Finite Math Examples

$3 x + 2 y + 2 z = 4$ , $- 6 x + 4 y - 2 z = - 9$ , $9 x - 2 y + 2 z = 10$

Step 1

Solve for $x$ in $3 x + 2 y + 2 z = 4$ .

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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 $2 y$ from both sides of the equation.

$3 x + 2 z = 4 - 2 y$

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

$9 x - 2 y + 2 z = 10$

Step 1.1.2

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

$3 x = 4 - 2 y - 2 z$

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

$9 x - 2 y + 2 z = 10$

$3 x = 4 - 2 y - 2 z$

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

$9 x - 2 y + 2 z = 10$

Step 1.2

Divide each term in $3 x = 4 - 2 y - 2 z$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 4 - 2 y - 2 z$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3} + \frac{- 2 y}{3} + \frac{- 2 z}{3}$

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

$9 x - 2 y + 2 z = 10$

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.

$\frac{3 x}{3} = \frac{4}{3} + \frac{- 2 y}{3} + \frac{- 2 z}{3}$

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

$9 x - 2 y + 2 z = 10$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3} + \frac{- 2 y}{3} + \frac{- 2 z}{3}$

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

$9 x - 2 y + 2 z = 10$

$x = \frac{4}{3} + \frac{- 2 y}{3} + \frac{- 2 z}{3}$

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

$9 x - 2 y + 2 z = 10$

$x = \frac{4}{3} + \frac{- 2 y}{3} + \frac{- 2 z}{3}$

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

$9 x - 2 y + 2 z = 10$

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.

$x = \frac{4}{3} - \frac{2 y}{3} + \frac{- 2 z}{3}$

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

$9 x - 2 y + 2 z = 10$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

$9 x - 2 y + 2 z = 10$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

$9 x - 2 y + 2 z = 10$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

$9 x - 2 y + 2 z = 10$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

$9 x - 2 y + 2 z = 10$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

$9 x - 2 y + 2 z = 10$

Step 2

Replace all occurrences of $x$ with $\frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 6 x + 4 y - 2 z = - 9$ with $\frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$ .

$- 6 (\frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 6 (\frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}) + 4 y - 2 z$ .

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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.

$- 6 (\frac{4}{3}) - 6 (- \frac{2 y}{3}) - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1.1

Factor $3$ out of $- 6$ .

$3 (- 2) (\frac{4}{3}) - 6 (- \frac{2 y}{3}) - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.1.2

Cancel the common factor.

$3 \cdot (- 2 (\frac{4}{3})) - 6 (- \frac{2 y}{3}) - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.1.3

Rewrite the expression.

$- 2 \cdot 4 - 6 (- \frac{2 y}{3}) - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

$- 2 \cdot 4 - 6 (- \frac{2 y}{3}) - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.2

Multiply $- 2$ by $4$ .

$- 8 - 6 (- \frac{2 y}{3}) - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.3.1

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

$- 8 - 6 \frac{- 2 y}{3} - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.3.2

Factor $3$ out of $- 6$ .

$- 8 + 3 (- 2) (\frac{- 2 y}{3}) - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.3.3

Cancel the common factor.

$- 8 + 3 \cdot (- 2 \frac{- 2 y}{3}) - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.3.4

Rewrite the expression.

$- 8 - 2 (- 2 y) - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

$- 8 - 2 (- 2 y) - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.4

Multiply $- 2$ by $- 2$ .

$- 8 + 4 y - 6 (- \frac{2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.5

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.5.1

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

$- 8 + 4 y - 6 \frac{- 2 z}{3} + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.5.2

Factor $3$ out of $- 6$ .

$- 8 + 4 y + 3 (- 2) (\frac{- 2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.5.3

Cancel the common factor.

$- 8 + 4 y + 3 \cdot (- 2 \frac{- 2 z}{3}) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.5.4

Rewrite the expression.

$- 8 + 4 y - 2 (- 2 z) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

$- 8 + 4 y - 2 (- 2 z) + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.1.2.6

Multiply $- 2$ by $- 2$ .

