Finite Math Examples

$2 x + y - 3 z = - 2$ , $- 2 x + 2 y + z = - 9$ , $3 x - 4 y - 3 z = 15$

Step 1

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

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Step 1.1

Subtract $2 x$ from both sides of the equation.

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

$- 2 x + 2 y + z = - 9$

$3 x - 4 y - 3 z = 15$

Step 1.2

Add $3 z$ to both sides of the equation.

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

$- 2 x + 2 y + z = - 9$

$3 x - 4 y - 3 z = 15$

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

$- 2 x + 2 y + z = - 9$

$3 x - 4 y - 3 z = 15$

Step 2

Replace all occurrences of $y$ with $- 2 - 2 x + 3 z$ in each equation.

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Step 2.1

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

$- 2 x + 2 (- 2 - 2 x + 3 z) + z = - 9$

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

$3 x - 4 y - 3 z = 15$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 2 x + 2 (- 2 - 2 x + 3 z) + z$ .

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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.

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

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

$3 x - 4 y - 3 z = 15$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $2$ by $- 2$ .

$- 2 x - 4 + 2 (- 2 x) + 2 (3 z) + z = - 9$

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

$3 x - 4 y - 3 z = 15$

Step 2.2.1.1.2.2

Multiply $- 2$ by $2$ .

$- 2 x - 4 - 4 x + 2 (3 z) + z = - 9$

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

$3 x - 4 y - 3 z = 15$

Step 2.2.1.1.2.3

Multiply $3$ by $2$ .

$- 2 x - 4 - 4 x + 6 z + z = - 9$

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

$3 x - 4 y - 3 z = 15$

$- 2 x - 4 - 4 x + 6 z + z = - 9$

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

$3 x - 4 y - 3 z = 15$

$- 2 x - 4 - 4 x + 6 z + z = - 9$

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

$3 x - 4 y - 3 z = 15$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $4 x$ from $- 2 x$ .

$- 6 x - 4 + 6 z + z = - 9$

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

$3 x - 4 y - 3 z = 15$

Step 2.2.1.2.2

Add $6 z$ and $z$ .

$- 6 x - 4 + 7 z = - 9$

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

$3 x - 4 y - 3 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

$3 x - 4 y - 3 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

$3 x - 4 y - 3 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

$3 x - 4 y - 3 z = 15$

Step 2.3

Replace all occurrences of $y$ in $3 x - 4 y - 3 z = 15$ with $- 2 - 2 x + 3 z$ .

$3 x - 4 (- 2 - 2 x + 3 z) - 3 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $3 x - 4 (- 2 - 2 x + 3 z) - 3 z$ .

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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.

$3 x - 4 \cdot - 2 - 4 (- 2 x) - 4 (3 z) - 3 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 4$ by $- 2$ .

$3 x + 8 - 4 (- 2 x) - 4 (3 z) - 3 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

Step 2.4.1.1.2.2

Multiply $- 2$ by $- 4$ .

$3 x + 8 + 8 x - 4 (3 z) - 3 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

Step 2.4.1.1.2.3

Multiply $3$ by $- 4$ .

$3 x + 8 + 8 x - 12 z - 3 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

$3 x + 8 + 8 x - 12 z - 3 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

$3 x + 8 + 8 x - 12 z - 3 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $3 x$ and $8 x$ .

$11 x + 8 - 12 z - 3 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

Step 2.4.1.2.2

Subtract $3 z$ from $- 12 z$ .

$11 x + 8 - 15 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

$11 x + 8 - 15 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

$11 x + 8 - 15 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

$11 x + 8 - 15 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

$11 x + 8 - 15 z = 15$

$- 6 x - 4 + 7 z = - 9$

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

Step 3

Solve for $x$ in $11 x + 8 - 15 z = 15$ .

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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 $8$ from both sides of the equation.

$11 x - 15 z = 15 - 8$

$- 6 x - 4 + 7 z = - 9$

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

Step 3.1.2

Add $15 z$ to both sides of the equation.

$11 x = 15 - 8 + 15 z$

$- 6 x - 4 + 7 z = - 9$

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

Step 3.1.3

Subtract $8$ from $15$ .

$11 x = 7 + 15 z$

$- 6 x - 4 + 7 z = - 9$

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

$11 x = 7 + 15 z$

$- 6 x - 4 + 7 z = - 9$

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

Step 3.2

Divide each term in $11 x = 7 + 15 z$ by $11$ and simplify.

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Step 3.2.1

Divide each term in $11 x = 7 + 15 z$ by $11$ .

