Finite Math Examples

$3 x - 9 y + 0 z = 1$ , $3 x + 9 y - 9 z = 3$ , $2 x + 0 y + 5 z = 0$

Step 1

Solve for $x$ in $3 x - 9 y = 1$ .

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Step 1.1

Add $9 y$ to both sides of the equation.

$3 x = 1 + 9 y$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

Step 1.2

Divide each term in $3 x = 1 + 9 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 1 + 9 y$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3} + \frac{9 y}{3}$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

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.

$\frac{3 x}{3} = \frac{1}{3} + \frac{9 y}{3}$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3} + \frac{9 y}{3}$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

$x = \frac{1}{3} + \frac{9 y}{3}$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

$x = \frac{1}{3} + \frac{9 y}{3}$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $9$ and $3$ .

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Step 1.2.3.1.1

Factor $3$ out of $9 y$ .

$x = \frac{1}{3} + \frac{3 (3 y)}{3}$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $3$ out of $3$ .

$x = \frac{1}{3} + \frac{3 (3 y)}{3 (1)}$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

Step 1.2.3.1.2.2

Cancel the common factor.

$x = \frac{1}{3} + \frac{3 (3 y)}{3 \cdot 1}$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

Step 1.2.3.1.2.3

Rewrite the expression.

$x = \frac{1}{3} + \frac{3 y}{1}$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

Step 1.2.3.1.2.4

Divide $3 y$ by $1$ .

$x = \frac{1}{3} + 3 y$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

$x = \frac{1}{3} + 3 y$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

$x = \frac{1}{3} + 3 y$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

$x = \frac{1}{3} + 3 y$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

$x = \frac{1}{3} + 3 y$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

$x = \frac{1}{3} + 3 y$

$3 x + 9 y - 9 z = 3$

$2 x + 5 z = 0$

Step 2

Replace all occurrences of $x$ with $\frac{1}{3} + 3 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x + 9 y - 9 z = 3$ with $\frac{1}{3} + 3 y$ .

$3 (\frac{1}{3} + 3 y) + 9 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$2 x + 5 z = 0$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (\frac{1}{3} + 3 y) + 9 y - 9 z$ .

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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.

$3 (\frac{1}{3}) + 3 (3 y) + 9 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$2 x + 5 z = 0$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$3 (\frac{1}{3}) + 3 (3 y) + 9 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$2 x + 5 z = 0$

Step 2.2.1.1.2.2

Rewrite the expression.

$1 + 3 (3 y) + 9 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$2 x + 5 z = 0$

$1 + 3 (3 y) + 9 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$2 x + 5 z = 0$

Step 2.2.1.1.3

Multiply $3$ by $3$ .

$1 + 9 y + 9 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$2 x + 5 z = 0$

$1 + 9 y + 9 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$2 x + 5 z = 0$

Step 2.2.1.2

Add $9 y$ and $9 y$ .

$1 + 18 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$2 x + 5 z = 0$

$1 + 18 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$2 x + 5 z = 0$

$1 + 18 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$2 x + 5 z = 0$

Step 2.3

Replace all occurrences of $x$ in $2 x + 5 z = 0$ with $\frac{1}{3} + 3 y$ .

$2 (\frac{1}{3} + 3 y) + 5 z = 0$

$1 + 18 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

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.

$2 (\frac{1}{3}) + 2 (3 y) + 5 z = 0$

$1 + 18 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

Step 2.4.1.2

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3} + 2 (3 y) + 5 z = 0$

$1 + 18 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

Step 2.4.1.3

Multiply $3$ by $2$ .

$\frac{2}{3} + 6 y + 5 z = 0$

$1 + 18 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$\frac{2}{3} + 6 y + 5 z = 0$

$1 + 18 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$\frac{2}{3} + 6 y + 5 z = 0$

$1 + 18 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

$\frac{2}{3} + 6 y + 5 z = 0$

$1 + 18 y - 9 z = 3$

$x = \frac{1}{3} + 3 y$

Step 3

Reorder $\frac{1}{3}$ and $3 y$ .

$x = 3 y + \frac{1}{3}$

$\frac{2}{3} + 6 y + 5 z = 0$

$1 + 18 y - 9 z = 3$

Step 4

Solve for $y$ in $\frac{2}{3} + 6 y + 5 z = 0$ .

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Step 4.1

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

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Step 4.1.1

Subtract $\frac{2}{3}$ from both sides of the equation.

