Precalculus Examples

$- 2 x + y - z = 6$ , $x - 3 y + 2 z = - 11$ , $x + 4 y - 5 z = 17$

Step 1

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

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Step 1.1

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

$y - z = 6 + 2 x$

$x - 3 y + 2 z = - 11$

$x + 4 y - 5 z = 17$

Step 1.2

Add $z$ to both sides of the equation.

$y = 6 + 2 x + z$

$x - 3 y + 2 z = - 11$

$x + 4 y - 5 z = 17$

$y = 6 + 2 x + z$

$x - 3 y + 2 z = - 11$

$x + 4 y - 5 z = 17$

Step 2

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

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Step 2.1

Replace all occurrences of $y$ in $x - 3 y + 2 z = - 11$ with $6 + 2 x + z$ .

$x - 3 (6 + 2 x + z) + 2 z = - 11$

$y = 6 + 2 x + z$

$x + 4 y - 5 z = 17$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$x - 3 \cdot 6 - 3 (2 x) - 3 z + 2 z = - 11$

$y = 6 + 2 x + z$

$x + 4 y - 5 z = 17$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 3$ by $6$ .

$x - 18 - 3 (2 x) - 3 z + 2 z = - 11$

$y = 6 + 2 x + z$

$x + 4 y - 5 z = 17$

Step 2.2.1.1.2.2

Multiply $2$ by $- 3$ .

$x - 18 - 6 x - 3 z + 2 z = - 11$

$y = 6 + 2 x + z$

$x + 4 y - 5 z = 17$

$x - 18 - 6 x - 3 z + 2 z = - 11$

$y = 6 + 2 x + z$

$x + 4 y - 5 z = 17$

$x - 18 - 6 x - 3 z + 2 z = - 11$

$y = 6 + 2 x + z$

$x + 4 y - 5 z = 17$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $6 x$ from $x$ .

$- 5 x - 18 - 3 z + 2 z = - 11$

$y = 6 + 2 x + z$

$x + 4 y - 5 z = 17$

Step 2.2.1.2.2

Add $- 3 z$ and $2 z$ .

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

$x + 4 y - 5 z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

$x + 4 y - 5 z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

$x + 4 y - 5 z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

$x + 4 y - 5 z = 17$

Step 2.3

Replace all occurrences of $y$ in $x + 4 y - 5 z = 17$ with $6 + 2 x + z$ .

$x + 4 (6 + 2 x + z) - 5 z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $x + 4 (6 + 2 x + z) - 5 z$ .

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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.

$x + 4 \cdot 6 + 4 (2 x) + 4 z - 5 z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $4$ by $6$ .

$x + 24 + 4 (2 x) + 4 z - 5 z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

Step 2.4.1.1.2.2

Multiply $2$ by $4$ .

$x + 24 + 8 x + 4 z - 5 z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

$x + 24 + 8 x + 4 z - 5 z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

$x + 24 + 8 x + 4 z - 5 z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $x$ and $8 x$ .

$9 x + 24 + 4 z - 5 z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

Step 2.4.1.2.2

Subtract $5 z$ from $4 z$ .

$9 x + 24 - z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

$9 x + 24 - z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

$9 x + 24 - z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

$9 x + 24 - z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

$9 x + 24 - z = 17$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

Step 3

Solve for $x$ in $9 x + 24 - z = 17$ .

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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 $24$ from both sides of the equation.

$9 x - z = 17 - 24$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

Step 3.1.2

Add $z$ to both sides of the equation.

$9 x = 17 - 24 + z$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

Step 3.1.3

Subtract $24$ from $17$ .

$9 x = - 7 + z$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

$9 x = - 7 + z$

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

Step 3.2

Divide each term in $9 x = - 7 + z$ by $9$ and simplify.

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Step 3.2.1

Divide each term in $9 x = - 7 + z$ by $9$ .

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

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

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.

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

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

Step 3.2.2.1.2

Divide $x$ by $1$ .

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

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

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

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

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

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

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.

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

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

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

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

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

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

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

$- 5 x - 18 - z = - 11$

$y = 6 + 2 x + z$

Step 4

Replace all occurrences of $x$ with $- \frac{7}{9} + \frac{z}{9}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $- 5 x - 18 - z = - 11$ with $- \frac{7}{9} + \frac{z}{9}$ .

