Algebra Examples

$3 x + 5 y + 8 z = - 20$ $- x + 2 y - 4 z = 18$ $- 6 x + 4 z = 0$

Step 1

Solve for $x$ in $3 x + 5 y + 8 z = - 20$ .

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Step 1.1

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

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Step 1.1.1

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

$3 x + 8 z = - 20 - 5 y$

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

Step 1.1.2

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

$3 x = - 20 - 5 y - 8 z$

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

$3 x = - 20 - 5 y - 8 z$

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

Step 1.2

Divide each term in $3 x = - 20 - 5 y - 8 z$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = - 20 - 5 y - 8 z$ by $3$ .

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

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.

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

Step 1.2.3.1.3

Move the negative in front of the fraction.

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

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

$- x + 2 y - 4 z = 18$

$- 6 x + 4 z = 0$

Step 2

Replace all occurrences of $x$ with $- \frac{20}{3} - \frac{5 y}{3} - \frac{8 z}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- x + 2 y - 4 z = 18$ with $- \frac{20}{3} - \frac{5 y}{3} - \frac{8 z}{3}$ .

$- (- \frac{20}{3} - \frac{5 y}{3} - \frac{8 z}{3}) + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- (- \frac{20}{3} - \frac{5 y}{3} - \frac{8 z}{3}) + 2 y - 4 z$ .

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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.

$\frac{20}{3} + \frac{5 y}{3} + \frac{8 z}{3} + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- - \frac{20}{3}$ .

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Step 2.2.1.1.2.1.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{20}{3}) + \frac{5 y}{3} + \frac{8 z}{3} + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.1.2.1.2

Multiply $\frac{20}{3}$ by $1$ .

$\frac{20}{3} + \frac{5 y}{3} + \frac{8 z}{3} + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

$\frac{20}{3} + \frac{5 y}{3} + \frac{8 z}{3} + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.1.2.2

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

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Step 2.2.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$\frac{20}{3} + 1 (\frac{5 y}{3}) + \frac{8 z}{3} + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.1.2.2.2

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

$\frac{20}{3} + \frac{5 y}{3} + \frac{8 z}{3} + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

$\frac{20}{3} + \frac{5 y}{3} + \frac{8 z}{3} + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.1.2.3

Multiply $- - \frac{8 z}{3}$ .

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Step 2.2.1.1.2.3.1

Multiply $- 1$ by $- 1$ .

$\frac{20}{3} + \frac{5 y}{3} + 1 (\frac{8 z}{3}) + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.1.2.3.2

Multiply $\frac{8 z}{3}$ by $1$ .

$\frac{20}{3} + \frac{5 y}{3} + \frac{8 z}{3} + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

$\frac{20}{3} + \frac{5 y}{3} + \frac{8 z}{3} + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

$\frac{20}{3} + \frac{5 y}{3} + \frac{8 z}{3} + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

$\frac{20}{3} + \frac{5 y}{3} + \frac{8 z}{3} + 2 y - 4 z = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.2

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{20}{3} + \frac{8 z}{3} + \frac{5 y}{3} + 2 y \cdot \frac{3}{3} - 4 z = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.3

Combine $2 y$ and $\frac{3}{3}$ .

$\frac{20}{3} + \frac{8 z}{3} + \frac{5 y}{3} + \frac{2 y \cdot 3}{3} - 4 z = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{20}{3} + \frac{8 z}{3} + \frac{5 y + 2 y \cdot 3}{3} - 4 z = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.5

Combine the numerators over the common denominator.

$- 4 z + \frac{20 + 8 z + 5 y + 2 y \cdot 3}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.6

Multiply $3$ by $2$ .

$- 4 z + \frac{20 + 8 z + 5 y + 6 y}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.7

Add $5 y$ and $6 y$ .

$- 4 z + \frac{20 + 8 z + 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.8

To write $- 4 z$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 4 z \cdot \frac{3}{3} + \frac{20 + 8 z + 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.9

Simplify terms.

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Step 2.2.1.9.1

Combine $- 4 z$ and $\frac{3}{3}$ .