$6 y + 5 z = - \frac{2}{3}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.1.2

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

$6 y = - \frac{2}{3} - 5 z$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

$6 y = - \frac{2}{3} - 5 z$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2

Divide each term in $6 y = - \frac{2}{3} - 5 z$ by $6$ and simplify.

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Step 4.2.1

Divide each term in $6 y = - \frac{2}{3} - 5 z$ by $6$ .

$\frac{6 y}{6} = \frac{- \frac{2}{3}}{6} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $6$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{- \frac{2}{3}}{6} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{2}{3}}{6} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

$y = \frac{- \frac{2}{3}}{6} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

$y = \frac{- \frac{2}{3}}{6} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

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

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{2}{3} \cdot \frac{1}{6} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2.3.1.2

Cancel the common factor of $2$ .

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Step 4.2.3.1.2.1

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

$y = \frac{- 2}{3} \cdot \frac{1}{6} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2.3.1.2.2

Factor $2$ out of $- 2$ .

$y = \frac{2 (- 1)}{3} \cdot \frac{1}{6} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2.3.1.2.3

Factor $2$ out of $6$ .

$y = \frac{2 \cdot - 1}{3} \cdot \frac{1}{2 \cdot 3} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2.3.1.2.4

Cancel the common factor.

$y = \frac{2 \cdot - 1}{3} \cdot \frac{1}{2 \cdot 3} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2.3.1.2.5

Rewrite the expression.

$y = \frac{- 1}{3} \cdot \frac{1}{3} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

$y = \frac{- 1}{3} \cdot \frac{1}{3} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2.3.1.3

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

$y = \frac{- 1}{3 \cdot 3} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2.3.1.4

Multiply $3$ by $3$ .

$y = \frac{- 1}{9} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2.3.1.5

Move the negative in front of the fraction.

$y = - \frac{1}{9} + \frac{- 5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 4.2.3.1.6

Move the negative in front of the fraction.

$y = - \frac{1}{9} - \frac{5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$x = 3 y + \frac{1}{3}$

$1 + 18 y - 9 z = 3$

Step 5

Replace all occurrences of $y$ with $- \frac{1}{9} - \frac{5 z}{6}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 3 y + \frac{1}{3}$ with $- \frac{1}{9} - \frac{5 z}{6}$ .

$x = 3 (- \frac{1}{9} - \frac{5 z}{6}) + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $3 (- \frac{1}{9} - \frac{5 z}{6}) + \frac{1}{3}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$x = 3 (- \frac{1}{9}) + 3 (- \frac{5 z}{6}) + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.1.2.1

Move the leading negative in $- \frac{1}{9}$ into the numerator.

$x = 3 (\frac{- 1}{9}) + 3 (- \frac{5 z}{6}) + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.1.2.2

Factor $3$ out of $9$ .

$x = 3 (\frac{- 1}{3 (3)}) + 3 (- \frac{5 z}{6}) + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.1.2.3

Cancel the common factor.

$x = 3 (\frac{- 1}{3 \cdot 3}) + 3 (- \frac{5 z}{6}) + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.1.2.4

Rewrite the expression.

$x = \frac{- 1}{3} + 3 (- \frac{5 z}{6}) + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

$x = \frac{- 1}{3} + 3 (- \frac{5 z}{6}) + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.1.3

Cancel the common factor of $3$ .

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Step 5.2.1.1.3.1

Move the leading negative in $- \frac{5 z}{6}$ into the numerator.

$x = \frac{- 1}{3} + 3 (\frac{- 5 z}{6}) + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.1.3.2

Factor $3$ out of $6$ .

$x = \frac{- 1}{3} + 3 (\frac{- 5 z}{3 (2)}) + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.1.3.3

Cancel the common factor.

$x = \frac{- 1}{3} + 3 (\frac{- 5 z}{3 \cdot 2}) + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.1.3.4

Rewrite the expression.

$x = \frac{- 1}{3} + \frac{- 5 z}{2} + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

$x = \frac{- 1}{3} + \frac{- 5 z}{2} + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.1.4

Simplify each term.

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Step 5.2.1.1.4.1

Move the negative in front of the fraction.

$x = - \frac{1}{3} + \frac{- 5 z}{2} + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.1.4.2

Move the negative in front of the fraction.

$x = - \frac{1}{3} - \frac{5 z}{2} + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

$x = - \frac{1}{3} - \frac{5 z}{2} + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

$x = - \frac{1}{3} - \frac{5 z}{2} + \frac{1}{3}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.2

Combine the opposite terms in $- \frac{1}{3} - \frac{5 z}{2} + \frac{1}{3}$ .

