Algebra Examples

$- 6 x + 8 y + 3 z = 9$ $- 6 x + y + 5 z = 2$ $x + 2 y - z = 5$

Step 1

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

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Step 1.1

Add $6 x$ to both sides of the equation.

$y + 5 z = 2 + 6 x$

$- 6 x + 8 y + 3 z = 9$

$x + 2 y - z = 5$

Step 1.2

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

$y = 2 + 6 x - 5 z$

$- 6 x + 8 y + 3 z = 9$

$x + 2 y - z = 5$

$y = 2 + 6 x - 5 z$

$- 6 x + 8 y + 3 z = 9$

$x + 2 y - z = 5$

Step 2

Replace all occurrences of $y$ with $2 + 6 x - 5 z$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $- 6 x + 8 y + 3 z = 9$ with $2 + 6 x - 5 z$ .

$- 6 x + 8 (2 + 6 x - 5 z) + 3 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 6 x + 8 (2 + 6 x - 5 z) + 3 z$ .

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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.

$- 6 x + 8 \cdot 2 + 8 (6 x) + 8 (- 5 z) + 3 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $8$ by $2$ .

$- 6 x + 16 + 8 (6 x) + 8 (- 5 z) + 3 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

Step 2.2.1.1.2.2

Multiply $6$ by $8$ .

$- 6 x + 16 + 48 x + 8 (- 5 z) + 3 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

Step 2.2.1.1.2.3

Multiply $- 5$ by $8$ .

$- 6 x + 16 + 48 x - 40 z + 3 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

$- 6 x + 16 + 48 x - 40 z + 3 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

$- 6 x + 16 + 48 x - 40 z + 3 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 6 x$ and $48 x$ .

$42 x + 16 - 40 z + 3 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

Step 2.2.1.2.2

Add $- 40 z$ and $3 z$ .

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$x + 2 y - z = 5$

Step 2.3

Replace all occurrences of $y$ in $x + 2 y - z = 5$ with $2 + 6 x - 5 z$ .

$x + 2 (2 + 6 x - 5 z) - z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $x + 2 (2 + 6 x - 5 z) - z$ .

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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.

$x + 2 \cdot 2 + 2 (6 x) + 2 (- 5 z) - z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $2$ .

$x + 4 + 2 (6 x) + 2 (- 5 z) - z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 2.4.1.1.2.2

Multiply $6$ by $2$ .

$x + 4 + 12 x + 2 (- 5 z) - z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 2.4.1.1.2.3

Multiply $- 5$ by $2$ .

$x + 4 + 12 x - 10 z - z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$x + 4 + 12 x - 10 z - z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$x + 4 + 12 x - 10 z - z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $x$ and $12 x$ .

$13 x + 4 - 10 z - z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 2.4.1.2.2

Subtract $z$ from $- 10 z$ .

$13 x + 4 - 11 z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$13 x + 4 - 11 z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$13 x + 4 - 11 z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$13 x + 4 - 11 z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$13 x + 4 - 11 z = 5$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 3

Solve for $x$ in $13 x + 4 - 11 z = 5$ .

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Step 3.1

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

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Step 3.1.1

Subtract $4$ from both sides of the equation.

$13 x - 11 z = 5 - 4$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 3.1.2

Add $11 z$ to both sides of the equation.

$13 x = 5 - 4 + 11 z$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 3.1.3

Subtract $4$ from $5$ .

$13 x = 1 + 11 z$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$13 x = 1 + 11 z$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 3.2

Divide each term in $13 x = 1 + 11 z$ by $13$ and simplify.

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Step 3.2.1

Divide each term in $13 x = 1 + 11 z$ by $13$ .

$\frac{13 x}{13} = \frac{1}{13} + \frac{11 z}{13}$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $13$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{13 x}{13} = \frac{1}{13} + \frac{11 z}{13}$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{13} + \frac{11 z}{13}$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$x = \frac{1}{13} + \frac{11 z}{13}$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$x = \frac{1}{13} + \frac{11 z}{13}$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$x = \frac{1}{13} + \frac{11 z}{13}$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

$x = \frac{1}{13} + \frac{11 z}{13}$

$42 x + 16 - 37 z = 9$

$y = 2 + 6 x - 5 z$

Step 4

Replace all occurrences of $x$ with $\frac{1}{13} + \frac{11 z}{13}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $42 x + 16 - 37 z = 9$ with $\frac{1}{13} + \frac{11 z}{13}$ .

$42 (\frac{1}{13} + \frac{11 z}{13}) + 16 - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $42 (\frac{1}{13} + \frac{11 z}{13}) + 16 - 37 z$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$42 (\frac{1}{13}) + 42 (\frac{11 z}{13}) + 16 - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.1.2

Combine $42$ and $\frac{1}{13}$ .

