Finite Math Examples

$- 5 x - y = 21$ , $3 x - 4 y + 5 z = - 8$ , $- 3 x - z = 12$

Step 1

Solve for $x$ in $- 5 x - y = 21$ .

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Step 1.1

Add $y$ to both sides of the equation.

$- 5 x = 21 + y$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

Step 1.2

Divide each term in $- 5 x = 21 + y$ by $- 5$ and simplify.

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Step 1.2.1

Divide each term in $- 5 x = 21 + y$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{21}{- 5} + \frac{y}{- 5}$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{21}{- 5} + \frac{y}{- 5}$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{21}{- 5} + \frac{y}{- 5}$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

$x = \frac{21}{- 5} + \frac{y}{- 5}$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

$x = \frac{21}{- 5} + \frac{y}{- 5}$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{21}{5} + \frac{y}{- 5}$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{21}{5} - \frac{y}{5}$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

$x = - \frac{21}{5} - \frac{y}{5}$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

$x = - \frac{21}{5} - \frac{y}{5}$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

$x = - \frac{21}{5} - \frac{y}{5}$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

$x = - \frac{21}{5} - \frac{y}{5}$

$3 x - 4 y + 5 z = - 8$

$- 3 x - z = 12$

Step 2

Replace all occurrences of $x$ with $- \frac{21}{5} - \frac{y}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x - 4 y + 5 z = - 8$ with $- \frac{21}{5} - \frac{y}{5}$ .

$3 (- \frac{21}{5} - \frac{y}{5}) - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (- \frac{21}{5} - \frac{y}{5}) - 4 y + 5 z$ .

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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.

$3 (- \frac{21}{5}) + 3 (- \frac{y}{5}) - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.1.2

Multiply $3 (- \frac{21}{5})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{21}{5}) + 3 (- \frac{y}{5}) - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.1.2.2

Combine $- 3$ and $\frac{21}{5}$ .

$\frac{- 3 \cdot 21}{5} + 3 (- \frac{y}{5}) - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.1.2.3

Multiply $- 3$ by $21$ .

$\frac{- 63}{5} + 3 (- \frac{y}{5}) - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

$\frac{- 63}{5} + 3 (- \frac{y}{5}) - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.1.3

Multiply $3 (- \frac{y}{5})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $3$ .

$\frac{- 63}{5} - 3 \frac{y}{5} - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.1.3.2

Combine $- 3$ and $\frac{y}{5}$ .

$\frac{- 63}{5} + \frac{- 3 y}{5} - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

$\frac{- 63}{5} + \frac{- 3 y}{5} - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Move the negative in front of the fraction.

$- \frac{63}{5} + \frac{- 3 y}{5} - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{63}{5} - \frac{3 y}{5} - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

$- \frac{63}{5} - \frac{3 y}{5} - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

$- \frac{63}{5} - \frac{3 y}{5} - 4 y + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.2

To write $- 4 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{63}{5} - \frac{3 y}{5} - 4 y \cdot \frac{5}{5} + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.3

Combine $- 4 y$ and $\frac{5}{5}$ .

$- \frac{63}{5} - \frac{3 y}{5} + \frac{- 4 y \cdot 5}{5} + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{63}{5} + \frac{- 3 y - 4 y \cdot 5}{5} + 5 z = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.5

Combine the numerators over the common denominator.

$5 z + \frac{- 63 - 3 y - 4 y \cdot 5}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.6

Multiply $5$ by $- 4$ .

$5 z + \frac{- 63 - 3 y - 20 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.7

Subtract $20 y$ from $- 3 y$ .

$5 z + \frac{- 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.8

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

$5 z \cdot \frac{5}{5} + \frac{- 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.9

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

$\frac{5 z \cdot 5}{5} + \frac{- 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.10

Combine the numerators over the common denominator.

$\frac{5 z \cdot 5 - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.2.1.11

Multiply $5$ by $5$ .

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$- 3 x - z = 12$

Step 2.3

Replace all occurrences of $x$ in $- 3 x - z = 12$ with $- \frac{21}{5} - \frac{y}{5}$ .

$- 3 (- \frac{21}{5} - \frac{y}{5}) - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Apply the distributive property.

$- 3 (- \frac{21}{5}) - 3 (- \frac{y}{5}) - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

Step 2.4.1.2

Multiply $- 3 (- \frac{21}{5})$ .