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Step 5.2.1.2.1

Add $- \frac{1}{3}$ and $\frac{1}{3}$ .

$x = - \frac{5 z}{2} + 0$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.2.1.2.2

Add $- \frac{5 z}{2}$ and $0$ .

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 18 y - 9 z = 3$

Step 5.3

Replace all occurrences of $y$ in $1 + 18 y - 9 z = 3$ with $- \frac{1}{9} - \frac{5 z}{6}$ .

$1 + 18 (- \frac{1}{9} - \frac{5 z}{6}) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $1 + 18 (- \frac{1}{9} - \frac{5 z}{6}) - 9 z$ .

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Step 5.4.1.1

Simplify each term.

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Step 5.4.1.1.1

Apply the distributive property.

$1 + 18 (- \frac{1}{9}) + 18 (- \frac{5 z}{6}) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.1.2

Cancel the common factor of $9$ .

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Step 5.4.1.1.2.1

Move the leading negative in $- \frac{1}{9}$ into the numerator.

$1 + 18 (\frac{- 1}{9}) + 18 (- \frac{5 z}{6}) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.1.2.2

Factor $9$ out of $18$ .

$1 + 9 (2) (\frac{- 1}{9}) + 18 (- \frac{5 z}{6}) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.1.2.3

Cancel the common factor.

$1 + 9 \cdot (2 (\frac{- 1}{9})) + 18 (- \frac{5 z}{6}) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.1.2.4

Rewrite the expression.

$1 + 2 \cdot - 1 + 18 (- \frac{5 z}{6}) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 + 2 \cdot - 1 + 18 (- \frac{5 z}{6}) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.1.3

Multiply $2$ by $- 1$ .

$1 - 2 + 18 (- \frac{5 z}{6}) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.1.4

Cancel the common factor of $6$ .

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Step 5.4.1.1.4.1

Move the leading negative in $- \frac{5 z}{6}$ into the numerator.

$1 - 2 + 18 (\frac{- 5 z}{6}) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.1.4.2

Factor $6$ out of $18$ .

$1 - 2 + 6 (3) (\frac{- 5 z}{6}) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.1.4.3

Cancel the common factor.

$1 - 2 + 6 \cdot (3 (\frac{- 5 z}{6})) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.1.4.4

Rewrite the expression.

$1 - 2 + 3 (- 5 z) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 - 2 + 3 (- 5 z) - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.1.5

Multiply $- 5$ by $3$ .

$1 - 2 - 15 z - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$1 - 2 - 15 z - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.2

Simplify by adding terms.

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Step 5.4.1.2.1

Subtract $2$ from $1$ .

$- 1 - 15 z - 9 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 5.4.1.2.2

Subtract $9 z$ from $- 15 z$ .

$- 1 - 24 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$- 1 - 24 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$- 1 - 24 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$- 1 - 24 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$- 1 - 24 z = 3$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 6

Solve for $z$ in $- 1 - 24 z = 3$ .

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Step 6.1

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

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Step 6.1.1

Add $1$ to both sides of the equation.

$- 24 z = 3 + 1$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 6.1.2

Add $3$ and $1$ .

$- 24 z = 4$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$- 24 z = 4$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 6.2

Divide each term in $- 24 z = 4$ by $- 24$ and simplify.

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Step 6.2.1

Divide each term in $- 24 z = 4$ by $- 24$ .

$\frac{- 24 z}{- 24} = \frac{4}{- 24}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $- 24$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{- 24 z}{- 24} = \frac{4}{- 24}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 6.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{4}{- 24}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$z = \frac{4}{- 24}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$z = \frac{4}{- 24}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Cancel the common factor of $4$ and $- 24$ .

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Step 6.2.3.1.1

Factor $4$ out of $4$ .

$z = \frac{4 (1)}{- 24}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 6.2.3.1.2

Cancel the common factors.

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Step 6.2.3.1.2.1

Factor $4$ out of $- 24$ .

$z = \frac{4 \cdot 1}{4 \cdot - 6}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 6.2.3.1.2.2

Cancel the common factor.

$z = \frac{4 \cdot 1}{4 \cdot - 6}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 6.2.3.1.2.3

Rewrite the expression.

$z = \frac{1}{- 6}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$z = \frac{1}{- 6}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$z = \frac{1}{- 6}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 6.2.3.2

Move the negative in front of the fraction.

$z = - \frac{1}{6}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$z = - \frac{1}{6}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$z = - \frac{1}{6}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$z = - \frac{1}{6}$

$x = - \frac{5 z}{2}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 7

Replace all occurrences of $z$ with $- \frac{1}{6}$ in each equation.

