Finite Math Examples

$2 x - 5 y + 3 z = 9$ , $2 x + 4 y - z = 13$ , $6 x - 6 y + 5 z = 31$

Step 1

Solve for $x$ in $2 x - 5 y + 3 z = 9$ .

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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

Add $5 y$ to both sides of the equation.

$2 x + 3 z = 9 + 5 y$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

Step 1.1.2

Subtract $3 z$ from both sides of the equation.

$2 x = 9 + 5 y - 3 z$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

$2 x = 9 + 5 y - 3 z$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

Step 1.2

Divide each term in $2 x = 9 + 5 y - 3 z$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 9 + 5 y - 3 z$ by $2$ .

$\frac{2 x}{2} = \frac{9}{2} + \frac{5 y}{2} + \frac{- 3 z}{2}$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{9}{2} + \frac{5 y}{2} + \frac{- 3 z}{2}$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{2} + \frac{5 y}{2} + \frac{- 3 z}{2}$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

$x = \frac{9}{2} + \frac{5 y}{2} + \frac{- 3 z}{2}$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

$x = \frac{9}{2} + \frac{5 y}{2} + \frac{- 3 z}{2}$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$2 x + 4 y - z = 13$

$6 x - 6 y + 5 z = 31$

Step 2

Replace all occurrences of $x$ with $\frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x + 4 y - z = 13$ with $\frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$ .

$2 (\frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}) + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (\frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}) + 4 y - z$ .

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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.

$2 (\frac{9}{2}) + 2 (\frac{5 y}{2}) + 2 (- \frac{3 z}{2}) + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1.1

Cancel the common factor.

$2 (\frac{9}{2}) + 2 (\frac{5 y}{2}) + 2 (- \frac{3 z}{2}) + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

Step 2.2.1.1.2.1.2

Rewrite the expression.

$9 + 2 (\frac{5 y}{2}) + 2 (- \frac{3 z}{2}) + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

$9 + 2 (\frac{5 y}{2}) + 2 (- \frac{3 z}{2}) + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

Step 2.2.1.1.2.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.2.1

Cancel the common factor.

$9 + 2 (\frac{5 y}{2}) + 2 (- \frac{3 z}{2}) + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

Step 2.2.1.1.2.2.2

Rewrite the expression.

$9 + 5 y + 2 (- \frac{3 z}{2}) + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

$9 + 5 y + 2 (- \frac{3 z}{2}) + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

Step 2.2.1.1.2.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.3.1

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

$9 + 5 y + 2 (\frac{- 3 z}{2}) + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

Step 2.2.1.1.2.3.2

Cancel the common factor.

$9 + 5 y + 2 (\frac{- 3 z}{2}) + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

Step 2.2.1.1.2.3.3

Rewrite the expression.

$9 + 5 y - 3 z + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

$9 + 5 y - 3 z + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

$9 + 5 y - 3 z + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

$9 + 5 y - 3 z + 4 y - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $5 y$ and $4 y$ .

$9 + 9 y - 3 z - z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

Step 2.2.1.2.2

Subtract $z$ from $- 3 z$ .

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$6 x - 6 y + 5 z = 31$

Step 2.3

Replace all occurrences of $x$ in $6 x - 6 y + 5 z = 31$ with $\frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$ .

$6 (\frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $6 (\frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}) - 6 y + 5 z$ .

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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.

$6 (\frac{9}{2}) + 6 (\frac{5 y}{2}) + 6 (- \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.1.1

Factor $2$ out of $6$ .

$2 (3) (\frac{9}{2}) + 6 (\frac{5 y}{2}) + 6 (- \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.1.2

Cancel the common factor.

$2 \cdot (3 (\frac{9}{2})) + 6 (\frac{5 y}{2}) + 6 (- \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.1.3

Rewrite the expression.

$3 \cdot 9 + 6 (\frac{5 y}{2}) + 6 (- \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$3 \cdot 9 + 6 (\frac{5 y}{2}) + 6 (- \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.2

Multiply $3$ by $9$ .

$27 + 6 (\frac{5 y}{2}) + 6 (- \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.3

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.3.1

Factor $2$ out of $6$ .

$27 + 2 (3) (\frac{5 y}{2}) + 6 (- \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.3.2

Cancel the common factor.