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Step 2.4.1.2.1

Multiply $- 1$ by $- 3$ .

$3 (\frac{21}{5}) - 3 (- \frac{y}{5}) - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

Step 2.4.1.2.2

Combine $3$ and $\frac{21}{5}$ .

$\frac{3 \cdot 21}{5} - 3 (- \frac{y}{5}) - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

Step 2.4.1.2.3

Multiply $3$ by $21$ .

$\frac{63}{5} - 3 (- \frac{y}{5}) - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$\frac{63}{5} - 3 (- \frac{y}{5}) - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

Step 2.4.1.3

Multiply $- 3 (- \frac{y}{5})$ .

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Step 2.4.1.3.1

Multiply $- 1$ by $- 3$ .

$\frac{63}{5} + 3 (\frac{y}{5}) - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

Step 2.4.1.3.2

Combine $3$ and $\frac{y}{5}$ .

$\frac{63}{5} + \frac{3 y}{5} - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$\frac{63}{5} + \frac{3 y}{5} - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$\frac{63}{5} + \frac{3 y}{5} - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$\frac{63}{5} + \frac{3 y}{5} - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

$\frac{63}{5} + \frac{3 y}{5} - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$x = - \frac{21}{5} - \frac{y}{5}$

Step 3

Reorder $- \frac{21}{5}$ and $- \frac{y}{5}$ .

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{63}{5} + \frac{3 y}{5} - z = 12$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4

Solve for $z$ in $\frac{63}{5} + \frac{3 y}{5} - z = 12$ .

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Step 4.1

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

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Step 4.1.1

Subtract $\frac{63}{5}$ from both sides of the equation.

$\frac{3 y}{5} - z = 12 - \frac{63}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.1.2

Subtract $\frac{3 y}{5}$ from both sides of the equation.

$- z = 12 - \frac{63}{5} - \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.1.3

To write $12$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- z = 12 \cdot \frac{5}{5} - \frac{63}{5} - \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.1.4

Combine $12$ and $\frac{5}{5}$ .

$- z = \frac{12 \cdot 5}{5} - \frac{63}{5} - \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.1.5

Combine the numerators over the common denominator.

$- z = \frac{12 \cdot 5 - 63}{5} - \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

Multiply $12$ by $5$ .

$- z = \frac{60 - 63}{5} - \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.1.6.2

Subtract $63$ from $60$ .

$- z = \frac{- 3}{5} - \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$- z = \frac{- 3}{5} - \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.1.7

Move the negative in front of the fraction.

$- z = - \frac{3}{5} - \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$- z = - \frac{3}{5} - \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.2

Divide each term in $- z = - \frac{3}{5} - \frac{3 y}{5}$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- z = - \frac{3}{5} - \frac{3 y}{5}$ by $- 1$ .

$\frac{- z}{- 1} = \frac{- \frac{3}{5}}{- 1} + \frac{- \frac{3 y}{5}}{- 1}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{z}{1} = \frac{- \frac{3}{5}}{- 1} + \frac{- \frac{3 y}{5}}{- 1}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.2.2.2

Divide $z$ by $1$ .

$z = \frac{- \frac{3}{5}}{- 1} + \frac{- \frac{3 y}{5}}{- 1}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$z = \frac{- \frac{3}{5}}{- 1} + \frac{- \frac{3 y}{5}}{- 1}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Dividing two negative values results in a positive value.

$z = \frac{\frac{3}{5}}{1} + \frac{- \frac{3 y}{5}}{- 1}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.2.3.1.2

Divide $\frac{3}{5}$ by $1$ .

$z = \frac{3}{5} + \frac{- \frac{3 y}{5}}{- 1}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.2.3.1.3

Dividing two negative values results in a positive value.

$z = \frac{3}{5} + \frac{\frac{3 y}{5}}{1}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 4.2.3.1.4

Divide $\frac{3 y}{5}$ by $1$ .

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{25 z - 63 - 23 y}{5} = - 8$

Step 5

Replace all occurrences of $z$ with $\frac{3}{5} + \frac{3 y}{5}$ in each equation.

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Step 5.1

Replace all occurrences of $z$ in $\frac{25 z - 63 - 23 y}{5} = - 8$ with $\frac{3}{5} + \frac{3 y}{5}$ .

$\frac{25 (\frac{3}{5} + \frac{3 y}{5}) - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $\frac{25 (\frac{3}{5} + \frac{3 y}{5}) - 63 - 23 y}{5}$ .