$\frac{42}{13} + 42 (\frac{11 z}{13}) + 16 - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.1.3

Multiply $42 \frac{11 z}{13}$ .

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Step 4.2.1.1.3.1

Combine $42$ and $\frac{11 z}{13}$ .

$\frac{42}{13} + \frac{42 (11 z)}{13} + 16 - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.1.3.2

Multiply $11$ by $42$ .

$\frac{42}{13} + \frac{462 z}{13} + 16 - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

$\frac{42}{13} + \frac{462 z}{13} + 16 - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

$\frac{42}{13} + \frac{462 z}{13} + 16 - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.2

To write $16$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$\frac{462 z}{13} + \frac{42}{13} + 16 \cdot \frac{13}{13} - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.3

Combine $16$ and $\frac{13}{13}$ .

$\frac{462 z}{13} + \frac{42}{13} + \frac{16 \cdot 13}{13} - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{462 z}{13} + \frac{42 + 16 \cdot 13}{13} - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $16$ by $13$ .

$\frac{462 z}{13} + \frac{42 + 208}{13} - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.5.2

Add $42$ and $208$ .

$\frac{462 z}{13} + \frac{250}{13} - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

$\frac{462 z}{13} + \frac{250}{13} - 37 z = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.6

To write $- 37 z$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$\frac{462 z}{13} - 37 z \cdot \frac{13}{13} + \frac{250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.7

Combine $- 37 z$ and $\frac{13}{13}$ .

$\frac{462 z}{13} + \frac{- 37 z \cdot 13}{13} + \frac{250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.8

Combine the numerators over the common denominator.

$\frac{462 z - 37 z \cdot 13}{13} + \frac{250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.9

Combine the numerators over the common denominator.

$\frac{462 z - 37 z \cdot 13 + 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.10

Multiply $13$ by $- 37$ .

$\frac{462 z - 481 z + 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.11

Subtract $481 z$ from $462 z$ .

$\frac{- 19 z + 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.12

Factor $- 1$ out of $- 19 z$ .

$\frac{- (19 z) + 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.13

Rewrite $250$ as $- 1 (- 250)$ .

$\frac{- (19 z) - 1 \cdot - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.14

Factor $- 1$ out of $- (19 z) - 1 (- 250)$ .

$\frac{- (19 z - 250)}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.15

Simplify the expression.

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Step 4.2.1.15.1

Rewrite $- (19 z - 250)$ as $- 1 (19 z - 250)$ .

$\frac{- 1 (19 z - 250)}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.2.1.15.2

Move the negative in front of the fraction.

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + 6 x - 5 z$

Step 4.3

Replace all occurrences of $x$ in $y = 2 + 6 x - 5 z$ with $\frac{1}{13} + \frac{11 z}{13}$ .

$y = 2 + 6 (\frac{1}{13} + \frac{11 z}{13}) - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $2 + 6 (\frac{1}{13} + \frac{11 z}{13}) - 5 z$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$y = 2 + 6 (\frac{1}{13}) + 6 (\frac{11 z}{13}) - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.1.2

Combine $6$ and $\frac{1}{13}$ .

$y = 2 + \frac{6}{13} + 6 (\frac{11 z}{13}) - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.1.3

Multiply $6 \frac{11 z}{13}$ .

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Step 4.4.1.1.3.1

Combine $6$ and $\frac{11 z}{13}$ .

$y = 2 + \frac{6}{13} + \frac{6 (11 z)}{13} - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.1.3.2

Multiply $11$ by $6$ .

$y = 2 + \frac{6}{13} + \frac{66 z}{13} - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + \frac{6}{13} + \frac{66 z}{13} - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 2 + \frac{6}{13} + \frac{66 z}{13} - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$y = 2 \cdot \frac{13}{13} + \frac{6}{13} + \frac{66 z}{13} - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.3

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

$y = \frac{2 \cdot 13}{13} + \frac{6}{13} + \frac{66 z}{13} - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$y = \frac{2 \cdot 13 + 6}{13} + \frac{66 z}{13} - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $2$ by $13$ .

$y = \frac{26 + 6}{13} + \frac{66 z}{13} - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.5.2

Add $26$ and $6$ .

$y = \frac{32}{13} + \frac{66 z}{13} - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = \frac{32}{13} + \frac{66 z}{13} - 5 z$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.6

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

$y = \frac{32}{13} + \frac{66 z}{13} - 5 z \cdot \frac{13}{13}$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.7

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

$y = \frac{32}{13} + \frac{66 z}{13} + \frac{- 5 z \cdot 13}{13}$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.8

Combine the numerators over the common denominator.

$y = \frac{32}{13} + \frac{66 z - 5 z \cdot 13}{13}$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.9

Combine the numerators over the common denominator.

$y = \frac{32 + 66 z - 5 z \cdot 13}{13}$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.10

Multiply $13$ by $- 5$ .

