2 b}{3}$ .

$\frac{162400}{3} + \frac{8 (- 2 b)}{3} + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.6

Multiply $- 2$ by $8$ .

$\frac{162400}{3} + \frac{- 16 b}{3} + 24 (- \frac{5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.7

Cancel the common factor of $6$ .

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Step 3.2.4.1.1.2.7.1

Move the leading negative in $- \frac{5 c}{18}$ into the numerator.

$\frac{162400}{3} + \frac{- 16 b}{3} + 24 (\frac{- 5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.7.2

Factor $6$ out of $24$ .

$\frac{162400}{3} + \frac{- 16 b}{3} + 6 (4) (\frac{- 5 c}{18}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.7.3

Factor $6$ out of $18$ .

$\frac{162400}{3} + \frac{- 16 b}{3} + 6 \cdot (4 (\frac{- 5 c}{6 \cdot 3})) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.7.4

Cancel the common factor.

$\frac{162400}{3} + \frac{- 16 b}{3} + 6 \cdot (4 (\frac{- 5 c}{6 \cdot 3})) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.7.5

Rewrite the expression.

$\frac{162400}{3} + \frac{- 16 b}{3} + 4 (\frac{- 5 c}{3}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$\frac{162400}{3} + \frac{- 16 b}{3} + 4 (\frac{- 5 c}{3}) + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.8

Combine $4$ and $\frac{- 5 c}{3}$ .

$\frac{162400}{3} + \frac{- 16 b}{3} + \frac{4 (- 5 c)}{3} + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.2.9

Multiply $- 5$ by $4$ .

$\frac{162400}{3} + \frac{- 16 b}{3} + \frac{- 20 c}{3} + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$\frac{162400}{3} + \frac{- 16 b}{3} + \frac{- 20 c}{3} + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.3

Simplify each term.

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Step 3.2.4.1.1.3.1

Move the negative in front of the fraction.

$\frac{162400}{3} - \frac{16 b}{3} + \frac{- 20 c}{3} + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.1.3.2

Move the negative in front of the fraction.

$\frac{162400}{3} - \frac{16 b}{3} - \frac{20 c}{3} + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$\frac{162400}{3} - \frac{16 b}{3} - \frac{20 c}{3} + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$\frac{162400}{3} - \frac{16 b}{3} - \frac{20 c}{3} + 3 b + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.2

To write $3 b$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{162400}{3} - \frac{20 c}{3} - \frac{16 b}{3} + 3 b \cdot \frac{3}{3} + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.3

Combine $3 b$ and $\frac{3}{3}$ .

$\frac{162400}{3} - \frac{20 c}{3} - \frac{16 b}{3} + \frac{3 b \cdot 3}{3} + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.4

Combine the numerators over the common denominator.

$\frac{162400}{3} - \frac{20 c}{3} + \frac{- 16 b + 3 b \cdot 3}{3} + 6 c = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.5

Combine the numerators over the common denominator.

$6 c + \frac{162400 - 20 c - 16 b + 3 b \cdot 3}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.6

Multiply $3$ by $3$ .

$6 c + \frac{162400 - 20 c - 16 b + 9 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.7

Add $- 16 b$ and $9 b$ .

$6 c + \frac{162400 - 20 c - 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.8

To write $6 c$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6 c \cdot \frac{3}{3} + \frac{162400 - 20 c - 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.9

Simplify terms.

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Step 3.2.4.1.9.1

Combine $6 c$ and $\frac{3}{3}$ .

$\frac{6 c \cdot 3}{3} + \frac{162400 - 20 c - 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.9.2

Combine the numerators over the common denominator.

$\frac{6 c \cdot 3 + 162400 - 20 c - 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$\frac{6 c \cdot 3 + 162400 - 20 c - 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.10

Simplify the numerator.

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Step 3.2.4.1.10.1

Multiply $3$ by $6$ .