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Step 5.2.1.1

Simplify the numerator.

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Step 5.2.1.1.1

Apply the distributive property.

$\frac{25 (\frac{3}{5}) + 25 (\frac{3 y}{5}) - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.2

Cancel the common factor of $5$ .

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Step 5.2.1.1.2.1

Factor $5$ out of $25$ .

$\frac{5 (5) (\frac{3}{5}) + 25 (\frac{3 y}{5}) - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.2.2

Cancel the common factor.

$\frac{5 \cdot (5 (\frac{3}{5})) + 25 (\frac{3 y}{5}) - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.2.3

Rewrite the expression.

$\frac{5 \cdot 3 + 25 (\frac{3 y}{5}) - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{5 \cdot 3 + 25 (\frac{3 y}{5}) - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.3

Multiply $5$ by $3$ .

$\frac{15 + 25 (\frac{3 y}{5}) - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.4

Cancel the common factor of $5$ .

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Step 5.2.1.1.4.1

Factor $5$ out of $25$ .

$\frac{15 + 5 (5) (\frac{3 y}{5}) - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.4.2

Cancel the common factor.

$\frac{15 + 5 \cdot (5 (\frac{3 y}{5})) - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.4.3

Rewrite the expression.

$\frac{15 + 5 (3 y) - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{15 + 5 (3 y) - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.5

Multiply $3$ by $5$ .

$\frac{15 + 15 y - 63 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.6

Subtract $63$ from $15$ .

$\frac{15 y - 48 - 23 y}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.7

Subtract $23 y$ from $15 y$ .

$\frac{- 8 y - 48}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.8

Factor $8$ out of $- 8 y - 48$ .

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Step 5.2.1.1.8.1

Factor $8$ out of $- 8 y$ .

$\frac{8 (- y) - 48}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.8.2

Factor $8$ out of $- 48$ .

$\frac{8 (- y) + 8 (- 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.1.8.3

Factor $8$ out of $8 (- y) + 8 (- 6)$ .

$\frac{8 (- y - 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{8 (- y - 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{8 (- y - 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.2

Simplify with factoring out.

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Step 5.2.1.2.1

Factor $- 1$ out of $- y$ .

$\frac{8 (- (y) - 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.2.2

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{8 (- (y) - 1 \cdot 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.2.3

Factor $- 1$ out of $- (y) - 1 (6)$ .

$\frac{8 (- (y + 6))}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.2.4

Simplify the expression.

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Step 5.2.1.2.4.1

Rewrite $- (y + 6)$ as $- 1 (y + 6)$ .

$\frac{8 (- 1 (y + 6))}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 5.2.1.2.4.2

Move the negative in front of the fraction.

$- \frac{8 (y + 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$- \frac{8 (y + 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$- \frac{8 (y + 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$- \frac{8 (y + 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$- \frac{8 (y + 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$- \frac{8 (y + 6)}{5} = - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6

Solve for $y$ in $- \frac{8 (y + 6)}{5} = - 8$ .

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Step 6.1

Multiply both sides of the equation by $- \frac{5}{8}$ .

$- \frac{5}{8} \cdot (- \frac{8 (y + 6)}{5}) = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2

Simplify both sides of the equation.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify $- \frac{5}{8} (- \frac{8 (y + 6)}{5})$ .

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Step 6.2.1.1.1

Cancel the common factor of $5$ .

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Step 6.2.1.1.1.1

Move the leading negative in $- \frac{5}{8}$ into the numerator.

$\frac{- 5}{8} \cdot (- \frac{8 (y + 6)}{5}) = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.1.1.1.2

Move the leading negative in $- \frac{8 (y + 6)}{5}$ into the numerator.

$\frac{- 5}{8} \cdot \frac{- 8 (y + 6)}{5} = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.1.1.1.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{8} \cdot \frac{- 8 (y + 6)}{5} = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.1.1.1.4

Cancel the common factor.

$\frac{5 \cdot - 1}{8} \cdot \frac{- 8 (y + 6)}{5} = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.1.1.1.5

Rewrite the expression.

$\frac{- 1}{8} \cdot (- 8 (y + 6)) = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$\frac{- 1}{8} \cdot (- 8 (y + 6)) = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.1.1.2

Cancel the common factor of $8$ .