$\frac{11 x}{11} = \frac{7}{11} + \frac{15 z}{11}$

$- 6 x - 4 + 7 z = - 9$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $11$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{7}{11} + \frac{15 z}{11}$

$- 6 x - 4 + 7 z = - 9$

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

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{11} + \frac{15 z}{11}$

$- 6 x - 4 + 7 z = - 9$

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

$x = \frac{7}{11} + \frac{15 z}{11}$

$- 6 x - 4 + 7 z = - 9$

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

$x = \frac{7}{11} + \frac{15 z}{11}$

$- 6 x - 4 + 7 z = - 9$

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

$x = \frac{7}{11} + \frac{15 z}{11}$

$- 6 x - 4 + 7 z = - 9$

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

$x = \frac{7}{11} + \frac{15 z}{11}$

$- 6 x - 4 + 7 z = - 9$

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

Step 4

Replace all occurrences of $x$ with $\frac{7}{11} + \frac{15 z}{11}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $- 6 x - 4 + 7 z = - 9$ with $\frac{7}{11} + \frac{15 z}{11}$ .

$- 6 (\frac{7}{11} + \frac{15 z}{11}) - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 6 (\frac{7}{11} + \frac{15 z}{11}) - 4 + 7 z$ .

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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.

$- 6 (\frac{7}{11}) - 6 \frac{15 z}{11} - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.1.2

Multiply $- 6 (\frac{7}{11})$ .

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Step 4.2.1.1.2.1

Combine $- 6$ and $\frac{7}{11}$ .

$\frac{- 6 \cdot 7}{11} - 6 \frac{15 z}{11} - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.1.2.2

Multiply $- 6$ by $7$ .

$\frac{- 42}{11} - 6 \frac{15 z}{11} - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

$\frac{- 42}{11} - 6 \frac{15 z}{11} - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.1.3

Multiply $- 6 \frac{15 z}{11}$ .

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Step 4.2.1.1.3.1

Combine $- 6$ and $\frac{15 z}{11}$ .

$\frac{- 42}{11} + \frac{- 6 (15 z)}{11} - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.1.3.2

Multiply $15$ by $- 6$ .

$\frac{- 42}{11} + \frac{- 90 z}{11} - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

$\frac{- 42}{11} + \frac{- 90 z}{11} - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.1.4

Simplify each term.

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Step 4.2.1.1.4.1

Move the negative in front of the fraction.

$- \frac{42}{11} + \frac{- 90 z}{11} - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{42}{11} - \frac{90 z}{11} - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

$- \frac{42}{11} - \frac{90 z}{11} - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

$- \frac{42}{11} - \frac{90 z}{11} - 4 + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$- \frac{90 z}{11} - \frac{42}{11} - 4 \cdot \frac{11}{11} + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.3

Combine $- 4$ and $\frac{11}{11}$ .

$- \frac{90 z}{11} - \frac{42}{11} + \frac{- 4 \cdot 11}{11} + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.4

Combine the numerators over the common denominator.

$- \frac{90 z}{11} + \frac{- 42 - 4 \cdot 11}{11} + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $- 4$ by $11$ .

$- \frac{90 z}{11} + \frac{- 42 - 44}{11} + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.5.2

Subtract $44$ from $- 42$ .

$- \frac{90 z}{11} + \frac{- 86}{11} + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

$- \frac{90 z}{11} + \frac{- 86}{11} + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.6

Move the negative in front of the fraction.

$- \frac{90 z}{11} - \frac{86}{11} + 7 z = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.7

To write $7 z$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$- \frac{90 z}{11} + 7 z \cdot \frac{11}{11} - \frac{86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.8

Combine $7 z$ and $\frac{11}{11}$ .

$- \frac{90 z}{11} + \frac{7 z \cdot 11}{11} - \frac{86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.9

Combine the numerators over the common denominator.

$\frac{- 90 z + 7 z \cdot 11}{11} - \frac{86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.10

Combine the numerators over the common denominator.

$\frac{- 90 z + 7 z \cdot 11 - 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.11

Multiply $11$ by $7$ .

$\frac{- 90 z + 77 z - 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.12

Add $- 90 z$ and $77 z$ .

$\frac{- 13 z - 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.13

Factor $- 1$ out of $- 13 z$ .

$\frac{- (13 z) - 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.14

Rewrite $- 86$ as $- 1 (86)$ .

$\frac{- (13 z) - 1 \cdot 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.15

Factor $- 1$ out of $- (13 z) - 1 (86)$ .

$\frac{- (13 z + 86)}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.16

Simplify the expression.