$\frac{- 4 z \cdot 3}{3} + \frac{20 + 8 z + 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.9.2

Combine the numerators over the common denominator.

$\frac{- 4 z \cdot 3 + 20 + 8 z + 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

$\frac{- 4 z \cdot 3 + 20 + 8 z + 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.10

Simplify the numerator.

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Step 2.2.1.10.1

Multiply $3$ by $- 4$ .

$\frac{- 12 z + 20 + 8 z + 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.10.2

Add $- 12 z$ and $8 z$ .

$\frac{- 4 z + 20 + 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

$\frac{- 4 z + 20 + 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.11

Simplify with factoring out.

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Step 2.2.1.11.1

Factor $- 1$ out of $- 4 z$ .

$\frac{- (4 z) + 20 + 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.11.2

Rewrite $20$ as $- 1 (- 20)$ .

$\frac{- (4 z) - 1 \cdot - 20 + 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.11.3

Factor $- 1$ out of $- (4 z) - 1 (- 20)$ .

$\frac{- (4 z - 20) + 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.11.4

Factor $- 1$ out of $11 y$ .

$\frac{- (4 z - 20) - (- 11 y)}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.11.5

Factor $- 1$ out of $- (4 z - 20) - (- 11 y)$ .

$\frac{- (4 z - 20 - 11 y)}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.11.6

Simplify the expression.

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Step 2.2.1.11.6.1

Rewrite $- (4 z - 20 - 11 y)$ as $- 1 (4 z - 20 - 11 y)$ .

$\frac{- 1 (4 z - 20 - 11 y)}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.2.1.11.6.2

Move the negative in front of the fraction.

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$- 6 x + 4 z = 0$

Step 2.3

Replace all occurrences of $x$ in $- 6 x + 4 z = 0$ with $- \frac{20}{3} - \frac{5 y}{3} - \frac{8 z}{3}$ .

$- 6 (- \frac{20}{3} - \frac{5 y}{3} - \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 6 (- \frac{20}{3} - \frac{5 y}{3} - \frac{8 z}{3}) + 4 z$ .

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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.

$- 6 (- \frac{20}{3}) - 6 (- \frac{5 y}{3}) - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Cancel the common factor of $3$ .

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Step 2.4.1.1.2.1.1

Move the leading negative in $- \frac{20}{3}$ into the numerator.

$- 6 (\frac{- 20}{3}) - 6 (- \frac{5 y}{3}) - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.1.2

Factor $3$ out of $- 6$ .

$3 (- 2) (\frac{- 20}{3}) - 6 (- \frac{5 y}{3}) - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.1.3

Cancel the common factor.

$3 \cdot (- 2 (\frac{- 20}{3})) - 6 (- \frac{5 y}{3}) - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.1.4

Rewrite the expression.

$- 2 \cdot - 20 - 6 (- \frac{5 y}{3}) - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$- 2 \cdot - 20 - 6 (- \frac{5 y}{3}) - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.2

Multiply $- 2$ by $- 20$ .

$40 - 6 (- \frac{5 y}{3}) - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.3

Cancel the common factor of $3$ .

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Step 2.4.1.1.2.3.1

Move the leading negative in $- \frac{5 y}{3}$ into the numerator.

$40 - 6 \frac{- 5 y}{3} - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.3.2

Factor $3$ out of $- 6$ .

$40 + 3 (- 2) (\frac{- 5 y}{3}) - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.3.3

Cancel the common factor.

$40 + 3 \cdot (- 2 \frac{- 5 y}{3}) - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.3.4

Rewrite the expression.

$40 - 2 (- 5 y) - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$40 - 2 (- 5 y) - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.4

Multiply $- 5$ by $- 2$ .

$40 + 10 y - 6 (- \frac{8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.5

Cancel the common factor of $3$ .

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Step 2.4.1.1.2.5.1

Move the leading negative in $- \frac{8 z}{3}$ into the numerator.

$40 + 10 y - 6 \frac{- 8 z}{3} + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.5.2

Factor $3$ out of $- 6$ .