$27 + 2 \cdot (3 (\frac{5 y}{2})) + 6 (- \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.3.3

Rewrite the expression.

$27 + 3 (5 y) + 6 (- \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$27 + 3 (5 y) + 6 (- \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.4

Multiply $5$ by $3$ .

$27 + 15 y + 6 (- \frac{3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.5

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.5.1

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

$27 + 15 y + 6 (\frac{- 3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.5.2

Factor $2$ out of $6$ .

$27 + 15 y + 2 (3) (\frac{- 3 z}{2}) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.5.3

Cancel the common factor.

$27 + 15 y + 2 \cdot (3 (\frac{- 3 z}{2})) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.5.4

Rewrite the expression.

$27 + 15 y + 3 (- 3 z) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$27 + 15 y + 3 (- 3 z) - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.1.2.6

Multiply $- 3$ by $3$ .

$27 + 15 y - 9 z - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$27 + 15 y - 9 z - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$27 + 15 y - 9 z - 6 y + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $6 y$ from $15 y$ .

$27 + 9 y - 9 z + 5 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 2.4.1.2.2

Add $- 9 z$ and $5 z$ .

$27 + 9 y - 4 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$27 + 9 y - 4 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$27 + 9 y - 4 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$27 + 9 y - 4 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$27 + 9 y - 4 z = 31$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 3

Solve for $y$ in $27 + 9 y - 4 z = 31$ .

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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 $27$ from both sides of the equation.

$9 y - 4 z = 31 - 27$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 3.1.2

Add $4 z$ to both sides of the equation.

$9 y = 31 - 27 + 4 z$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 3.1.3

Subtract $27$ from $31$ .

$9 y = 4 + 4 z$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$9 y = 4 + 4 z$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 3.2

Divide each term in $9 y = 4 + 4 z$ by $9$ and simplify.

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Step 3.2.1

Divide each term in $9 y = 4 + 4 z$ by $9$ .

$\frac{9 y}{9} = \frac{4}{9} + \frac{4 z}{9}$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

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.

$\frac{9 y}{9} = \frac{4}{9} + \frac{4 z}{9}$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{4}{9} + \frac{4 z}{9}$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$y = \frac{4}{9} + \frac{4 z}{9}$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$y = \frac{4}{9} + \frac{4 z}{9}$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$y = \frac{4}{9} + \frac{4 z}{9}$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$y = \frac{4}{9} + \frac{4 z}{9}$

$9 + 9 y - 4 z = 13$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 4

Replace all occurrences of $y$ with $\frac{4}{9} + \frac{4 z}{9}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $9 + 9 y - 4 z = 13$ with $\frac{4}{9} + \frac{4 z}{9}$ .

$9 + 9 (\frac{4}{9} + \frac{4 z}{9}) - 4 z = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $9 + 9 (\frac{4}{9} + \frac{4 z}{9}) - 4 z$ .

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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.

$9 + 9 (\frac{4}{9}) + 9 (\frac{4 z}{9}) - 4 z = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 4.2.1.1.2

Cancel the common factor of $9$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

$9 + 9 (\frac{4}{9}) + 9 (\frac{4 z}{9}) - 4 z = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

$9 + 4 + 9 (\frac{4 z}{9}) - 4 z = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$9 + 4 + 9 (\frac{4 z}{9}) - 4 z = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 4.2.1.1.3

Cancel the common factor of $9$ .

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Step 4.2.1.1.3.1

Cancel the common factor.

$9 + 4 + 9 (\frac{4 z}{9}) - 4 z = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 4.2.1.1.3.2

Rewrite the expression.

$9 + 4 + 4 z - 4 z = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$9 + 4 + 4 z - 4 z = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$9 + 4 + 4 z - 4 z = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $9 + 4 + 4 z - 4 z$ .

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Step 4.2.1.2.1.1

Subtract $4 z$ from $4 z$ .

$9 + 4 + 0 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 4.2.1.2.1.2

Add $9 + 4$ and $0$ .

$9 + 4 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$9 + 4 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 4.2.1.2.2

Add $9$ and $4$ .

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$

Step 4.3

Replace all occurrences of $y$ in $x = \frac{9}{2} + \frac{5 y}{2} - \frac{3 z}{2}$ with $\frac{4}{9} + \frac{4 z}{9}$ .