$- 5 (- \frac{7}{9} + \frac{z}{9}) - 18 - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 5 (- \frac{7}{9} + \frac{z}{9}) - 18 - z$ .

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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.

$- 5 (- \frac{7}{9}) - 5 \frac{z}{9} - 18 - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.1.2

Multiply $- 5 (- \frac{7}{9})$ .

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Step 4.2.1.1.2.1

Multiply $- 1$ by $- 5$ .

$5 (\frac{7}{9}) - 5 \frac{z}{9} - 18 - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.1.2.2

Combine $5$ and $\frac{7}{9}$ .

$\frac{5 \cdot 7}{9} - 5 \frac{z}{9} - 18 - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.1.2.3

Multiply $5$ by $7$ .

$\frac{35}{9} - 5 \frac{z}{9} - 18 - z = - 11$

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

$y = 6 + 2 x + z$

$\frac{35}{9} - 5 \frac{z}{9} - 18 - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.1.3

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

$\frac{35}{9} + \frac{- 5 z}{9} - 18 - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.1.4

Move the negative in front of the fraction.

$\frac{35}{9} - \frac{5 z}{9} - 18 - z = - 11$

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

$y = 6 + 2 x + z$

$\frac{35}{9} - \frac{5 z}{9} - 18 - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.2

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- \frac{5 z}{9} + \frac{35}{9} - 18 \cdot \frac{9}{9} - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.3

Combine $- 18$ and $\frac{9}{9}$ .

$- \frac{5 z}{9} + \frac{35}{9} + \frac{- 18 \cdot 9}{9} - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.4

Combine the numerators over the common denominator.

$- \frac{5 z}{9} + \frac{35 - 18 \cdot 9}{9} - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $- 18$ by $9$ .

$- \frac{5 z}{9} + \frac{35 - 162}{9} - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.5.2

Subtract $162$ from $35$ .

$- \frac{5 z}{9} + \frac{- 127}{9} - z = - 11$

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

$y = 6 + 2 x + z$

$- \frac{5 z}{9} + \frac{- 127}{9} - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.6

Move the negative in front of the fraction.

$- \frac{5 z}{9} - \frac{127}{9} - z = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.7

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

$- \frac{5 z}{9} - z \cdot \frac{9}{9} - \frac{127}{9} = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.8

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

$- \frac{5 z}{9} + \frac{- z \cdot 9}{9} - \frac{127}{9} = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.9

Combine the numerators over the common denominator.

$\frac{- 5 z - z \cdot 9}{9} - \frac{127}{9} = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.10

Combine the numerators over the common denominator.

$\frac{- 5 z - z \cdot 9 - 127}{9} = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.11

Multiply $9$ by $- 1$ .

$\frac{- 5 z - 9 z - 127}{9} = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.12

Subtract $9 z$ from $- 5 z$ .

$\frac{- 14 z - 127}{9} = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.13

Factor $- 1$ out of $- 14 z$ .

$\frac{- (14 z) - 127}{9} = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.14

Rewrite $- 127$ as $- 1 (127)$ .

$\frac{- (14 z) - 1 \cdot 127}{9} = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.15

Factor $- 1$ out of $- (14 z) - 1 (127)$ .

$\frac{- (14 z + 127)}{9} = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.16

Simplify the expression.

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Step 4.2.1.16.1

Rewrite $- (14 z + 127)$ as $- 1 (14 z + 127)$ .

$\frac{- 1 (14 z + 127)}{9} = - 11$

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

$y = 6 + 2 x + z$

Step 4.2.1.16.2

Move the negative in front of the fraction.

$- \frac{14 z + 127}{9} = - 11$

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

$y = 6 + 2 x + z$

$- \frac{14 z + 127}{9} = - 11$

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

$y = 6 + 2 x + z$

$- \frac{14 z + 127}{9} = - 11$

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

$y = 6 + 2 x + z$

$- \frac{14 z + 127}{9} = - 11$

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

$y = 6 + 2 x + z$

Step 4.3

Replace all occurrences of $x$ in $y = 6 + 2 x + z$ with $- \frac{7}{9} + \frac{z}{9}$ .

$y = 6 + 2 (- \frac{7}{9} + \frac{z}{9}) + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $6 + 2 (- \frac{7}{9} + \frac{z}{9}) + z$ .

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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.

$y = 6 + 2 (- \frac{7}{9}) + 2 (\frac{z}{9}) + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.1.2

Multiply $2 (- \frac{7}{9})$ .