$\frac{18 c + 162400 - 20 c - 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.10.2

Subtract $20 c$ from $18 c$ .

$\frac{- 2 c + 162400 - 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$\frac{- 2 c + 162400 - 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.11

Simplify with factoring out.

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Step 3.2.4.1.11.1

Factor $- 1$ out of $- 2 c$ .

$\frac{- (2 c) + 162400 - 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.11.2

Rewrite $162400$ as $- 1 (- 162400)$ .

$\frac{- (2 c) - 1 \cdot - 162400 - 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.11.3

Factor $- 1$ out of $- (2 c) - 1 (- 162400)$ .

$\frac{- (2 c - 162400) - 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.11.4

Factor $- 1$ out of $- 7 b$ .

$\frac{- (2 c - 162400) - (7 b)}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.11.5

Factor $- 1$ out of $- (2 c - 162400) - (7 b)$ .

$\frac{- (2 c - 162400 + 7 b)}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.11.6

Simplify the expression.

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Step 3.2.4.1.11.6.1

Rewrite $- (2 c - 162400 + 7 b)$ as $- 1 (2 c - 162400 + 7 b)$ .

$\frac{- 1 (2 c - 162400 + 7 b)}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.2.4.1.11.6.2

Move the negative in front of the fraction.

$- \frac{2 c - 162400 + 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{2 c - 162400 + 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{2 c - 162400 + 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{2 c - 162400 + 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{2 c - 162400 + 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{2 c - 162400 + 7 b}{3} = 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3

Solve for $c$ in $- \frac{2 c - 162400 + 7 b}{3} = 9700$ .

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Step 3.3.1

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{2 c - 162400 + 7 b}{3}) = - 3 \cdot 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.2

Simplify both sides of the equation.

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Step 3.3.2.1

Simplify the left side.

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Step 3.3.2.1.1

Simplify $- 3 (- \frac{2 c - 162400 + 7 b}{3})$ .

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Step 3.3.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1.1.1

Move the leading negative in $- \frac{2 c - 162400 + 7 b}{3}$ into the numerator.

$- 3 \frac{- (2 c - 162400 + 7 b)}{3} = - 3 \cdot 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.2.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- (2 c - 162400 + 7 b)}{3}) = - 3 \cdot 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.2.1.1.1.3

Cancel the common factor.

$3 \cdot (- 1 \frac{- (2 c - 162400 + 7 b)}{3}) = - 3 \cdot 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.2.1.1.1.4

Rewrite the expression.

$2 c - 162400 + 7 b = - 3 \cdot 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$2 c - 162400 + 7 b = - 3 \cdot 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.2.1.1.2

Multiply.

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Step 3.3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (2 c - 162400 + 7 b) = - 3 \cdot 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.2.1.1.2.2

Multiply $2 c - 162400 + 7 b$ by $1$ .

$2 c - 162400 + 7 b = - 3 \cdot 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$2 c - 162400 + 7 b = - 3 \cdot 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$2 c - 162400 + 7 b = - 3 \cdot 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$2 c - 162400 + 7 b = - 3 \cdot 9700$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Multiply $- 3$ by $9700$ .

$2 c - 162400 + 7 b = - 29100$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$2 c - 162400 + 7 b = - 29100$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$2 c - 162400 + 7 b = - 29100$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.3

Move all terms not containing $c$ to the right side of the equation.

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Step 3.3.3.1

Add $162400$ to both sides of the equation.

$2 c + 7 b = - 29100 + 162400$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.3.2

Subtract $7 b$ from both sides of the equation.

$2 c = - 29100 + 162400 - 7 b$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.3.3

Add $- 29100$ and $162400$ .

$2 c = 133300 - 7 b$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$2 c = 133300 - 7 b$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.4

Divide each term in $2 c = 133300 - 7 b$ by $2$ and simplify.

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Step 3.3.4.1

Divide each term in $2 c = 133300 - 7 b$ by $2$ .

