Finite Math Examples

$x + 5 - z = 10$ , $- x + 7 y - z = 18$ , $5 x - 4 y - 2 z = - 14$

Step 1

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

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Step 1.1

Subtract $5$ from both sides of the equation.

$x - z = 10 - 5$

$- x + 7 y - z = 18$

$5 x - 4 y - 2 z = - 14$

Step 1.2

Add $z$ to both sides of the equation.

$x = 10 - 5 + z$

$- x + 7 y - z = 18$

$5 x - 4 y - 2 z = - 14$

Step 1.3

Subtract $5$ from $10$ .

$x = 5 + z$

$- x + 7 y - z = 18$

$5 x - 4 y - 2 z = - 14$

$x = 5 + z$

$- x + 7 y - z = 18$

$5 x - 4 y - 2 z = - 14$

Step 2

Replace all occurrences of $x$ with $5 + z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- x + 7 y - z = 18$ with $5 + z$ .

$- (5 + z) + 7 y - z = 18$

$x = 5 + z$

$5 x - 4 y - 2 z = - 14$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- (5 + z) + 7 y - z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$- 1 \cdot 5 - z + 7 y - z = 18$

$x = 5 + z$

$5 x - 4 y - 2 z = - 14$

Step 2.2.1.1.2

Multiply $- 1$ by $5$ .

$- 5 - z + 7 y - z = 18$

$x = 5 + z$

$5 x - 4 y - 2 z = - 14$

$- 5 - z + 7 y - z = 18$

$x = 5 + z$

$5 x - 4 y - 2 z = - 14$

Step 2.2.1.2

Subtract $z$ from $- z$ .

$- 5 + 7 y - 2 z = 18$

$x = 5 + z$

$5 x - 4 y - 2 z = - 14$

$- 5 + 7 y - 2 z = 18$

$x = 5 + z$

$5 x - 4 y - 2 z = - 14$

$- 5 + 7 y - 2 z = 18$

$x = 5 + z$

$5 x - 4 y - 2 z = - 14$

Step 2.3

Replace all occurrences of $x$ in $5 x - 4 y - 2 z = - 14$ with $5 + z$ .

$5 (5 + z) - 4 y - 2 z = - 14$

$- 5 + 7 y - 2 z = 18$

$x = 5 + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $5 (5 + z) - 4 y - 2 z$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$5 \cdot 5 + 5 z - 4 y - 2 z = - 14$

$- 5 + 7 y - 2 z = 18$

$x = 5 + z$

Step 2.4.1.1.2

Multiply $5$ by $5$ .

$25 + 5 z - 4 y - 2 z = - 14$

$- 5 + 7 y - 2 z = 18$

$x = 5 + z$

$25 + 5 z - 4 y - 2 z = - 14$

$- 5 + 7 y - 2 z = 18$

$x = 5 + z$

Step 2.4.1.2

Subtract $2 z$ from $5 z$ .

$25 - 4 y + 3 z = - 14$

$- 5 + 7 y - 2 z = 18$

$x = 5 + z$

$25 - 4 y + 3 z = - 14$

$- 5 + 7 y - 2 z = 18$

$x = 5 + z$

$25 - 4 y + 3 z = - 14$

$- 5 + 7 y - 2 z = 18$

$x = 5 + z$

$25 - 4 y + 3 z = - 14$

$- 5 + 7 y - 2 z = 18$

$x = 5 + z$

Step 3

Reorder $5$ and $z$ .

$x = z + 5$

$25 - 4 y + 3 z = - 14$

$- 5 + 7 y - 2 z = 18$

Step 4

Solve for $z$ in $x = z + 5$ .

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Step 4.1

Rewrite the equation as $z + 5 = x$ .

$z + 5 = x$

$25 - 4 y + 3 z = - 14$

$- 5 + 7 y - 2 z = 18$

Step 4.2

Subtract $5$ from both sides of the equation.

$z = x - 5$

$25 - 4 y + 3 z = - 14$

$- 5 + 7 y - 2 z = 18$

$z = x - 5$

$25 - 4 y + 3 z = - 14$

$- 5 + 7 y - 2 z = 18$

Step 5

Replace all occurrences of $z$ with $x - 5$ in each equation.

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Step 5.1

Replace all occurrences of $z$ in $25 - 4 y + 3 z = - 14$ with $x - 5$ .

$25 - 4 y + 3 (x - 5) = - 14$

$z = x - 5$

$- 5 + 7 y - 2 z = 18$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $25 - 4 y + 3 (x - 5)$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$25 - 4 y + 3 x + 3 \cdot - 5 = - 14$

$z = x - 5$

$- 5 + 7 y - 2 z = 18$

Step 5.2.1.1.2

Multiply $3$ by $- 5$ .