$40 + 10 y + 3 (- 2) (\frac{- 8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.5.3

Cancel the common factor.

$40 + 10 y + 3 \cdot (- 2 \frac{- 8 z}{3}) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.5.4

Rewrite the expression.

$40 + 10 y - 2 (- 8 z) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$40 + 10 y - 2 (- 8 z) + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.1.2.6

Multiply $- 8$ by $- 2$ .

$40 + 10 y + 16 z + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$40 + 10 y + 16 z + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$40 + 10 y + 16 z + 4 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 2.4.1.2

Add $16 z$ and $4 z$ .

$40 + 10 y + 20 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$40 + 10 y + 20 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$40 + 10 y + 20 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$40 + 10 y + 20 z = 0$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 3

Solve for $y$ in $40 + 10 y + 20 z = 0$ .

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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

Subtract $40$ from both sides of the equation.

$10 y + 20 z = - 40$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 3.1.2

Subtract $20 z$ from both sides of the equation.

$10 y = - 40 - 20 z$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$10 y = - 40 - 20 z$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 3.2

Divide each term in $10 y = - 40 - 20 z$ by $10$ and simplify.

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Step 3.2.1

Divide each term in $10 y = - 40 - 20 z$ by $10$ .

$\frac{10 y}{10} = \frac{- 40}{10} + \frac{- 20 z}{10}$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $10$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{10 y}{10} = \frac{- 40}{10} + \frac{- 20 z}{10}$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 40}{10} + \frac{- 20 z}{10}$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$y = \frac{- 40}{10} + \frac{- 20 z}{10}$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$y = \frac{- 40}{10} + \frac{- 20 z}{10}$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Divide $- 40$ by $10$ .

$y = - 4 + \frac{- 20 z}{10}$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 3.2.3.1.2

Cancel the common factor of $- 20$ and $10$ .

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Step 3.2.3.1.2.1

Factor $10$ out of $- 20 z$ .

$y = - 4 + \frac{10 (- 2 z)}{10}$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $10$ out of $10$ .

$y = - 4 + \frac{10 (- 2 z)}{10 (1)}$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 3.2.3.1.2.2.2

Cancel the common factor.

$y = - 4 + \frac{10 (- 2 z)}{10 \cdot 1}$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 3.2.3.1.2.2.3

Rewrite the expression.

$y = - 4 + \frac{- 2 z}{1}$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 3.2.3.1.2.2.4

Divide $- 2 z$ by $1$ .

$y = - 4 - 2 z$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$y = - 4 - 2 z$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$y = - 4 - 2 z$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$y = - 4 - 2 z$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$y = - 4 - 2 z$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$y = - 4 - 2 z$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

$y = - 4 - 2 z$

$- \frac{4 z - 20 - 11 y}{3} = 18$

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

Step 4

Replace all occurrences of $y$ with $- 4 - 2 z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- \frac{4 z - 20 - 11 y}{3} = 18$ with $- 4 - 2 z$ .

$- \frac{4 z - 20 - 11 (- 4 - 2 z)}{3} = 18$

$y = - 4 - 2 z$

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

Apply the distributive property.

$- \frac{4 z - 20 - 11 \cdot - 4 - 11 (- 2 z)}{3} = 18$

$y = - 4 - 2 z$

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

Step 4.2.1.2

Multiply $- 11$ by $- 4$ .

$- \frac{4 z - 20 + 44 - 11 (- 2 z)}{3} = 18$

$y = - 4 - 2 z$

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

Step 4.2.1.3

Multiply $- 2$ by $- 11$ .

$- \frac{4 z - 20 + 44 + 22 z}{3} = 18$

$y = - 4 - 2 z$

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

Step 4.2.1.4

Add $4 z$ and $22 z$ .

$- \frac{26 z - 20 + 44}{3} = 18$

$y = - 4 - 2 z$

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

Step 4.2.1.5

Add $- 20$ and $44$ .

$- \frac{26 z + 24}{3} = 18$

$y = - 4 - 2 z$

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

Step 4.2.1.6

Factor $2$ out of $26 z + 24$ .