$\frac{2 c}{2} = \frac{133300}{2} + \frac{- 7 b}{2}$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.4.2

Simplify the left side.

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Step 3.3.4.2.1

Cancel the common factor of $2$ .

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Step 3.3.4.2.1.1

Cancel the common factor.

$\frac{2 c}{2} = \frac{133300}{2} + \frac{- 7 b}{2}$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.4.2.1.2

Divide $c$ by $1$ .

$c = \frac{133300}{2} + \frac{- 7 b}{2}$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$c = \frac{133300}{2} + \frac{- 7 b}{2}$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$c = \frac{133300}{2} + \frac{- 7 b}{2}$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.4.3

Simplify the right side.

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Step 3.3.4.3.1

Simplify each term.

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Step 3.3.4.3.1.1

Divide $133300$ by $2$ .

$c = 66650 + \frac{- 7 b}{2}$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.3.4.3.1.2

Move the negative in front of the fraction.

$c = 66650 - \frac{7 b}{2}$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$c = 66650 - \frac{7 b}{2}$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$c = 66650 - \frac{7 b}{2}$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$c = 66650 - \frac{7 b}{2}$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$c = 66650 - \frac{7 b}{2}$

$- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4

Replace all occurrences of $c$ with $66650 - \frac{7 b}{2}$ in each equation.

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Step 3.4.1

Replace all occurrences of $c$ in $- \frac{2 (7 c - 203000 + 2 b)}{9} = 7500$ with $66650 - \frac{7 b}{2}$ .

$- \frac{2 (7 (66650 - \frac{7 b}{2}) - 203000 + 2 b)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify $- \frac{2 (7 (66650 - \frac{7 b}{2}) - 203000 + 2 b)}{9}$ .

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Step 3.4.2.1.1

Simplify the numerator.

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Step 3.4.2.1.1.1

Apply the distributive property.

$- \frac{2 (7 \cdot 66650 + 7 (- \frac{7 b}{2}) - 203000 + 2 b)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.2

Multiply $7$ by $66650$ .

$- \frac{2 (466550 + 7 (- \frac{7 b}{2}) - 203000 + 2 b)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.3

Multiply $7 (- \frac{7 b}{2})$ .

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Step 3.4.2.1.1.3.1

Multiply $- 1$ by $7$ .

$- \frac{2 (466550 - 7 \frac{7 b}{2} - 203000 + 2 b)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.3.2

Combine $- 7$ and $\frac{7 b}{2}$ .

$- \frac{2 (466550 + \frac{- 7 (7 b)}{2} - 203000 + 2 b)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.3.3

Multiply $7$ by $- 7$ .

$- \frac{2 (466550 + \frac{- 49 b}{2} - 203000 + 2 b)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{2 (466550 + \frac{- 49 b}{2} - 203000 + 2 b)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.4

Move the negative in front of the fraction.

$- \frac{2 (466550 - \frac{49 b}{2} - 203000 + 2 b)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.5

Subtract $203000$ from $466550$ .

$- \frac{2 (- \frac{49 b}{2} + 263550 + 2 b)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.6

To write $2 b$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{2 (- \frac{49 b}{2} + 2 b \cdot \frac{2}{2} + 263550)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.7

Combine $2 b$ and $\frac{2}{2}$ .

$- \frac{2 (- \frac{49 b}{2} + \frac{2 b \cdot 2}{2} + 263550)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.8

Combine the numerators over the common denominator.

$- \frac{2 (\frac{- 49 b + 2 b \cdot 2}{2} + 263550)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.9

To write $263550$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{2 (\frac{- 49 b + 2 b \cdot 2}{2} + 263550 \cdot \frac{2}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.10

Combine $263550$ and $\frac{2}{2}$ .

$- \frac{2 (\frac{- 49 b + 2 b \cdot 2}{2} + \frac{263550 \cdot 2}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.11

Combine the numerators over the common denominator.