$25 - 4 y + 3 x - 15 = - 14$

$z = x - 5$

$- 5 + 7 y - 2 z = 18$

$25 - 4 y + 3 x - 15 = - 14$

$z = x - 5$

$- 5 + 7 y - 2 z = 18$

Step 5.2.1.2

Subtract $15$ from $25$ .

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$- 5 + 7 y - 2 z = 18$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$- 5 + 7 y - 2 z = 18$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$- 5 + 7 y - 2 z = 18$

Step 5.3

Replace all occurrences of $z$ in $- 5 + 7 y - 2 z = 18$ with $x - 5$ .

$- 5 + 7 y - 2 (x - 5) = 18$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $- 5 + 7 y - 2 (x - 5)$ .

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Step 5.4.1.1

Simplify each term.

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Step 5.4.1.1.1

Apply the distributive property.

$- 5 + 7 y - 2 x - 2 \cdot - 5 = 18$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

Step 5.4.1.1.2

Multiply $- 2$ by $- 5$ .

$- 5 + 7 y - 2 x + 10 = 18$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$- 5 + 7 y - 2 x + 10 = 18$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

Step 5.4.1.2

Add $- 5$ and $10$ .

$7 y - 2 x + 5 = 18$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$7 y - 2 x + 5 = 18$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$7 y - 2 x + 5 = 18$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$7 y - 2 x + 5 = 18$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

Step 6

Solve for $y$ in $7 y - 2 x + 5 = 18$ .

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Step 6.1

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

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Step 6.1.1

Add $2 x$ to both sides of the equation.

$7 y + 5 = 18 + 2 x$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

Step 6.1.2

Subtract $5$ from both sides of the equation.

$7 y = 18 + 2 x - 5$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

Step 6.1.3

Subtract $5$ from $18$ .

$7 y = 2 x + 13$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$7 y = 2 x + 13$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

Step 6.2

Divide each term in $7 y = 2 x + 13$ by $7$ and simplify.

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Step 6.2.1

Divide each term in $7 y = 2 x + 13$ by $7$ .

$\frac{7 y}{7} = \frac{2 x}{7} + \frac{13}{7}$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $7$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{2 x}{7} + \frac{13}{7}$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

Step 6.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x}{7} + \frac{13}{7}$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$y = \frac{2 x}{7} + \frac{13}{7}$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$y = \frac{2 x}{7} + \frac{13}{7}$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$y = \frac{2 x}{7} + \frac{13}{7}$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

$y = \frac{2 x}{7} + \frac{13}{7}$

$- 4 y + 3 x + 10 = - 14$

$z = x - 5$

Step 7

Replace all occurrences of $y$ with $\frac{2 x}{7} + \frac{13}{7}$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $- 4 y + 3 x + 10 = - 14$ with $\frac{2 x}{7} + \frac{13}{7}$ .

$- 4 (\frac{2 x}{7} + \frac{13}{7}) + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2

Simplify the left side.

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Step 7.2.1

Simplify $- 4 (\frac{2 x}{7} + \frac{13}{7}) + 3 x + 10$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Apply the distributive property.

$- 4 \frac{2 x}{7} - 4 (\frac{13}{7}) + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.1.2

Multiply $- 4 \frac{2 x}{7}$ .

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Step 7.2.1.1.2.1

Combine $- 4$ and $\frac{2 x}{7}$ .

$\frac{- 4 (2 x)}{7} - 4 (\frac{13}{7}) + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.1.2.2

Multiply $2$ by $- 4$ .

$\frac{- 8 x}{7} - 4 (\frac{13}{7}) + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$\frac{- 8 x}{7} - 4 (\frac{13}{7}) + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.1.3

Multiply $- 4 (\frac{13}{7})$ .

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Step 7.2.1.1.3.1

Combine $- 4$ and $\frac{13}{7}$ .

$\frac{- 8 x}{7} + \frac{- 4 \cdot 13}{7} + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.1.3.2

Multiply $- 4$ by $13$ .

$\frac{- 8 x}{7} + \frac{- 52}{7} + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$\frac{- 8 x}{7} + \frac{- 52}{7} + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.1.4

Simplify each term.

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Step 7.2.1.1.4.1

Move the negative in front of the fraction.