$x = \frac{9}{2} + \frac{5 (\frac{4}{9} + \frac{4 z}{9})}{2} - \frac{3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $\frac{9}{2} + \frac{5 (\frac{4}{9} + \frac{4 z}{9})}{2} - \frac{3 z}{2}$ .

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Step 4.4.1.1

Combine the numerators over the common denominator.

$x = \frac{9 + 5 (\frac{4}{9} + \frac{4 z}{9}) - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.2

Simplify each term.

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Step 4.4.1.2.1

Apply the distributive property.

$x = \frac{9 + 5 (\frac{4}{9}) + 5 (\frac{4 z}{9}) - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.2.2

Multiply $5 (\frac{4}{9})$ .

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Step 4.4.1.2.2.1

Combine $5$ and $\frac{4}{9}$ .

$x = \frac{9 + \frac{5 \cdot 4}{9} + 5 (\frac{4 z}{9}) - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.2.2.2

Multiply $5$ by $4$ .

$x = \frac{9 + \frac{20}{9} + 5 (\frac{4 z}{9}) - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9 + \frac{20}{9} + 5 (\frac{4 z}{9}) - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.2.3

Multiply $5 \frac{4 z}{9}$ .

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Step 4.4.1.2.3.1

Combine $5$ and $\frac{4 z}{9}$ .

$x = \frac{9 + \frac{20}{9} + \frac{5 (4 z)}{9} - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.2.3.2

Multiply $4$ by $5$ .

$x = \frac{9 + \frac{20}{9} + \frac{20 z}{9} - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9 + \frac{20}{9} + \frac{20 z}{9} - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{9 + \frac{20}{9} + \frac{20 z}{9} - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.3

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

$x = \frac{9 \cdot \frac{9}{9} + \frac{20}{9} + \frac{20 z}{9} - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.4

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

$x = \frac{\frac{9 \cdot 9}{9} + \frac{20}{9} + \frac{20 z}{9} - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{9 \cdot 9 + 20}{9} + \frac{20 z}{9} - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.6

Simplify the numerator.

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Step 4.4.1.6.1

Multiply $9$ by $9$ .

$x = \frac{\frac{81 + 20}{9} + \frac{20 z}{9} - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.6.2

Add $81$ and $20$ .

$x = \frac{\frac{101}{9} + \frac{20 z}{9} - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{\frac{101}{9} + \frac{20 z}{9} - 3 z}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.7

To write $- 3 z$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x = \frac{\frac{101}{9} + \frac{20 z}{9} - 3 z \cdot \frac{9}{9}}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.8

Combine $- 3 z$ and $\frac{9}{9}$ .

$x = \frac{\frac{101}{9} + \frac{20 z}{9} + \frac{- 3 z \cdot 9}{9}}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.9

Combine the numerators over the common denominator.

$x = \frac{\frac{101}{9} + \frac{20 z - 3 z \cdot 9}{9}}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.10

Combine the numerators over the common denominator.

$x = \frac{\frac{101 + 20 z - 3 z \cdot 9}{9}}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.11

Multiply $9$ by $- 3$ .

$x = \frac{\frac{101 + 20 z - 27 z}{9}}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.12

Subtract $27 z$ from $20 z$ .

$x = \frac{\frac{101 - 7 z}{9}}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.13

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{101 - 7 z}{9} \cdot \frac{1}{2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.14

Multiply $\frac{101 - 7 z}{9} \cdot \frac{1}{2}$ .

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Step 4.4.1.14.1

Multiply $\frac{101 - 7 z}{9}$ by $\frac{1}{2}$ .

$x = \frac{101 - 7 z}{9 \cdot 2}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 4.4.1.14.2

Multiply $9$ by $2$ .

$x = \frac{101 - 7 z}{18}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{101 - 7 z}{18}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{101 - 7 z}{18}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{101 - 7 z}{18}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

$x = \frac{101 - 7 z}{18}$

$13 = 13$

$y = \frac{4}{9} + \frac{4 z}{9}$

Step 5

Remove any equations from the system that are always true.

$x = \frac{101 - 7 z}{18}$

$y = \frac{4}{9} + \frac{4 z}{9}$