Finite Math Examples

$3 x + 2 y - z = - 5$ , $x + 4 y + 3 z = 5$ , $- 2 x + y + 2 z = 3$

Step 1

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

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Step 1.1

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

$x + 3 z = 5 - 4 y$

$3 x + 2 y - z = - 5$

$- 2 x + y + 2 z = 3$

Step 1.2

Subtract $3 z$ from both sides of the equation.

$x = 5 - 4 y - 3 z$

$3 x + 2 y - z = - 5$

$- 2 x + y + 2 z = 3$

$x = 5 - 4 y - 3 z$

$3 x + 2 y - z = - 5$

$- 2 x + y + 2 z = 3$

Step 2

Replace all occurrences of $x$ with $5 - 4 y - 3 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x + 2 y - z = - 5$ with $5 - 4 y - 3 z$ .

$3 (5 - 4 y - 3 z) + 2 y - z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (5 - 4 y - 3 z) + 2 y - z$ .

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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.

$3 \cdot 5 + 3 (- 4 y) + 3 (- 3 z) + 2 y - z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $5$ .

$15 + 3 (- 4 y) + 3 (- 3 z) + 2 y - z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

Step 2.2.1.1.2.2

Multiply $- 4$ by $3$ .

$15 - 12 y + 3 (- 3 z) + 2 y - z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

Step 2.2.1.1.2.3

Multiply $- 3$ by $3$ .

$15 - 12 y - 9 z + 2 y - z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

$15 - 12 y - 9 z + 2 y - z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

$15 - 12 y - 9 z + 2 y - z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 12 y$ and $2 y$ .

$15 - 10 y - 9 z - z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

Step 2.2.1.2.2

Subtract $z$ from $- 9 z$ .

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$- 2 x + y + 2 z = 3$

Step 2.3

Replace all occurrences of $x$ in $- 2 x + y + 2 z = 3$ with $5 - 4 y - 3 z$ .

$- 2 (5 - 4 y - 3 z) + y + 2 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 2 (5 - 4 y - 3 z) + y + 2 z$ .

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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.

$- 2 \cdot 5 - 2 (- 4 y) - 2 (- 3 z) + y + 2 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 2$ by $5$ .

$- 10 - 2 (- 4 y) - 2 (- 3 z) + y + 2 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 2.4.1.1.2.2

Multiply $- 4$ by $- 2$ .

$- 10 + 8 y - 2 (- 3 z) + y + 2 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 2.4.1.1.2.3

Multiply $- 3$ by $- 2$ .

$- 10 + 8 y + 6 z + y + 2 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$- 10 + 8 y + 6 z + y + 2 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$- 10 + 8 y + 6 z + y + 2 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $8 y$ and $y$ .

$- 10 + 9 y + 6 z + 2 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 2.4.1.2.2

Add $6 z$ and $2 z$ .

$- 10 + 9 y + 8 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$- 10 + 9 y + 8 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$- 10 + 9 y + 8 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$- 10 + 9 y + 8 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$- 10 + 9 y + 8 z = 3$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 3

Solve for $y$ in $- 10 + 9 y + 8 z = 3$ .

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Step 3.1

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

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Step 3.1.1

Add $10$ to both sides of the equation.

$9 y + 8 z = 3 + 10$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 3.1.2

Subtract $8 z$ from both sides of the equation.

$9 y = 3 + 10 - 8 z$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 3.1.3

Add $3$ and $10$ .

$9 y = 13 - 8 z$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$9 y = 13 - 8 z$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 3.2

Divide each term in $9 y = 13 - 8 z$ by $9$ and simplify.

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Step 3.2.1

Divide each term in $9 y = 13 - 8 z$ by $9$ .