$- \frac{2 (\frac{- 49 b + 2 b \cdot 2 + 263550 \cdot 2}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.12

Rewrite $\frac{- 49 b + 2 b \cdot 2 + 263550 \cdot 2}{2}$ in a factored form.

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Step 3.4.2.1.1.12.1

Multiply $2$ by $2$ .

$- \frac{2 (\frac{- 49 b + 4 b + 263550 \cdot 2}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.12.2

Multiply $263550$ by $2$ .

$- \frac{2 (\frac{- 49 b + 4 b + 527100}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.12.3

Add $- 49 b$ and $4 b$ .

$- \frac{2 (\frac{- 45 b + 527100}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.12.4

Factor $15$ out of $- 45 b + 527100$ .

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Step 3.4.2.1.1.12.4.1

Factor $15$ out of $- 45 b$ .

$- \frac{2 (\frac{15 (- 3 b) + 527100}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.12.4.2

Factor $15$ out of $527100$ .

$- \frac{2 (\frac{15 (- 3 b) + 15 (35140)}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.1.12.4.3

Factor $15$ out of $15 (- 3 b) + 15 (35140)$ .

$- \frac{2 (\frac{15 (- 3 b + 35140)}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{2 (\frac{15 (- 3 b + 35140)}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{2 (\frac{15 (- 3 b + 35140)}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{2 (\frac{15 (- 3 b + 35140)}{2})}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.2

Combine $2$ and $\frac{15 (- 3 b + 35140)}{2}$ .

$- \frac{\frac{2 (15 (- 3 b + 35140))}{2}}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.3

Multiply $2$ by $15$ .

$- \frac{\frac{30 (- 3 b + 35140)}{2}}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.4

Reduce the expression by cancelling the common factors.

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Step 3.4.2.1.4.1

Reduce the expression $\frac{30 (- 3 b + 35140)}{2}$ by cancelling the common factors.

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Step 3.4.2.1.4.1.1

Factor $2$ out of $30 (- 3 b + 35140)$ .

$- \frac{\frac{2 (15 (- 3 b + 35140))}{2}}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.4.1.2

Factor $2$ out of $2$ .

$- \frac{\frac{2 (15 (- 3 b + 35140))}{2 (1)}}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.4.1.3

Cancel the common factor.

$- \frac{\frac{2 (15 (- 3 b + 35140))}{2 \cdot 1}}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.4.1.4

Rewrite the expression.

$- \frac{\frac{15 (- 3 b + 35140)}{1}}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{\frac{15 (- 3 b + 35140)}{1}}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.4.2

Divide $15 (- 3 b + 35140)$ by $1$ .

$- \frac{15 (- 3 b + 35140)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{15 (- 3 b + 35140)}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.5

Cancel the common factor of $15$ and $9$ .

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Step 3.4.2.1.5.1

Factor $3$ out of $15 (- 3 b + 35140)$ .

$- \frac{3 (5 (- 3 b + 35140))}{9} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.5.2

Cancel the common factors.

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Step 3.4.2.1.5.2.1

Factor $3$ out of $9$ .

$- \frac{3 (5 (- 3 b + 35140))}{3 (3)} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.5.2.2

Cancel the common factor.

$- \frac{3 (5 (- 3 b + 35140))}{3 \cdot 3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.5.2.3

Rewrite the expression.

$- \frac{5 (- 3 b + 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{5 (- 3 b + 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$- \frac{5 (- 3 b + 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.6

Factor $- 1$ out of $- 3 b$ .

$- \frac{5 (- (3 b) + 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.7

Rewrite $35140$ as $- 1 (- 35140)$ .

$- \frac{5 (- (3 b) - 1 \cdot - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.8

Factor $- 1$ out of $- (3 b) - 1 (- 35140)$ .

$- \frac{5 (- (3 b - 35140))}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.9

Simplify the expression.