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Step 4.2.1.16.1

Rewrite $- (13 z + 86)$ as $- 1 (13 z + 86)$ .

$\frac{- 1 (13 z + 86)}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.2.1.16.2

Move the negative in front of the fraction.

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

Step 4.3

Replace all occurrences of $x$ in $y = - 2 - 2 x + 3 z$ with $\frac{7}{11} + \frac{15 z}{11}$ .

$y = - 2 - 2 (\frac{7}{11} + \frac{15 z}{11}) + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 2 - 2 (\frac{7}{11} + \frac{15 z}{11}) + 3 z$ .

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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.

$y = - 2 - 2 (\frac{7}{11}) - 2 \frac{15 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.1.2

Multiply $- 2 (\frac{7}{11})$ .

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Step 4.4.1.1.2.1

Combine $- 2$ and $\frac{7}{11}$ .

$y = - 2 + \frac{- 2 \cdot 7}{11} - 2 \frac{15 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.1.2.2

Multiply $- 2$ by $7$ .

$y = - 2 + \frac{- 14}{11} - 2 \frac{15 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = - 2 + \frac{- 14}{11} - 2 \frac{15 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.1.3

Multiply $- 2 \frac{15 z}{11}$ .

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Step 4.4.1.1.3.1

Combine $- 2$ and $\frac{15 z}{11}$ .

$y = - 2 + \frac{- 14}{11} + \frac{- 2 (15 z)}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.1.3.2

Multiply $15$ by $- 2$ .

$y = - 2 + \frac{- 14}{11} + \frac{- 30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = - 2 + \frac{- 14}{11} + \frac{- 30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.1.4

Simplify each term.

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Step 4.4.1.1.4.1

Move the negative in front of the fraction.

$y = - 2 - \frac{14}{11} + \frac{- 30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.1.4.2

Move the negative in front of the fraction.

$y = - 2 - \frac{14}{11} - \frac{30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = - 2 - \frac{14}{11} - \frac{30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = - 2 - \frac{14}{11} - \frac{30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$y = - 2 \cdot \frac{11}{11} - \frac{14}{11} - \frac{30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.3

Combine $- 2$ and $\frac{11}{11}$ .

$y = \frac{- 2 \cdot 11}{11} - \frac{14}{11} - \frac{30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$y = \frac{- 2 \cdot 11 - 14}{11} - \frac{30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $- 2$ by $11$ .

$y = \frac{- 22 - 14}{11} - \frac{30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.5.2

Subtract $14$ from $- 22$ .

$y = \frac{- 36}{11} - \frac{30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = \frac{- 36}{11} - \frac{30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.6

Move the negative in front of the fraction.

$y = - \frac{36}{11} - \frac{30 z}{11} + 3 z$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.7

To write $3 z$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$y = - \frac{36}{11} - \frac{30 z}{11} + 3 z \cdot \frac{11}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.8

Combine $3 z$ and $\frac{11}{11}$ .

$y = - \frac{36}{11} - \frac{30 z}{11} + \frac{3 z \cdot 11}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.9

Combine the numerators over the common denominator.

$y = - \frac{36}{11} + \frac{- 30 z + 3 z \cdot 11}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.10

Combine the numerators over the common denominator.

$y = \frac{- 36 - 30 z + 3 z \cdot 11}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.11

Multiply $11$ by $3$ .

$y = \frac{- 36 - 30 z + 33 z}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.12

Add $- 30 z$ and $33 z$ .

$y = \frac{- 36 + 3 z}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.13

Factor $3$ out of $- 36 + 3 z$ .

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Step 4.4.1.13.1

Factor $3$ out of $- 36$ .

$y = \frac{3 \cdot - 12 + 3 z}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.13.2

Factor $3$ out of $3 \cdot - 12 + 3 z$ .

$y = \frac{3 (- 12 + z)}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = \frac{3 (- 12 + z)}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.14

Rewrite $- 12$ as $- 1 (12)$ .

$y = \frac{3 (- 1 \cdot 12 + z)}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.15

Factor $- 1$ out of $z$ .

$y = \frac{3 (- 1 \cdot 12 - 1 (- z))}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.16

Factor $- 1$ out of $- 1 (12) - 1 (- z)$ .

$y = \frac{3 (- 1 (12 - z))}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 4.4.1.17

Move the negative in front of the fraction.

$y = - \frac{3 (12 - z)}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = - \frac{3 (12 - z)}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = - \frac{3 (12 - z)}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = - \frac{3 (12 - z)}{11}$

$- \frac{13 z + 86}{11} = - 9$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5

Solve for $z$ in $- \frac{13 z + 86}{11} = - 9$ .