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Step 4.4.1.1.2.1

Multiply $- 1$ by $2$ .

$y = 6 - 2 (\frac{7}{9}) + 2 (\frac{z}{9}) + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.1.2.2

Combine $- 2$ and $\frac{7}{9}$ .

$y = 6 + \frac{- 2 \cdot 7}{9} + 2 (\frac{z}{9}) + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.1.2.3

Multiply $- 2$ by $7$ .

$y = 6 + \frac{- 14}{9} + 2 (\frac{z}{9}) + z$

$- \frac{14 z + 127}{9} = - 11$

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

$y = 6 + \frac{- 14}{9} + 2 (\frac{z}{9}) + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.1.3

Combine $2$ and $\frac{z}{9}$ .

$y = 6 + \frac{- 14}{9} + \frac{2 z}{9} + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.1.4

Move the negative in front of the fraction.

$y = 6 - \frac{14}{9} + \frac{2 z}{9} + z$

$- \frac{14 z + 127}{9} = - 11$

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

$y = 6 - \frac{14}{9} + \frac{2 z}{9} + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = 6 \cdot \frac{9}{9} - \frac{14}{9} + \frac{2 z}{9} + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.3

Combine $6$ and $\frac{9}{9}$ .

$y = \frac{6 \cdot 9}{9} - \frac{14}{9} + \frac{2 z}{9} + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.4

Combine the numerators over the common denominator.

$y = \frac{6 \cdot 9 - 14}{9} + \frac{2 z}{9} + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $6$ by $9$ .

$y = \frac{54 - 14}{9} + \frac{2 z}{9} + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.5.2

Subtract $14$ from $54$ .

$y = \frac{40}{9} + \frac{2 z}{9} + z$

$- \frac{14 z + 127}{9} = - 11$

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

$y = \frac{40}{9} + \frac{2 z}{9} + z$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.6

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

$y = \frac{40}{9} + \frac{2 z}{9} + z \cdot \frac{9}{9}$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.7

Simplify terms.

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Step 4.4.1.7.1

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

$y = \frac{40}{9} + \frac{2 z}{9} + \frac{z \cdot 9}{9}$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.7.2

Combine the numerators over the common denominator.

$y = \frac{40}{9} + \frac{2 z + z \cdot 9}{9}$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.7.3

Combine the numerators over the common denominator.

$y = \frac{40 + 2 z + z \cdot 9}{9}$

$- \frac{14 z + 127}{9} = - 11$

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

$y = \frac{40 + 2 z + z \cdot 9}{9}$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.8

Move $9$ to the left of $z$ .

$y = \frac{40 + 2 z + 9 z}{9}$

$- \frac{14 z + 127}{9} = - 11$

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

Step 4.4.1.9

Add $2 z$ and $9 z$ .

$y = \frac{40 + 11 z}{9}$

$- \frac{14 z + 127}{9} = - 11$

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

$y = \frac{40 + 11 z}{9}$

$- \frac{14 z + 127}{9} = - 11$

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

$y = \frac{40 + 11 z}{9}$

$- \frac{14 z + 127}{9} = - 11$

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

$y = \frac{40 + 11 z}{9}$

$- \frac{14 z + 127}{9} = - 11$

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

Step 5

Solve for $z$ in $- \frac{14 z + 127}{9} = - 11$ .

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Step 5.1

Multiply both sides of the equation by $- 9$ .

$- 9 (- \frac{14 z + 127}{9}) = - 9 \cdot - 11$

$y = \frac{40 + 11 z}{9}$

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

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 $- 9 (- \frac{14 z + 127}{9})$ .

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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

Move the leading negative in $- \frac{14 z + 127}{9}$ into the numerator.

$- 9 \frac{- (14 z + 127)}{9} = - 9 \cdot - 11$

$y = \frac{40 + 11 z}{9}$

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

Step 5.2.1.1.1.2

Factor $9$ out of $- 9$ .

$9 (- 1) (\frac{- (14 z + 127)}{9}) = - 9 \cdot - 11$

$y = \frac{40 + 11 z}{9}$

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

Step 5.2.1.1.1.3

Cancel the common factor.

$9 \cdot (- 1 \frac{- (14 z + 127)}{9}) = - 9 \cdot - 11$

$y = \frac{40 + 11 z}{9}$

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

Step 5.2.1.1.1.4

Rewrite the expression.