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Step 3.4.2.1.9.1

Rewrite $- (3 b - 35140)$ as $- 1 (3 b - 35140)$ .

$- \frac{5 (- 1 (3 b - 35140))}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.9.2

Move the negative in front of the fraction.

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.9.3

Multiply $- 1$ by $- 1$ .

$1 (\frac{5 (3 b - 35140)}{3}) = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.2.1.9.4

Multiply $\frac{5 (3 b - 35140)}{3}$ by $1$ .

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$

Step 3.4.3

Replace all occurrences of $c$ in $a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 c}{18}$ with $66650 - \frac{7 b}{2}$ .

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 (66650 - \frac{7 b}{2})}{18}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $\frac{20300}{9} - \frac{2 b}{9} - \frac{5 (66650 - \frac{7 b}{2})}{18}$ .

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Step 3.4.4.1.1

Simplify each term.

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Step 3.4.4.1.1.1

Simplify the numerator.

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Step 3.4.4.1.1.1.1

To write $66650$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 (66650 \cdot \frac{2}{2} - \frac{7 b}{2})}{18}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.1.1.2

Combine $66650$ and $\frac{2}{2}$ .

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 (\frac{66650 \cdot 2}{2} - \frac{7 b}{2})}{18}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.1.1.3

Combine the numerators over the common denominator.

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 (\frac{66650 \cdot 2 - 7 b}{2})}{18}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.1.1.4

Multiply $66650$ by $2$ .

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 (\frac{133300 - 7 b}{2})}{18}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 (\frac{133300 - 7 b}{2})}{18}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.1.2

Combine $5$ and $\frac{133300 - 7 b}{2}$ .

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{\frac{5 (133300 - 7 b)}{2}}{18}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$a = \frac{20300}{9} - \frac{2 b}{9} - (\frac{5 (133300 - 7 b)}{2} \cdot \frac{1}{18})$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.1.4

Multiply $\frac{5 (133300 - 7 b)}{2} \cdot \frac{1}{18}$ .

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Step 3.4.4.1.1.4.1

Multiply $\frac{5 (133300 - 7 b)}{2}$ by $\frac{1}{18}$ .

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 (133300 - 7 b)}{2 \cdot 18}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.1.4.2

Multiply $2$ by $18$ .

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 (133300 - 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 (133300 - 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{20300}{9} - \frac{2 b}{9} - \frac{5 (133300 - 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.2

To write $\frac{20300}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$a = - \frac{2 b}{9} + \frac{20300}{9} \cdot \frac{4}{4} - \frac{5 (133300 - 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.3

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.4.1.3.1

Multiply $\frac{20300}{9}$ by $\frac{4}{4}$ .

$a = - \frac{2 b}{9} + \frac{20300 \cdot 4}{9 \cdot 4} - \frac{5 (133300 - 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.3.2

Multiply $9$ by $4$ .

$a = - \frac{2 b}{9} + \frac{20300 \cdot 4}{36} - \frac{5 (133300 - 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = - \frac{2 b}{9} + \frac{20300 \cdot 4}{36} - \frac{5 (133300 - 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.4

Combine into one fraction.

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Step 3.4.4.1.4.1

Combine the numerators over the common denominator.

$a = - \frac{2 b}{9} + \frac{20300 \cdot 4 - 5 (133300 - 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.4.2

Multiply $20300$ by $4$ .

$a = - \frac{2 b}{9} + \frac{81200 - 5 (133300 - 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = - \frac{2 b}{9} + \frac{81200 - 5 (133300 - 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.5

Simplify the numerator.

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Step 3.4.4.1.5.1

Factor $5$ out of $81200 - 5 (133300 - 7 b)$ .

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Step 3.4.4.1.5.1.1

Factor $5$ out of $81200$ .

$a = - \frac{2 b}{9} + \frac{5 (16240) - 5 (133300 - 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.5.1.2

Factor $5$ out of $- 5 (133300 - 7 b)$ .