$- 8 + 4 y + 4 z + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

$- 8 + 4 y + 4 z + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

$- 8 + 4 y + 4 z + 4 y - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $4 y$ and $4 y$ .

$- 8 + 8 y + 4 z - 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.2.1.2.2

Subtract $2 z$ from $4 z$ .

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$9 x - 2 y + 2 z = 10$

Step 2.3

Replace all occurrences of $x$ in $9 x - 2 y + 2 z = 10$ with $\frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$ .

$9 (\frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4

Simplify the left side.

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Step 2.4.1

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

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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.

$9 (\frac{4}{3}) + 9 (- \frac{2 y}{3}) + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

Factor $3$ out of $9$ .

$3 (3) (\frac{4}{3}) + 9 (- \frac{2 y}{3}) + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.1.2.1.2

Cancel the common factor.

$3 \cdot (3 (\frac{4}{3})) + 9 (- \frac{2 y}{3}) + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.1.2.1.3

Rewrite the expression.

$3 \cdot 4 + 9 (- \frac{2 y}{3}) + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$3 \cdot 4 + 9 (- \frac{2 y}{3}) + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.1.2.2

Multiply $3$ by $4$ .

$12 + 9 (- \frac{2 y}{3}) + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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 $- \frac{2 y}{3}$ into the numerator.

$12 + 9 (\frac{- 2 y}{3}) + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.1.2.3.2

Factor $3$ out of $9$ .

$12 + 3 (3) (\frac{- 2 y}{3}) + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.1.2.3.3

Cancel the common factor.

$12 + 3 \cdot (3 (\frac{- 2 y}{3})) + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.1.2.3.4

Rewrite the expression.

$12 + 3 (- 2 y) + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$12 + 3 (- 2 y) + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.1.2.4

Multiply $- 2$ by $3$ .

$12 - 6 y + 9 (- \frac{2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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 $- \frac{2 z}{3}$ into the numerator.

$12 - 6 y + 9 (\frac{- 2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.1.2.5.2

Factor $3$ out of $9$ .

$12 - 6 y + 3 (3) (\frac{- 2 z}{3}) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.1.2.5.3

Cancel the common factor.

$12 - 6 y + 3 \cdot (3 (\frac{- 2 z}{3})) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.1.2.5.4

Rewrite the expression.

$12 - 6 y + 3 (- 2 z) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$12 - 6 y + 3 (- 2 z) - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.1.2.6

Multiply $- 2$ by $3$ .

$12 - 6 y - 6 z - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$12 - 6 y - 6 z - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$12 - 6 y - 6 z - 2 y + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $2 y$ from $- 6 y$ .

$12 - 8 y - 6 z + 2 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 2.4.1.2.2

Add $- 6 z$ and $2 z$ .

$12 - 8 y - 4 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$12 - 8 y - 4 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$12 - 8 y - 4 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$12 - 8 y - 4 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$12 - 8 y - 4 z = 10$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3

Solve for $y$ in $12 - 8 y - 4 z = 10$ .

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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 $12$ from both sides of the equation.

$- 8 y - 4 z = 10 - 12$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.1.2

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

$- 8 y = 10 - 12 + 4 z$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.1.3

Subtract $12$ from $10$ .

$- 8 y = - 2 + 4 z$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$- 8 y = - 2 + 4 z$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.2

Divide each term in $- 8 y = - 2 + 4 z$ by $- 8$ and simplify.

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Step 3.2.1

Divide each term in $- 8 y = - 2 + 4 z$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{- 2}{- 8} + \frac{4 z}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 8$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{- 2}{- 8} + \frac{4 z}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 2}{- 8} + \frac{4 z}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$y = \frac{- 2}{- 8} + \frac{4 z}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$y = \frac{- 2}{- 8} + \frac{4 z}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

Cancel the common factor of $- 2$ and $- 8$ .

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Step 3.2.3.1.1.1

Factor $- 2$ out of $- 2$ .

$y = \frac{- 2 \cdot 1}{- 8} + \frac{4 z}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.2.3.1.1.2

Cancel the common factors.