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Step 6.2.1.1.2.1

Factor $8$ out of $- 8 (y + 6)$ .

$\frac{- 1}{8} \cdot (8 (- (y + 6))) = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.1.1.2.2

Cancel the common factor.

$\frac{- 1}{8} \cdot (8 (- (y + 6))) = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.1.1.2.3

Rewrite the expression.

$y + 6 = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$y + 6 = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.1.1.3

Multiply.

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Step 6.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 (y + 6) = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.1.1.3.2

Multiply $y + 6$ by $1$ .

$y + 6 = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$y + 6 = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$y + 6 = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$y + 6 = - \frac{5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $- \frac{5}{8} \cdot - 8$ .

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Step 6.2.2.1.1

Cancel the common factor of $8$ .

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Step 6.2.2.1.1.1

Move the leading negative in $- \frac{5}{8}$ into the numerator.

$y + 6 = \frac{- 5}{8} \cdot - 8$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.2.1.1.2

Factor $8$ out of $- 8$ .

$y + 6 = \frac{- 5}{8} \cdot (8 (- 1))$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.2.1.1.3

Cancel the common factor.

$y + 6 = \frac{- 5}{8} \cdot (8 \cdot - 1)$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.2.1.1.4

Rewrite the expression.

$y + 6 = - 5 \cdot - 1$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$y + 6 = - 5 \cdot - 1$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.2.2.1.2

Multiply $- 5$ by $- 1$ .

$y + 6 = 5$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$y + 6 = 5$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$y + 6 = 5$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$y + 6 = 5$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.3

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

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Step 6.3.1

Subtract $6$ from both sides of the equation.

$y = 5 - 6$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 6.3.2

Subtract $6$ from $5$ .

$y = - 1$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$y = - 1$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

$y = - 1$

$z = \frac{3}{5} + \frac{3 y}{5}$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 7

Replace all occurrences of $y$ with $- 1$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $z = \frac{3}{5} + \frac{3 y}{5}$ with $- 1$ .

$z = \frac{3}{5} + \frac{3 (- 1)}{5}$

$y = - 1$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{3}{5} + \frac{3 (- 1)}{5}$ .

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Step 7.2.1.1

Combine the numerators over the common denominator.

$z = \frac{3 + 3 (- 1)}{5}$

$y = - 1$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 7.2.1.2

Simplify the expression.

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Step 7.2.1.2.1

Multiply $3$ by $- 1$ .

$z = \frac{3 - 3}{5}$

$y = - 1$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 7.2.1.2.2

Subtract $3$ from $3$ .

$z = \frac{0}{5}$

$y = - 1$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 7.2.1.2.3

Divide $0$ by $5$ .

$z = 0$

$y = - 1$

$x = - \frac{y}{5} - \frac{21}{5}$

$z = 0$

$y = - 1$

$x = - \frac{y}{5} - \frac{21}{5}$

$z = 0$

$y = - 1$

$x = - \frac{y}{5} - \frac{21}{5}$

$z = 0$

$y = - 1$

$x = - \frac{y}{5} - \frac{21}{5}$

Step 7.3

Replace all occurrences of $y$ in $x = - \frac{y}{5} - \frac{21}{5}$ with $- 1$ .

$x = - \frac{- 1}{5} - \frac{21}{5}$

$z = 0$

$y = - 1$

Step 7.4

Simplify the right side.

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Step 7.4.1

Simplify $- \frac{- 1}{5} - \frac{21}{5}$ .

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Step 7.4.1.1

Combine the numerators over the common denominator.

$x = \frac{1 - 21}{5}$

$z = 0$

$y = - 1$

Step 7.4.1.2

Simplify the expression.

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Step 7.4.1.2.1

Subtract $21$ from $1$ .

$x = \frac{- 20}{5}$

$z = 0$

$y = - 1$

Step 7.4.1.2.2

Divide $- 20$ by $5$ .

$x = - 4$

$z = 0$

$y = - 1$

$x = - 4$

$z = 0$

$y = - 1$

$x = - 4$

$z = 0$

$y = - 1$

$x = - 4$

$z = 0$

$y = - 1$

$x = - 4$

$z = 0$

$y = - 1$

Step 8

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

$(- 4, - 1, 0)$

Step 9

The result can be shown in multiple forms.

Point Form:

$(- 4, - 1, 0)$

Equation Form:

$x = - 4, y = - 1, z = 0$