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Step 7.1

Replace all occurrences of $z$ in $x = - \frac{5 z}{2}$ with $- \frac{1}{6}$ .

$x = - \frac{5 (- \frac{1}{6})}{2}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- \frac{5 (- \frac{1}{6})}{2}$ .

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Step 7.2.1.1

Simplify the numerator.

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Step 7.2.1.1.1

Multiply $- 1$ by $5$ .

$x = - \frac{- 5 (\frac{1}{6})}{2}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 7.2.1.1.2

Combine $- 5$ and $\frac{1}{6}$ .

$x = - \frac{\frac{- 5}{6}}{2}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$x = - \frac{\frac{- 5}{6}}{2}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 7.2.1.2

Move the negative in front of the fraction.

$x = - \frac{- \frac{5}{6}}{2}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 7.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = - (- \frac{5}{6} \cdot \frac{1}{2})$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 7.2.1.4

Multiply $- \frac{5}{6} \cdot \frac{1}{2}$ .

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Step 7.2.1.4.1

Multiply $\frac{1}{2}$ by $\frac{5}{6}$ .

$x = \frac{5}{2 \cdot 6}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 7.2.1.4.2

Multiply $2$ by $6$ .

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 7.2.1.5

Multiply $- - \frac{5}{12}$ .

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Step 7.2.1.5.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{5}{12})$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 7.2.1.5.2

Multiply $\frac{5}{12}$ by $1$ .

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{5 z}{6}$

Step 7.3

Replace all occurrences of $z$ in $y = - \frac{1}{9} - \frac{5 z}{6}$ with $- \frac{1}{6}$ .

$y = - \frac{1}{9} - \frac{5 (- \frac{1}{6})}{6}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4

Simplify the right side.

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Step 7.4.1

Simplify $- \frac{1}{9} - \frac{5 (- \frac{1}{6})}{6}$ .

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Step 7.4.1.1

Simplify each term.

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Step 7.4.1.1.1

Simplify the numerator.

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Step 7.4.1.1.1.1

Multiply $- 1$ by $5$ .

$y = - \frac{1}{9} - \frac{- 5 (\frac{1}{6})}{6}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.1.1.2

Combine $- 5$ and $\frac{1}{6}$ .

$y = - \frac{1}{9} - \frac{\frac{- 5}{6}}{6}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} - \frac{\frac{- 5}{6}}{6}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.1.2

Move the negative in front of the fraction.

$y = - \frac{1}{9} - \frac{- \frac{5}{6}}{6}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{1}{9} - (- \frac{5}{6} \cdot \frac{1}{6})$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.1.4

Multiply $- \frac{5}{6} \cdot \frac{1}{6}$ .

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Step 7.4.1.1.4.1

Multiply $\frac{1}{6}$ by $\frac{5}{6}$ .

$y = - \frac{1}{9} + \frac{5}{6 \cdot 6}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.1.4.2

Multiply $6$ by $6$ .

$y = - \frac{1}{9} + \frac{5}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} + \frac{5}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.1.5

Multiply $- - \frac{5}{36}$ .

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Step 7.4.1.1.5.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{1}{9} + 1 (\frac{5}{36})$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.1.5.2

Multiply $\frac{5}{36}$ by $1$ .

$y = - \frac{1}{9} + \frac{5}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} + \frac{5}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = - \frac{1}{9} + \frac{5}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.2

To write $- \frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - \frac{1}{9} \cdot \frac{4}{4} + \frac{5}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.3

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.4.1.3.1

Multiply $\frac{1}{9}$ by $\frac{4}{4}$ .

$y = - \frac{4}{9 \cdot 4} + \frac{5}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.3.2

Multiply $9$ by $4$ .

$y = - \frac{4}{36} + \frac{5}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = - \frac{4}{36} + \frac{5}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.4

Combine the numerators over the common denominator.

$y = \frac{- 4 + 5}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 7.4.1.5

Add $- 4$ and $5$ .

$y = \frac{1}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = \frac{1}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = \frac{1}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

$y = \frac{1}{36}$

$x = \frac{5}{12}$

$z = - \frac{1}{6}$

Step 8

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

$(\frac{5}{12}, \frac{1}{36}, - \frac{1}{6})$

Step 9

The result can be shown in multiple forms.

Point Form:

$(\frac{5}{12}, \frac{1}{36}, - \frac{1}{6})$

Equation Form:

$x = \frac{5}{12}, y = \frac{1}{36}, z = - \frac{1}{6}$