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Step 4.2.1.6.1

Factor $2$ out of $26 z$ .

$- \frac{2 (13 z) + 24}{3} = 18$

$y = - 4 - 2 z$

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

Step 4.2.1.6.2

Factor $2$ out of $24$ .

$- \frac{2 (13 z) + 2 (12)}{3} = 18$

$y = - 4 - 2 z$

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

Step 4.2.1.6.3

Factor $2$ out of $2 (13 z) + 2 (12)$ .

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

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

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

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

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

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

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

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

Step 4.3

Replace all occurrences of $y$ in $x = - \frac{20}{3} - \frac{5 y}{3} - \frac{8 z}{3}$ with $- 4 - 2 z$ .

$x = - \frac{20}{3} - \frac{5 (- 4 - 2 z)}{3} - \frac{8 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- \frac{20}{3} - \frac{5 (- 4 - 2 z)}{3} - \frac{8 z}{3}$ .

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Step 4.4.1.1

Combine the numerators over the common denominator.

$x = \frac{- 20 - 5 (- 4 - 2 z) - 8 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

Step 4.4.1.2

Simplify each term.

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Step 4.4.1.2.1

Apply the distributive property.

$x = \frac{- 20 - 5 \cdot - 4 - 5 (- 2 z) - 8 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

Step 4.4.1.2.2

Multiply $- 5$ by $- 4$ .

$x = \frac{- 20 + 20 - 5 (- 2 z) - 8 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

Step 4.4.1.2.3

Multiply $- 2$ by $- 5$ .

$x = \frac{- 20 + 20 + 10 z - 8 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

$x = \frac{- 20 + 20 + 10 z - 8 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

Step 4.4.1.3

Simplify by adding terms.

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Step 4.4.1.3.1

Add $- 20$ and $20$ .

$x = \frac{0 + 10 z - 8 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

Step 4.4.1.3.2

Add $0$ and $10 z$ .

$x = \frac{10 z - 8 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

Step 4.4.1.3.3

Subtract $8 z$ from $10 z$ .

$x = \frac{2 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

$x = \frac{2 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

$x = \frac{2 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

$x = \frac{2 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

$x = \frac{2 z}{3}$

$- \frac{2 (13 z + 12)}{3} = 18$

$y = - 4 - 2 z$

Step 5

Solve for $z$ in $- \frac{2 (13 z + 12)}{3} = 18$ .

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Step 5.1

Multiply both sides of the equation by $- \frac{3}{2}$ .

$- \frac{3}{2} \cdot (- \frac{2 (13 z + 12)}{3}) = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

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 $- \frac{3}{2} (- \frac{2 (13 z + 12)}{3})$ .

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Step 5.2.1.1.1

Cancel the common factor of $3$ .

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Step 5.2.1.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\frac{- 3}{2} \cdot (- \frac{2 (13 z + 12)}{3}) = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.1.1.1.2

Move the leading negative in $- \frac{2 (13 z + 12)}{3}$ into the numerator.

$\frac{- 3}{2} \cdot \frac{- 2 (13 z + 12)}{3} = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 (13 z + 12)}{3} = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 (13 z + 12)}{3} = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} \cdot (- 2 (13 z + 12)) = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$\frac{- 1}{2} \cdot (- 2 (13 z + 12)) = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Factor $2$ out of $- 2 (13 z + 12)$ .

$\frac{- 1}{2} \cdot (2 (- (13 z + 12))) = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} \cdot (2 (- (13 z + 12))) = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.1.1.2.3

Rewrite the expression.

$13 z + 12 = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$13 z + 12 = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.1.1.3

Multiply.

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Step 5.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 (13 z + 12) = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.1.1.3.2

Multiply $13 z + 12$ by $1$ .

$13 z + 12 = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$13 z + 12 = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$13 z + 12 = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$13 z + 12 = - \frac{3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $- \frac{3}{2} \cdot 18$ .

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Step 5.2.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$13 z + 12 = \frac{- 3}{2} \cdot 18$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.2.1.1.2

Factor $2$ out of $18$ .