$\frac{9 y}{9} = \frac{13}{9} + \frac{- 8 z}{9}$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $9$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{9 y}{9} = \frac{13}{9} + \frac{- 8 z}{9}$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{13}{9} + \frac{- 8 z}{9}$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$y = \frac{13}{9} + \frac{- 8 z}{9}$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$y = \frac{13}{9} + \frac{- 8 z}{9}$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$y = \frac{13}{9} - \frac{8 z}{9}$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$y = \frac{13}{9} - \frac{8 z}{9}$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$y = \frac{13}{9} - \frac{8 z}{9}$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

$y = \frac{13}{9} - \frac{8 z}{9}$

$15 - 10 y - 10 z = - 5$

$x = 5 - 4 y - 3 z$

Step 4

Replace all occurrences of $y$ with $\frac{13}{9} - \frac{8 z}{9}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $15 - 10 y - 10 z = - 5$ with $\frac{13}{9} - \frac{8 z}{9}$ .

$15 - 10 (\frac{13}{9} - \frac{8 z}{9}) - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $15 - 10 (\frac{13}{9} - \frac{8 z}{9}) - 10 z$ .

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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.

$15 - 10 (\frac{13}{9}) - 10 (- \frac{8 z}{9}) - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.1.2

Multiply $- 10 (\frac{13}{9})$ .

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Step 4.2.1.1.2.1

Combine $- 10$ and $\frac{13}{9}$ .

$15 + \frac{- 10 \cdot 13}{9} - 10 (- \frac{8 z}{9}) - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.1.2.2

Multiply $- 10$ by $13$ .

$15 + \frac{- 130}{9} - 10 (- \frac{8 z}{9}) - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

$15 + \frac{- 130}{9} - 10 (- \frac{8 z}{9}) - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.1.3

Multiply $- 10 (- \frac{8 z}{9})$ .

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Step 4.2.1.1.3.1

Multiply $- 1$ by $- 10$ .

$15 + \frac{- 130}{9} + 10 (\frac{8 z}{9}) - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.1.3.2

Combine $10$ and $\frac{8 z}{9}$ .

$15 + \frac{- 130}{9} + \frac{10 (8 z)}{9} - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.1.3.3

Multiply $8$ by $10$ .

$15 + \frac{- 130}{9} + \frac{80 z}{9} - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

$15 + \frac{- 130}{9} + \frac{80 z}{9} - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.1.4

Move the negative in front of the fraction.

$15 - \frac{130}{9} + \frac{80 z}{9} - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

$15 - \frac{130}{9} + \frac{80 z}{9} - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.2

To write $15$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$15 \cdot \frac{9}{9} - \frac{130}{9} + \frac{80 z}{9} - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.3

Combine $15$ and $\frac{9}{9}$ .

$\frac{15 \cdot 9}{9} - \frac{130}{9} + \frac{80 z}{9} - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{15 \cdot 9 - 130}{9} + \frac{80 z}{9} - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $15$ by $9$ .

$\frac{135 - 130}{9} + \frac{80 z}{9} - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.5.2

Subtract $130$ from $135$ .

$\frac{5}{9} + \frac{80 z}{9} - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

$\frac{5}{9} + \frac{80 z}{9} - 10 z = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.6

To write $- 10 z$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{5}{9} + \frac{80 z}{9} - 10 z \cdot \frac{9}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.7

Combine $- 10 z$ and $\frac{9}{9}$ .

$\frac{5}{9} + \frac{80 z}{9} + \frac{- 10 z \cdot 9}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.8

Combine the numerators over the common denominator.

$\frac{5}{9} + \frac{80 z - 10 z \cdot 9}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.9

Combine the numerators over the common denominator.

$\frac{5 + 80 z - 10 z \cdot 9}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.10

Multiply $9$ by $- 10$ .

$\frac{5 + 80 z - 90 z}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.11

Subtract $90 z$ from $80 z$ .

$\frac{5 - 10 z}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.12

Factor $5$ out of $5 - 10 z$ .

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Step 4.2.1.12.1

Factor $5$ out of $5$ .

$\frac{5 (1) - 10 z}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.12.2

Factor $5$ out of $- 10 z$ .

$\frac{5 (1) + 5 (- 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.2.1.12.3

Factor $5$ out of $5 (1) + 5 (- 2 z)$ .

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - 4 y - 3 z$

Step 4.3

Replace all occurrences of $y$ in $x = 5 - 4 y - 3 z$ with $\frac{13}{9} - \frac{8 z}{9}$ .

$x = 5 - 4 (\frac{13}{9} - \frac{8 z}{9}) - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $5 - 4 (\frac{13}{9} - \frac{8 z}{9}) - 3 z$ .