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Step 3.2.3.1.1.2.1

Factor $- 2$ out of $- 8$ .

$y = \frac{- 2 \cdot 1}{- 2 \cdot 4} + \frac{4 z}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 1}{- 2 \cdot 4} + \frac{4 z}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{1}{4} + \frac{4 z}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$y = \frac{1}{4} + \frac{4 z}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$y = \frac{1}{4} + \frac{4 z}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.2.3.1.2

Cancel the common factor of $4$ and $- 8$ .

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Step 3.2.3.1.2.1

Factor $4$ out of $4 z$ .

$y = \frac{1}{4} + \frac{4 (z)}{- 8}$

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $4$ out of $- 8$ .

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

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.2.3.1.2.2.2

Cancel the common factor.

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

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.2.3.1.2.2.3

Rewrite the expression.

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

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 3.2.3.1.3

Move the negative in front of the fraction.

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

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

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

$- 8 + 8 y + 2 z = - 9$

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4

Replace all occurrences of $y$ with $\frac{1}{4} - \frac{z}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 8 + 8 y + 2 z = - 9$ with $\frac{1}{4} - \frac{z}{2}$ .

$- 8 + 8 (\frac{1}{4} - \frac{z}{2}) + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 8 + 8 (\frac{1}{4} - \frac{z}{2}) + 2 z$ .

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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.

$- 8 + 8 (\frac{1}{4}) + 8 (- \frac{z}{2}) + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.2.1.1.2

Cancel the common factor of $4$ .

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Step 4.2.1.1.2.1

Factor $4$ out of $8$ .

$- 8 + 4 (2) (\frac{1}{4}) + 8 (- \frac{z}{2}) + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.2.1.1.2.2

Cancel the common factor.

$- 8 + 4 \cdot (2 (\frac{1}{4})) + 8 (- \frac{z}{2}) + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.2.1.1.2.3

Rewrite the expression.

$- 8 + 2 + 8 (- \frac{z}{2}) + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$- 8 + 2 + 8 (- \frac{z}{2}) + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.2.1.1.3

Cancel the common factor of $2$ .

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Step 4.2.1.1.3.1

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

$- 8 + 2 + 8 (\frac{- z}{2}) + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.2.1.1.3.2

Factor $2$ out of $8$ .

$- 8 + 2 + 2 (4) (\frac{- z}{2}) + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.2.1.1.3.3

Cancel the common factor.

$- 8 + 2 + 2 \cdot (4 (\frac{- z}{2})) + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.2.1.1.3.4

Rewrite the expression.

$- 8 + 2 + 4 (- z) + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$- 8 + 2 + 4 (- z) + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.2.1.1.4

Multiply $- 1$ by $4$ .

$- 8 + 2 - 4 z + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$- 8 + 2 - 4 z + 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $- 8$ and $2$ .

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

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.2.1.2.2

Add $- 4 z$ and $2 z$ .

$- 6 - 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$- 6 - 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$- 6 - 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

$- 6 - 2 z = - 9$

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

$x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$

Step 4.3

Replace all occurrences of $y$ in $x = \frac{4}{3} - \frac{2 y}{3} - \frac{2 z}{3}$ with $\frac{1}{4} - \frac{z}{2}$ .

$x = \frac{4}{3} - \frac{2 (\frac{1}{4} - \frac{z}{2})}{3} - \frac{2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $\frac{4}{3} - \frac{2 (\frac{1}{4} - \frac{z}{2})}{3} - \frac{2 z}{3}$ .

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Step 4.4.1.1

Combine the numerators over the common denominator.

$x = \frac{4 - 2 (\frac{1}{4} - \frac{z}{2}) - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2

Simplify each term.

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Step 4.4.1.2.1

Apply the distributive property.

$x = \frac{4 - 2 (\frac{1}{4}) - 2 (- \frac{z}{2}) - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2.2

Cancel the common factor of $2$ .

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Step 4.4.1.2.2.1

Factor $2$ out of $- 2$ .