$14 z + 127 = - 9 \cdot - 11$

$y = \frac{40 + 11 z}{9}$

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

$14 z + 127 = - 9 \cdot - 11$

$y = \frac{40 + 11 z}{9}$

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

Step 5.2.1.1.2

Multiply.

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Step 5.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (14 z + 127) = - 9 \cdot - 11$

$y = \frac{40 + 11 z}{9}$

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

Step 5.2.1.1.2.2

Multiply $14 z + 127$ by $1$ .

$14 z + 127 = - 9 \cdot - 11$

$y = \frac{40 + 11 z}{9}$

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

$14 z + 127 = - 9 \cdot - 11$

$y = \frac{40 + 11 z}{9}$

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

$14 z + 127 = - 9 \cdot - 11$

$y = \frac{40 + 11 z}{9}$

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

$14 z + 127 = - 9 \cdot - 11$

$y = \frac{40 + 11 z}{9}$

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $- 9$ by $- 11$ .

$14 z + 127 = 99$

$y = \frac{40 + 11 z}{9}$

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

$14 z + 127 = 99$

$y = \frac{40 + 11 z}{9}$

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

$14 z + 127 = 99$

$y = \frac{40 + 11 z}{9}$

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

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 $127$ from both sides of the equation.

$14 z = 99 - 127$

$y = \frac{40 + 11 z}{9}$

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

Step 5.3.2

Subtract $127$ from $99$ .

$14 z = - 28$

$y = \frac{40 + 11 z}{9}$

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

$14 z = - 28$

$y = \frac{40 + 11 z}{9}$

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

Step 5.4

Divide each term in $14 z = - 28$ by $14$ and simplify.

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Step 5.4.1

Divide each term in $14 z = - 28$ by $14$ .

$\frac{14 z}{14} = \frac{- 28}{14}$

$y = \frac{40 + 11 z}{9}$

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $14$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{14 z}{14} = \frac{- 28}{14}$

$y = \frac{40 + 11 z}{9}$

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

Step 5.4.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 28}{14}$

$y = \frac{40 + 11 z}{9}$

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

$z = \frac{- 28}{14}$

$y = \frac{40 + 11 z}{9}$

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

$z = \frac{- 28}{14}$

$y = \frac{40 + 11 z}{9}$

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $- 28$ by $14$ .

$z = - 2$

$y = \frac{40 + 11 z}{9}$

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

$z = - 2$

$y = \frac{40 + 11 z}{9}$

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

$z = - 2$

$y = \frac{40 + 11 z}{9}$

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

$z = - 2$

$y = \frac{40 + 11 z}{9}$

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

Step 6

Replace all occurrences of $z$ with $- 2$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $y = \frac{40 + 11 z}{9}$ with $- 2$ .

$y = \frac{40 + 11 (- 2)}{9}$

$z = - 2$

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

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{40 + 11 (- 2)}{9}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Multiply $11$ by $- 2$ .

$y = \frac{40 - 22}{9}$

$z = - 2$

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

Step 6.2.1.1.2

Subtract $22$ from $40$ .

$y = \frac{18}{9}$

$z = - 2$

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

$y = \frac{18}{9}$

$z = - 2$

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

Step 6.2.1.2

Divide $18$ by $9$ .

$y = 2$

$z = - 2$

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

$y = 2$

$z = - 2$

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

$y = 2$

$z = - 2$

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

Step 6.3

Replace all occurrences of $z$ in $x = - \frac{7}{9} + \frac{z}{9}$ with $- 2$ .

$x = - \frac{7}{9} + \frac{- 2}{9}$

$y = 2$

$z = - 2$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{7}{9} + \frac{- 2}{9}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$x = \frac{- 7 - 2}{9}$

$y = 2$

$z = - 2$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Subtract $2$ from $- 7$ .

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

$y = 2$

$z = - 2$

Step 6.4.1.2.2

Divide $- 9$ by $9$ .

$x = - 1$

$y = 2$

$z = - 2$

$x = - 1$

$y = 2$

$z = - 2$

$x = - 1$

$y = 2$

$z = - 2$

$x = - 1$

$y = 2$

$z = - 2$

$x = - 1$

$y = 2$

$z = - 2$

Step 7

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

$(- 1, 2, - 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 1, 2, - 2)$

Equation Form:

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