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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.

$x = 5 - 4 (\frac{13}{9}) - 4 (- \frac{8 z}{9}) - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.1.2

Multiply $- 4 (\frac{13}{9})$ .

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Step 4.4.1.1.2.1

Combine $- 4$ and $\frac{13}{9}$ .

$x = 5 + \frac{- 4 \cdot 13}{9} - 4 (- \frac{8 z}{9}) - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.1.2.2

Multiply $- 4$ by $13$ .

$x = 5 + \frac{- 52}{9} - 4 (- \frac{8 z}{9}) - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 + \frac{- 52}{9} - 4 (- \frac{8 z}{9}) - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.1.3

Multiply $- 4 (- \frac{8 z}{9})$ .

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Step 4.4.1.1.3.1

Multiply $- 1$ by $- 4$ .

$x = 5 + \frac{- 52}{9} + 4 (\frac{8 z}{9}) - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.1.3.2

Combine $4$ and $\frac{8 z}{9}$ .

$x = 5 + \frac{- 52}{9} + \frac{4 (8 z)}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.1.3.3

Multiply $8$ by $4$ .

$x = 5 + \frac{- 52}{9} + \frac{32 z}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 + \frac{- 52}{9} + \frac{32 z}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.1.4

Move the negative in front of the fraction.

$x = 5 - \frac{52}{9} + \frac{32 z}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 5 - \frac{52}{9} + \frac{32 z}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x = 5 \cdot \frac{9}{9} - \frac{52}{9} + \frac{32 z}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.3

Combine $5$ and $\frac{9}{9}$ .

$x = \frac{5 \cdot 9}{9} - \frac{52}{9} + \frac{32 z}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$x = \frac{5 \cdot 9 - 52}{9} + \frac{32 z}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $5$ by $9$ .

$x = \frac{45 - 52}{9} + \frac{32 z}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.5.2

Subtract $52$ from $45$ .

$x = \frac{- 7}{9} + \frac{32 z}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = \frac{- 7}{9} + \frac{32 z}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.6

Move the negative in front of the fraction.

$x = - \frac{7}{9} + \frac{32 z}{9} - 3 z$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.7

To write $- 3 z$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x = - \frac{7}{9} + \frac{32 z}{9} - 3 z \cdot \frac{9}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.8

Combine $- 3 z$ and $\frac{9}{9}$ .

$x = - \frac{7}{9} + \frac{32 z}{9} + \frac{- 3 z \cdot 9}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.9

Combine the numerators over the common denominator.

$x = - \frac{7}{9} + \frac{32 z - 3 z \cdot 9}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.10

Combine the numerators over the common denominator.

$x = \frac{- 7 + 32 z - 3 z \cdot 9}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.11

Multiply $9$ by $- 3$ .

$x = \frac{- 7 + 32 z - 27 z}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.12

Subtract $27 z$ from $32 z$ .

$x = \frac{- 7 + 5 z}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.13

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + 5 z}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.14

Factor $- 1$ out of $5 z$ .

$x = \frac{- 1 \cdot 7 - (- 5 z)}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.15

Factor $- 1$ out of $- 1 (7) - (- 5 z)$ .

$x = \frac{- 1 (7 - 5 z)}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 4.4.1.16

Move the negative in front of the fraction.

$x = - \frac{7 - 5 z}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = - \frac{7 - 5 z}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = - \frac{7 - 5 z}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = - \frac{7 - 5 z}{9}$

$\frac{5 (1 - 2 z)}{9} = - 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5

Solve for $z$ in $\frac{5 (1 - 2 z)}{9} = - 5$ .

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Step 5.1

Multiply both sides by $9$ .

$\frac{5 (1 - 2 z)}{9} \cdot 9 = - 5 \cdot 9$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $\frac{5 (1 - 2 z)}{9} \cdot 9$ .

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Step 5.2.1.1.1

Cancel the common factor of $9$ .

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Step 5.2.1.1.1.1

Cancel the common factor.

$\frac{5 (1 - 2 z)}{9} \cdot 9 = - 5 \cdot 9$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.2.1.1.1.2

Rewrite the expression.