$a = - \frac{2 b}{9} + \frac{5 (16240) + 5 (- (133300 - 7 b))}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.5.1.3

Factor $5$ out of $5 (16240) + 5 (- (133300 - 7 b))$ .

$a = - \frac{2 b}{9} + \frac{5 (16240 - (133300 - 7 b))}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = - \frac{2 b}{9} + \frac{5 (16240 - (133300 - 7 b))}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.5.2

Apply the distributive property.

$a = - \frac{2 b}{9} + \frac{5 (16240 - 1 \cdot 133300 - (- 7 b))}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.5.3

Multiply $- 1$ by $133300$ .

$a = - \frac{2 b}{9} + \frac{5 (16240 - 133300 - (- 7 b))}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.5.4

Multiply $- 7$ by $- 1$ .

$a = - \frac{2 b}{9} + \frac{5 (16240 - 133300 + 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.5.5

Subtract $133300$ from $16240$ .

$a = - \frac{2 b}{9} + \frac{5 (- 117060 + 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = - \frac{2 b}{9} + \frac{5 (- 117060 + 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.6

To write $- \frac{2 b}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$a = - \frac{2 b}{9} \cdot \frac{4}{4} + \frac{5 (- 117060 + 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.7

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.4.1.7.1

Multiply $\frac{2 b}{9}$ by $\frac{4}{4}$ .

$a = - \frac{2 b \cdot 4}{9 \cdot 4} + \frac{5 (- 117060 + 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.7.2

Multiply $9$ by $4$ .

$a = - \frac{2 b \cdot 4}{36} + \frac{5 (- 117060 + 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = - \frac{2 b \cdot 4}{36} + \frac{5 (- 117060 + 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.8

Combine the numerators over the common denominator.

$a = \frac{- 2 b \cdot 4 + 5 (- 117060 + 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.9

Simplify the numerator.

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Step 3.4.4.1.9.1

Multiply $4$ by $- 2$ .

$a = \frac{- 8 b + 5 (- 117060 + 7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.9.2

Apply the distributive property.

$a = \frac{- 8 b + 5 \cdot - 117060 + 5 (7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.9.3

Multiply $5$ by $- 117060$ .

$a = \frac{- 8 b - 585300 + 5 (7 b)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.9.4

Multiply $7$ by $5$ .

$a = \frac{- 8 b - 585300 + 35 b}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.9.5

Add $- 8 b$ and $35 b$ .

$a = \frac{27 b - 585300}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.9.6

Factor $3$ out of $27 b - 585300$ .

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Step 3.4.4.1.9.6.1

Factor $3$ out of $27 b$ .

$a = \frac{3 (9 b) - 585300}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.9.6.2

Factor $3$ out of $- 585300$ .

$a = \frac{3 (9 b) + 3 (- 195100)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.9.6.3

Factor $3$ out of $3 (9 b) + 3 (- 195100)$ .

$a = \frac{3 (9 b - 195100)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{3 (9 b - 195100)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{3 (9 b - 195100)}{36}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.10

Cancel the common factors.

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Step 3.4.4.1.10.1

Factor $3$ out of $36$ .

$a = \frac{3 (9 b - 195100)}{3 \cdot 12}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.10.2

Cancel the common factor.

$a = \frac{3 (9 b - 195100)}{3 \cdot 12}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.4.4.1.10.3

Rewrite the expression.

$a = \frac{9 b - 195100}{12}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{9 b - 195100}{12}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{9 b - 195100}{12}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{9 b - 195100}{12}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{9 b - 195100}{12}$

$\frac{5 (3 b - 35140)}{3} = 7500$

$c = 66650 - \frac{7 b}{2}$

Step 3.5

Solve for $b$ in $\frac{5 (3 b - 35140)}{3} = 7500$ .

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Step 3.5.1

Multiply both sides by $3$ .

$\frac{5 (3 b - 35140)}{3} \cdot 3 = 7500 \cdot 3$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.2

Simplify.