$- \frac{8 x}{7} + \frac{- 52}{7} + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{8 x}{7} - \frac{52}{7} + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$- \frac{8 x}{7} - \frac{52}{7} + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$- \frac{8 x}{7} - \frac{52}{7} + 3 x + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.2

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{8 x}{7} + 3 x \cdot \frac{7}{7} - \frac{52}{7} + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.3

Combine $3 x$ and $\frac{7}{7}$ .

$- \frac{8 x}{7} + \frac{3 x \cdot 7}{7} - \frac{52}{7} + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.4

Combine the numerators over the common denominator.

$\frac{- 8 x + 3 x \cdot 7}{7} - \frac{52}{7} + 10 = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.5

Combine the numerators over the common denominator.

$10 + \frac{- 8 x + 3 x \cdot 7 - 52}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.6

Multiply $7$ by $3$ .

$10 + \frac{- 8 x + 21 x - 52}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.7

Add $- 8 x$ and $21 x$ .

$10 + \frac{13 x - 52}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.8

Factor $13$ out of $13 x - 52$ .

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Step 7.2.1.8.1

Factor $13$ out of $13 x$ .

$10 + \frac{13 (x) - 52}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.8.2

Factor $13$ out of $- 52$ .

$10 + \frac{13 x + 13 \cdot - 4}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.8.3

Factor $13$ out of $13 x + 13 \cdot - 4$ .

$10 + \frac{13 (x - 4)}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$10 + \frac{13 (x - 4)}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.9

To write $10$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$10 \cdot \frac{7}{7} + \frac{13 (x - 4)}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.10

Simplify terms.

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Step 7.2.1.10.1

Combine $10$ and $\frac{7}{7}$ .

$\frac{10 \cdot 7}{7} + \frac{13 (x - 4)}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.10.2

Combine the numerators over the common denominator.

$\frac{10 \cdot 7 + 13 (x - 4)}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$\frac{10 \cdot 7 + 13 (x - 4)}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.11

Simplify the numerator.

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Step 7.2.1.11.1

Multiply $10$ by $7$ .

$\frac{70 + 13 (x - 4)}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.11.2

Apply the distributive property.

$\frac{70 + 13 x + 13 \cdot - 4}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.11.3

Multiply $13$ by $- 4$ .

$\frac{70 + 13 x - 52}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 7.2.1.11.4

Subtract $52$ from $70$ .

$\frac{13 x + 18}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$\frac{13 x + 18}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$\frac{13 x + 18}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$\frac{13 x + 18}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$\frac{13 x + 18}{7} = - 14$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 8

Solve for $x$ in $\frac{13 x + 18}{7} = - 14$ .

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Step 8.1

Multiply both sides by $7$ .

$\frac{13 x + 18}{7} \cdot 7 = - 14 \cdot 7$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 8.2

Simplify.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

Cancel the common factor of $7$ .

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Step 8.2.1.1.1

Cancel the common factor.

$\frac{13 x + 18}{7} \cdot 7 = - 14 \cdot 7$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 8.2.1.1.2

Rewrite the expression.

$13 x + 18 = - 14 \cdot 7$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$13 x + 18 = - 14 \cdot 7$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$13 x + 18 = - 14 \cdot 7$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Multiply $- 14$ by $7$ .

$13 x + 18 = - 98$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$13 x + 18 = - 98$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$13 x + 18 = - 98$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 8.3

Solve for $x$ .

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Step 8.3.1

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

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Step 8.3.1.1

Subtract $18$ from both sides of the equation.

$13 x = - 98 - 18$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 8.3.1.2

Subtract $18$ from $- 98$ .

$13 x = - 116$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$13 x = - 116$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 8.3.2

Divide each term in $13 x = - 116$ by $13$ and simplify.

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Step 8.3.2.1

Divide each term in $13 x = - 116$ by $13$ .

$\frac{13 x}{13} = \frac{- 116}{13}$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 8.3.2.2

Simplify the left side.

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Step 8.3.2.2.1

Cancel the common factor of $13$ .

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Step 8.3.2.2.1.1

Cancel the common factor.

$\frac{13 x}{13} = \frac{- 116}{13}$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 8.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 116}{13}$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$x = \frac{- 116}{13}$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$x = \frac{- 116}{13}$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 8.3.2.3

Simplify the right side.

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Step 8.3.2.3.1

Move the negative in front of the fraction.

$x = - \frac{116}{13}$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$x = - \frac{116}{13}$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$x = - \frac{116}{13}$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$x = - \frac{116}{13}$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

$x = - \frac{116}{13}$

$y = \frac{2 x}{7} + \frac{13}{7}$

$z = x - 5$

Step 9

Replace all occurrences of $x$ with $- \frac{116}{13}$ in each equation.