$5 (1 - 2 z) = - 5 \cdot 9$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$5 (1 - 2 z) = - 5 \cdot 9$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.2.1.1.2

Apply the distributive property.

$5 \cdot 1 + 5 (- 2 z) = - 5 \cdot 9$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.2.1.1.3

Simplify the expression.

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Step 5.2.1.1.3.1

Multiply $5$ by $1$ .

$5 + 5 (- 2 z) = - 5 \cdot 9$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.2.1.1.3.2

Multiply $- 2$ by $5$ .

$5 - 10 z = - 5 \cdot 9$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.2.1.1.3.3

Reorder $5$ and $- 10 z$ .

$- 10 z + 5 = - 5 \cdot 9$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$- 10 z + 5 = - 5 \cdot 9$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$- 10 z + 5 = - 5 \cdot 9$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$- 10 z + 5 = - 5 \cdot 9$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $- 5$ by $9$ .

$- 10 z + 5 = - 45$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$- 10 z + 5 = - 45$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$- 10 z + 5 = - 45$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.3

Solve for $z$ .

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Step 5.3.1

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

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Step 5.3.1.1

Subtract $5$ from both sides of the equation.

$- 10 z = - 45 - 5$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.3.1.2

Subtract $5$ from $- 45$ .

$- 10 z = - 50$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$- 10 z = - 50$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.3.2

Divide each term in $- 10 z = - 50$ by $- 10$ and simplify.

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Step 5.3.2.1

Divide each term in $- 10 z = - 50$ by $- 10$ .

$\frac{- 10 z}{- 10} = \frac{- 50}{- 10}$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $- 10$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{- 10 z}{- 10} = \frac{- 50}{- 10}$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.3.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 50}{- 10}$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$z = \frac{- 50}{- 10}$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$z = \frac{- 50}{- 10}$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $- 50$ by $- 10$ .

$z = 5$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$z = 5$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$z = 5$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$z = 5$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

$z = 5$

$x = - \frac{7 - 5 z}{9}$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 6

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

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Step 6.1

Replace all occurrences of $z$ in $x = - \frac{7 - 5 z}{9}$ with $5$ .

$x = - \frac{7 - 5 \cdot 5}{9}$

$z = 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{7 - 5 \cdot 5}{9}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Multiply $- 5$ by $5$ .

$x = - \frac{7 - 25}{9}$

$z = 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 6.2.1.1.2

Subtract $25$ from $7$ .

$x = - \frac{- 18}{9}$

$z = 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = - \frac{- 18}{9}$

$z = 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 6.2.1.2

Simplify the expression.

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Step 6.2.1.2.1

Divide $- 18$ by $9$ .

$x = 2$

$z = 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 6.2.1.2.2

Multiply $- 1$ by $- 2$ .

$x = 2$

$z = 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 2$

$z = 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 2$

$z = 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

$x = 2$

$z = 5$

$y = \frac{13}{9} - \frac{8 z}{9}$

Step 6.3

Replace all occurrences of $z$ in $y = \frac{13}{9} - \frac{8 z}{9}$ with $5$ .

$y = \frac{13}{9} - \frac{8 (5)}{9}$

$x = 2$

$z = 5$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $\frac{13}{9} - \frac{8 (5)}{9}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$y = \frac{13 - 8 \cdot 5}{9}$

$x = 2$

$z = 5$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Multiply $- 8$ by $5$ .

$y = \frac{13 - 40}{9}$

$x = 2$

$z = 5$

Step 6.4.1.2.2

Subtract $40$ from $13$ .

$y = \frac{- 27}{9}$

$x = 2$

$z = 5$

Step 6.4.1.2.3

Divide $- 27$ by $9$ .

$y = - 3$

$x = 2$

$z = 5$

$y = - 3$

$x = 2$

$z = 5$

$y = - 3$

$x = 2$

$z = 5$

$y = - 3$

$x = 2$

$z = 5$

$y = - 3$

$x = 2$

$z = 5$

Step 7

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

$(2, - 3, 5)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(2, - 3, 5)$

Equation Form:

$x = 2, y = - 3, z = 5$