$y = \frac{32 + 66 z - 65 z}{13}$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 4.4.1.11

Subtract $65 z$ from $66 z$ .

$y = \frac{32 + z}{13}$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = \frac{32 + z}{13}$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = \frac{32 + z}{13}$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = \frac{32 + z}{13}$

$- \frac{19 z - 250}{13} = 9$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5

Solve for $z$ in $- \frac{19 z - 250}{13} = 9$ .

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Step 5.1

Multiply both sides of the equation by $- 13$ .

$- 13 (- \frac{19 z - 250}{13}) = - 13 \cdot 9$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $- 13 (- \frac{19 z - 250}{13})$ .

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Step 5.2.1.1.1

Cancel the common factor of $13$ .

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Step 5.2.1.1.1.1

Move the leading negative in $- \frac{19 z - 250}{13}$ into the numerator.

$- 13 \frac{- (19 z - 250)}{13} = - 13 \cdot 9$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.2.1.1.1.2

Factor $13$ out of $- 13$ .

$13 (- 1) (\frac{- (19 z - 250)}{13}) = - 13 \cdot 9$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.2.1.1.1.3

Cancel the common factor.

$13 \cdot (- 1 \frac{- (19 z - 250)}{13}) = - 13 \cdot 9$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.2.1.1.1.4

Rewrite the expression.

$19 z - 250 = - 13 \cdot 9$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$19 z - 250 = - 13 \cdot 9$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.2.1.1.2

Multiply.

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Step 5.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (19 z - 250) = - 13 \cdot 9$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.2.1.1.2.2

Multiply $19 z - 250$ by $1$ .

$19 z - 250 = - 13 \cdot 9$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$19 z - 250 = - 13 \cdot 9$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$19 z - 250 = - 13 \cdot 9$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$19 z - 250 = - 13 \cdot 9$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $- 13$ by $9$ .

$19 z - 250 = - 117$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$19 z - 250 = - 117$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$19 z - 250 = - 117$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.3

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

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Step 5.3.1

Add $250$ to both sides of the equation.

$19 z = - 117 + 250$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.3.2

Add $- 117$ and $250$ .

$19 z = 133$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$19 z = 133$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.4

Divide each term in $19 z = 133$ by $19$ and simplify.

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Step 5.4.1

Divide each term in $19 z = 133$ by $19$ .

$\frac{19 z}{19} = \frac{133}{19}$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $19$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{19 z}{19} = \frac{133}{19}$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.4.2.1.2

Divide $z$ by $1$ .

$z = \frac{133}{19}$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$z = \frac{133}{19}$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$z = \frac{133}{19}$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $133$ by $19$ .

$z = 7$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$z = 7$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$z = 7$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

$z = 7$

$y = \frac{32 + z}{13}$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 6

Replace all occurrences of $z$ with $7$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $y = \frac{32 + z}{13}$ with $7$ .

$y = \frac{32 + 7}{13}$

$z = 7$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 6.2

Simplify $y = \frac{32 + 7}{13}$ .

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Remove parentheses.

$y = \frac{32 + 7}{13}$

$z = 7$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = \frac{32 + 7}{13}$

$z = 7$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $\frac{32 + 7}{13}$ .

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Step 6.2.2.1.1

Add $32$ and $7$ .

$y = \frac{39}{13}$

$z = 7$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 6.2.2.1.2

Divide $39$ by $13$ .

$y = 3$

$z = 7$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 3$

$z = 7$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 3$

$z = 7$

$x = \frac{1}{13} + \frac{11 z}{13}$

$y = 3$

$z = 7$

$x = \frac{1}{13} + \frac{11 z}{13}$

Step 6.3

Replace all occurrences of $z$ in $x = \frac{1}{13} + \frac{11 z}{13}$ with $7$ .

$x = \frac{1}{13} + \frac{11 (7)}{13}$

$y = 3$

$z = 7$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $\frac{1}{13} + \frac{11 (7)}{13}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$x = \frac{1 + 11 (7)}{13}$

$y = 3$

$z = 7$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Multiply $11$ by $7$ .

$x = \frac{1 + 77}{13}$

$y = 3$

$z = 7$

Step 6.4.1.2.2

Add $1$ and $77$ .

$x = \frac{78}{13}$

$y = 3$

$z = 7$

Step 6.4.1.2.3

Divide $78$ by $13$ .

$x = 6$

$y = 3$

$z = 7$

$x = 6$

$y = 3$

$z = 7$

$x = 6$

$y = 3$

$z = 7$

$x = 6$

$y = 3$

$z = 7$

$x = 6$

$y = 3$

$z = 7$

Step 7

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

$(6, 3, 7)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(6, 3, 7)$

Equation Form:

$x = 6, y = 3, z = 7$