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Step 9.1

Replace all occurrences of $x$ in $y = \frac{2 x}{7} + \frac{13}{7}$ with $- \frac{116}{13}$ .

$y = \frac{2 (- \frac{116}{13})}{7} + \frac{13}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $\frac{2 (- \frac{116}{13})}{7} + \frac{13}{7}$ .

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Step 9.2.1.1

Combine the numerators over the common denominator.

$y = \frac{2 (- \frac{116}{13}) + 13}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.2

Simplify each term.

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Step 9.2.1.2.1

Multiply $2 (- \frac{116}{13})$ .

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Step 9.2.1.2.1.1

Multiply $- 1$ by $2$ .

$y = \frac{- 2 (\frac{116}{13}) + 13}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.2.1.2

Combine $- 2$ and $\frac{116}{13}$ .

$y = \frac{\frac{- 2 \cdot 116}{13} + 13}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.2.1.3

Multiply $- 2$ by $116$ .

$y = \frac{\frac{- 232}{13} + 13}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

$y = \frac{\frac{- 232}{13} + 13}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.2.2

Move the negative in front of the fraction.

$y = \frac{- \frac{232}{13} + 13}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

$y = \frac{- \frac{232}{13} + 13}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.3

To write $13$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$y = \frac{- \frac{232}{13} + 13 \cdot \frac{13}{13}}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.4

Combine $13$ and $\frac{13}{13}$ .

$y = \frac{- \frac{232}{13} + \frac{13 \cdot 13}{13}}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.5

Combine the numerators over the common denominator.

$y = \frac{\frac{- 232 + 13 \cdot 13}{13}}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.6

Simplify the numerator.

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Step 9.2.1.6.1

Multiply $13$ by $13$ .

$y = \frac{\frac{- 232 + 169}{13}}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.6.2

Add $- 232$ and $169$ .

$y = \frac{\frac{- 63}{13}}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

$y = \frac{\frac{- 63}{13}}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.7

Move the negative in front of the fraction.

$y = \frac{- \frac{63}{13}}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.8

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{63}{13} \cdot \frac{1}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.9

Cancel the common factor of $7$ .

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Step 9.2.1.9.1

Move the leading negative in $- \frac{63}{13}$ into the numerator.

$y = \frac{- 63}{13} \cdot \frac{1}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.9.2

Factor $7$ out of $- 63$ .

$y = \frac{7 (- 9)}{13} \cdot \frac{1}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.9.3

Cancel the common factor.

$y = \frac{7 \cdot - 9}{13} \cdot \frac{1}{7}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.9.4

Rewrite the expression.

$y = \frac{- 9}{13}$

$x = - \frac{116}{13}$

$z = x - 5$

$y = \frac{- 9}{13}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.2.1.10

Move the negative in front of the fraction.

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

$z = x - 5$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

$z = x - 5$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

$z = x - 5$

Step 9.3

Replace all occurrences of $x$ in $z = x - 5$ with $- \frac{116}{13}$ .

$z = (- \frac{116}{13}) - 5$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

Step 9.4

Simplify the right side.

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Step 9.4.1

Simplify $(- \frac{116}{13}) - 5$ .

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Step 9.4.1.1

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$z = - \frac{116}{13} - 5 \cdot \frac{13}{13}$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

Step 9.4.1.2

Combine $- 5$ and $\frac{13}{13}$ .

$z = - \frac{116}{13} + \frac{- 5 \cdot 13}{13}$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

Step 9.4.1.3

Combine the numerators over the common denominator.

$z = \frac{- 116 - 5 \cdot 13}{13}$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

Step 9.4.1.4

Simplify the numerator.

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Step 9.4.1.4.1

Multiply $- 5$ by $13$ .

$z = \frac{- 116 - 65}{13}$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

Step 9.4.1.4.2

Subtract $65$ from $- 116$ .

$z = \frac{- 181}{13}$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

$z = \frac{- 181}{13}$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

Step 9.4.1.5

Move the negative in front of the fraction.

$z = - \frac{181}{13}$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

$z = - \frac{181}{13}$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

$z = - \frac{181}{13}$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

$z = - \frac{181}{13}$

$y = - \frac{9}{13}$

$x = - \frac{116}{13}$

Step 10

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- \frac{116}{13}, - \frac{9}{13}, - \frac{181}{13})$

Step 11

The result can be shown in multiple forms.

Point Form:

$(- \frac{116}{13}, - \frac{9}{13}, - \frac{181}{13})$

Equation Form:

$x = - \frac{116}{13}, y = - \frac{9}{13}, z = - \frac{181}{13}$