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Step 5.1

Multiply both sides of the equation by $- 11$ .

$- 11 (- \frac{13 z + 86}{11}) = - 11 \cdot - 9$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

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 $- 11 (- \frac{13 z + 86}{11})$ .

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Step 5.2.1.1.1

Cancel the common factor of $11$ .

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Step 5.2.1.1.1.1

Move the leading negative in $- \frac{13 z + 86}{11}$ into the numerator.

$- 11 \frac{- (13 z + 86)}{11} = - 11 \cdot - 9$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.2.1.1.1.2

Factor $11$ out of $- 11$ .

$11 (- 1) (\frac{- (13 z + 86)}{11}) = - 11 \cdot - 9$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.2.1.1.1.3

Cancel the common factor.

$11 \cdot (- 1 \frac{- (13 z + 86)}{11}) = - 11 \cdot - 9$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.2.1.1.1.4

Rewrite the expression.

$13 z + 86 = - 11 \cdot - 9$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

$13 z + 86 = - 11 \cdot - 9$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.2.1.1.2

Multiply.

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Step 5.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (13 z + 86) = - 11 \cdot - 9$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.2.1.1.2.2

Multiply $13 z + 86$ by $1$ .

$13 z + 86 = - 11 \cdot - 9$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

$13 z + 86 = - 11 \cdot - 9$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

$13 z + 86 = - 11 \cdot - 9$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

$13 z + 86 = - 11 \cdot - 9$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $- 11$ by $- 9$ .

$13 z + 86 = 99$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

$13 z + 86 = 99$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

$13 z + 86 = 99$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.3

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

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Step 5.3.1

Subtract $86$ from both sides of the equation.

$13 z = 99 - 86$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.3.2

Subtract $86$ from $99$ .

$13 z = 13$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

$13 z = 13$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.4

Divide each term in $13 z = 13$ by $13$ and simplify.

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Step 5.4.1

Divide each term in $13 z = 13$ by $13$ .

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

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $13$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.4.2.1.2

Divide $z$ by $1$ .

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

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

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

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $13$ by $13$ .

$z = 1$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

$z = 1$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

$z = 1$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

$z = 1$

$y = - \frac{3 (12 - z)}{11}$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 6

Replace all occurrences of $z$ with $1$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $y = - \frac{3 (12 - z)}{11}$ with $1$ .

$y = - \frac{3 (12 - (1))}{11}$

$z = 1$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{3 (12 - (1))}{11}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Multiply $- 1$ by $1$ .

$y = - \frac{3 (12 - 1)}{11}$

$z = 1$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 6.2.1.1.2

Subtract $1$ from $12$ .

$y = - \frac{3 \cdot 11}{11}$

$z = 1$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = - \frac{3 \cdot 11}{11}$

$z = 1$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 6.2.1.2

Simplify the expression.

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Step 6.2.1.2.1

Multiply $3$ by $11$ .

$y = - \frac{33}{11}$

$z = 1$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 6.2.1.2.2

Divide $33$ by $11$ .

$y = - 1 \cdot 3$

$z = 1$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 6.2.1.2.3

Multiply $- 1$ by $3$ .

$y = - 3$

$z = 1$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = - 3$

$z = 1$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = - 3$

$z = 1$

$x = \frac{7}{11} + \frac{15 z}{11}$

$y = - 3$

$z = 1$

$x = \frac{7}{11} + \frac{15 z}{11}$

Step 6.3

Replace all occurrences of $z$ in $x = \frac{7}{11} + \frac{15 z}{11}$ with $1$ .

$x = \frac{7}{11} + \frac{15 (1)}{11}$

$y = - 3$

$z = 1$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $\frac{7}{11} + \frac{15 (1)}{11}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$x = \frac{7 + 15 (1)}{11}$

$y = - 3$

$z = 1$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Multiply $15$ by $1$ .

$x = \frac{7 + 15}{11}$

$y = - 3$

$z = 1$

Step 6.4.1.2.2

Add $7$ and $15$ .

$x = \frac{22}{11}$

$y = - 3$

$z = 1$

Step 6.4.1.2.3

Divide $22$ by $11$ .

$x = 2$

$y = - 3$

$z = 1$

$x = 2$

$y = - 3$

$z = 1$

$x = 2$

$y = - 3$

$z = 1$

$x = 2$

$y = - 3$

$z = 1$

$x = 2$

$y = - 3$

$z = 1$

Step 7

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

$(2, - 3, 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(2, - 3, 1)$

Equation Form:

$x = 2, y = - 3, z = 1$