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Step 3.5.2.1

Simplify the left side.

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Step 3.5.2.1.1

Simplify $\frac{5 (3 b - 35140)}{3} \cdot 3$ .

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Step 3.5.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.5.2.1.1.1.1

Cancel the common factor.

$\frac{5 (3 b - 35140)}{3} \cdot 3 = 7500 \cdot 3$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.2.1.1.1.2

Rewrite the expression.

$5 (3 b - 35140) = 7500 \cdot 3$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$5 (3 b - 35140) = 7500 \cdot 3$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.2.1.1.2

Apply the distributive property.

$5 (3 b) + 5 \cdot - 35140 = 7500 \cdot 3$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.2.1.1.3

Multiply.

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Step 3.5.2.1.1.3.1

Multiply $3$ by $5$ .

$15 b + 5 \cdot - 35140 = 7500 \cdot 3$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.2.1.1.3.2

Multiply $5$ by $- 35140$ .

$15 b - 175700 = 7500 \cdot 3$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$15 b - 175700 = 7500 \cdot 3$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$15 b - 175700 = 7500 \cdot 3$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$15 b - 175700 = 7500 \cdot 3$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.2.2

Simplify the right side.

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Step 3.5.2.2.1

Multiply $7500$ by $3$ .

$15 b - 175700 = 22500$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$15 b - 175700 = 22500$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$15 b - 175700 = 22500$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.3

Solve for $b$ .

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Step 3.5.3.1

Move all terms not containing $b$ to the right side of the equation.

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Step 3.5.3.1.1

Add $175700$ to both sides of the equation.

$15 b = 22500 + 175700$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.3.1.2

Add $22500$ and $175700$ .

$15 b = 198200$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$15 b = 198200$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.3.2

Divide each term in $15 b = 198200$ by $15$ and simplify.

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Step 3.5.3.2.1

Divide each term in $15 b = 198200$ by $15$ .

$\frac{15 b}{15} = \frac{198200}{15}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.3.2.2

Simplify the left side.

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Step 3.5.3.2.2.1

Cancel the common factor of $15$ .

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Step 3.5.3.2.2.1.1

Cancel the common factor.

$\frac{15 b}{15} = \frac{198200}{15}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.3.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{198200}{15}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$b = \frac{198200}{15}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$b = \frac{198200}{15}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.3.2.3

Simplify the right side.

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Step 3.5.3.2.3.1

Cancel the common factor of $198200$ and $15$ .

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Step 3.5.3.2.3.1.1

Factor $5$ out of $198200$ .

$b = \frac{5 (39640)}{15}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.3.2.3.1.2

Cancel the common factors.

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Step 3.5.3.2.3.1.2.1

Factor $5$ out of $15$ .

$b = \frac{5 \cdot 39640}{5 \cdot 3}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.3.2.3.1.2.2

Cancel the common factor.

$b = \frac{5 \cdot 39640}{5 \cdot 3}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.5.3.2.3.1.2.3

Rewrite the expression.

$b = \frac{39640}{3}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$b = \frac{39640}{3}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$b = \frac{39640}{3}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$b = \frac{39640}{3}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$b = \frac{39640}{3}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$b = \frac{39640}{3}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

$b = \frac{39640}{3}$

$a = \frac{9 b - 195100}{12}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6

Replace all occurrences of $b$ with $\frac{39640}{3}$ in each equation.

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Step 3.6.1

Replace all occurrences of $b$ in $a = \frac{9 b - 195100}{12}$ with $\frac{39640}{3}$ .

$a = \frac{9 (\frac{39640}{3}) - 195100}{12}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $\frac{9 (\frac{39640}{3}) - 195100}{12}$ .

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Step 3.6.2.1.1

Simplify the numerator.

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Step 3.6.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.6.2.1.1.1.1

Factor $3$ out of $9$ .