$13 z + 12 = \frac{- 3}{2} \cdot (2 (9))$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.2.1.1.3

Cancel the common factor.

$13 z + 12 = \frac{- 3}{2} \cdot (2 \cdot 9)$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.2.1.1.4

Rewrite the expression.

$13 z + 12 = - 3 \cdot 9$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$13 z + 12 = - 3 \cdot 9$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.2.2.1.2

Multiply $- 3$ by $9$ .

$13 z + 12 = - 27$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$13 z + 12 = - 27$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$13 z + 12 = - 27$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$13 z + 12 = - 27$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

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 $12$ from both sides of the equation.

$13 z = - 27 - 12$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.3.2

Subtract $12$ from $- 27$ .

$13 z = - 39$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$13 z = - 39$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.4

Divide each term in $13 z = - 39$ by $13$ and simplify.

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Step 5.4.1

Divide each term in $13 z = - 39$ by $13$ .

$\frac{13 z}{13} = \frac{- 39}{13}$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

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.

$\frac{13 z}{13} = \frac{- 39}{13}$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.4.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 39}{13}$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$z = \frac{- 39}{13}$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$z = \frac{- 39}{13}$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $- 39$ by $13$ .

$z = - 3$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$z = - 3$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$z = - 3$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

$z = - 3$

$x = \frac{2 z}{3}$

$y = - 4 - 2 z$

Step 6

Replace all occurrences of $z$ with $- 3$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = \frac{2 z}{3}$ with $- 3$ .

$x = \frac{2 (- 3)}{3}$

$z = - 3$

$y = - 4 - 2 z$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{2 (- 3)}{3}$ .

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Step 6.2.1.1

Cancel the common factor of $- 3$ and $3$ .

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Step 6.2.1.1.1

Factor $3$ out of $2 (- 3)$ .

$x = \frac{3 (2 \cdot (- 1))}{3}$

$z = - 3$

$y = - 4 - 2 z$

Step 6.2.1.1.2

Cancel the common factors.

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Step 6.2.1.1.2.1

Factor $3$ out of $3$ .

$x = \frac{3 (2 \cdot (- 1))}{3 (1)}$

$z = - 3$

$y = - 4 - 2 z$

Step 6.2.1.1.2.2

Cancel the common factor.

$x = \frac{3 (2 \cdot (- 1))}{3 \cdot 1}$

$z = - 3$

$y = - 4 - 2 z$

Step 6.2.1.1.2.3

Rewrite the expression.

$x = \frac{2 \cdot (- 1)}{1}$

$z = - 3$

$y = - 4 - 2 z$

Step 6.2.1.1.2.4

Divide $2 \cdot (- 1)$ by $1$ .

$x = 2 \cdot (- 1)$

$z = - 3$

$y = - 4 - 2 z$

$x = 2 \cdot (- 1)$

$z = - 3$

$y = - 4 - 2 z$

$x = 2 \cdot (- 1)$

$z = - 3$

$y = - 4 - 2 z$

Step 6.2.1.2

Multiply $2$ by $- 1$ .

$x = - 2$

$z = - 3$

$y = - 4 - 2 z$

$x = - 2$

$z = - 3$

$y = - 4 - 2 z$

$x = - 2$

$z = - 3$

$y = - 4 - 2 z$

Step 6.3

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

$y = - 4 - 2 \cdot - 3$

$x = - 2$

$z = - 3$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 4 - 2 \cdot - 3$ .

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Step 6.4.1.1

Multiply $- 2$ by $- 3$ .

$y = - 4 + 6$

$x = - 2$

$z = - 3$

Step 6.4.1.2

Add $- 4$ and $6$ .

$y = 2$

$x = - 2$

$z = - 3$

$y = 2$

$x = - 2$

$z = - 3$

$y = 2$

$x = - 2$

$z = - 3$

$y = 2$

$x = - 2$

$z = - 3$

Step 7

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

$(- 2, 2, - 3)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 2, 2, - 3)$

Equation Form:

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