$a = \frac{3 (3) (\frac{39640}{3}) - 195100}{12}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6.2.1.1.1.2

Cancel the common factor.

$a = \frac{3 \cdot (3 (\frac{39640}{3})) - 195100}{12}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6.2.1.1.1.3

Rewrite the expression.

$a = \frac{3 \cdot 39640 - 195100}{12}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{3 \cdot 39640 - 195100}{12}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6.2.1.1.2

Multiply $3$ by $39640$ .

$a = \frac{118920 - 195100}{12}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6.2.1.1.3

Subtract $195100$ from $118920$ .

$a = \frac{- 76180}{12}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{- 76180}{12}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6.2.1.2

Reduce the expression by cancelling the common factors.

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Step 3.6.2.1.2.1

Cancel the common factor of $- 76180$ and $12$ .

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Step 3.6.2.1.2.1.1

Factor $4$ out of $- 76180$ .

$a = \frac{4 (- 19045)}{12}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6.2.1.2.1.2

Cancel the common factors.

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Step 3.6.2.1.2.1.2.1

Factor $4$ out of $12$ .

$a = \frac{4 \cdot - 19045}{4 \cdot 3}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6.2.1.2.1.2.2

Cancel the common factor.

$a = \frac{4 \cdot - 19045}{4 \cdot 3}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6.2.1.2.1.2.3

Rewrite the expression.

$a = \frac{- 19045}{3}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{- 19045}{3}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

$a = \frac{- 19045}{3}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6.2.1.2.2

Move the negative in front of the fraction.

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{7 b}{2}$

Step 3.6.3

Replace all occurrences of $b$ in $c = 66650 - \frac{7 b}{2}$ with $\frac{39640}{3}$ .

$c = 66650 - \frac{7 (\frac{39640}{3})}{2}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $66650 - \frac{7 (\frac{39640}{3})}{2}$ .

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Step 3.6.4.1.1

Simplify each term.

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Step 3.6.4.1.1.1

Combine $7$ and $\frac{39640}{3}$ .

$c = 66650 - \frac{\frac{7 \cdot 39640}{3}}{2}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.6.4.1.1.2

Multiply $7$ by $39640$ .

$c = 66650 - \frac{\frac{277480}{3}}{2}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.6.4.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$c = 66650 - (\frac{277480}{3} \cdot \frac{1}{2})$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.6.4.1.1.4

Cancel the common factor of $2$ .

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Step 3.6.4.1.1.4.1

Factor $2$ out of $277480$ .

$c = 66650 - (\frac{2 (138740)}{3} \cdot \frac{1}{2})$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.6.4.1.1.4.2

Cancel the common factor.

$c = 66650 - (\frac{2 \cdot 138740}{3} \cdot \frac{1}{2})$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.6.4.1.1.4.3

Rewrite the expression.

$c = 66650 - \frac{138740}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{138740}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

$c = 66650 - \frac{138740}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.6.4.1.2

To write $66650$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$c = 66650 \cdot \frac{3}{3} - \frac{138740}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.6.4.1.3

Combine $66650$ and $\frac{3}{3}$ .

$c = \frac{66650 \cdot 3}{3} - \frac{138740}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.6.4.1.4

Combine the numerators over the common denominator.

$c = \frac{66650 \cdot 3 - 138740}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.6.4.1.5

Simplify the numerator.

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Step 3.6.4.1.5.1

Multiply $66650$ by $3$ .

$c = \frac{199950 - 138740}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.6.4.1.5.2

Subtract $138740$ from $199950$ .

$c = \frac{61210}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

$c = \frac{61210}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

$c = \frac{61210}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

$c = \frac{61210}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

$c = \frac{61210}{3}$

$a = - \frac{19045}{3}$

$b = \frac{39640}{3}$

Step 3.7

List all of the solutions.

$c = \frac{61210}{3}, a = - \frac{19045}{3}, b = \frac{39640}{3}$