$x = \frac{4 + 2 (- 1) (\frac{1}{4}) - 2 (- \frac{z}{2}) - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2.2.2

Factor $2$ out of $4$ .

$x = \frac{4 + 2 \cdot (- 1 \frac{1}{2 \cdot 2}) - 2 (- \frac{z}{2}) - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2.2.3

Cancel the common factor.

$x = \frac{4 + 2 \cdot (- 1 \frac{1}{2 \cdot 2}) - 2 (- \frac{z}{2}) - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2.2.4

Rewrite the expression.

$x = \frac{4 - 1 (\frac{1}{2}) - 2 (- \frac{z}{2}) - 2 z}{3}$

$- 6 - 2 z = - 9$

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

$x = \frac{4 - 1 (\frac{1}{2}) - 2 (- \frac{z}{2}) - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2.3

Cancel the common factor of $2$ .

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Step 4.4.1.2.3.1

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

$x = \frac{4 - 1 (\frac{1}{2}) - 2 \frac{- z}{2} - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2.3.2

Factor $2$ out of $- 2$ .

$x = \frac{4 - 1 (\frac{1}{2}) + 2 (- 1) (\frac{- z}{2}) - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2.3.3

Cancel the common factor.

$x = \frac{4 - 1 (\frac{1}{2}) + 2 \cdot (- 1 \frac{- z}{2}) - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2.3.4

Rewrite the expression.

$x = \frac{4 - 1 (\frac{1}{2}) - 1 (- z) - 2 z}{3}$

$- 6 - 2 z = - 9$

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

$x = \frac{4 - 1 (\frac{1}{2}) - 1 (- z) - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2.4

Multiply $- 1$ by $- 1$ .

$x = \frac{4 - 1 (\frac{1}{2}) + 1 z - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2.5

Multiply $z$ by $1$ .

$x = \frac{4 - 1 (\frac{1}{2}) + z - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.2.6

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$x = \frac{4 - \frac{1}{2} + z - 2 z}{3}$

$- 6 - 2 z = - 9$

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

$x = \frac{4 - \frac{1}{2} + z - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{4 \cdot \frac{2}{2} - \frac{1}{2} + z - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.4

Combine $4$ and $\frac{2}{2}$ .

$x = \frac{\frac{4 \cdot 2}{2} - \frac{1}{2} + z - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{4 \cdot 2 - 1}{2} + z - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.6

Simplify the numerator.

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Step 4.4.1.6.1

Multiply $4$ by $2$ .

$x = \frac{\frac{8 - 1}{2} + z - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.6.2

Subtract $1$ from $8$ .

$x = \frac{\frac{7}{2} + z - 2 z}{3}$

$- 6 - 2 z = - 9$

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

$x = \frac{\frac{7}{2} + z - 2 z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.7

Subtract $2 z$ from $z$ .

$x = \frac{\frac{7}{2} - z}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.8

Simplify the numerator.

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Step 4.4.1.8.1

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

$x = \frac{\frac{7}{2} - z \cdot \frac{2}{2}}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.8.2

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

$x = \frac{\frac{7}{2} + \frac{- z \cdot 2}{2}}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.8.3

Combine the numerators over the common denominator.

$x = \frac{\frac{7 - z \cdot 2}{2}}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.8.4

Multiply $2$ by $- 1$ .

$x = \frac{\frac{7 - 2 z}{2}}{3}$

$- 6 - 2 z = - 9$

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

$x = \frac{\frac{7 - 2 z}{2}}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.9

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 - 2 z}{2} \cdot \frac{1}{3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.10

Multiply $\frac{7 - 2 z}{2} \cdot \frac{1}{3}$ .

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Step 4.4.1.10.1

Multiply $\frac{7 - 2 z}{2}$ by $\frac{1}{3}$ .

$x = \frac{7 - 2 z}{2 \cdot 3}$

$- 6 - 2 z = - 9$

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

Step 4.4.1.10.2

Multiply $2$ by $3$ .

$x = \frac{7 - 2 z}{6}$

$- 6 - 2 z = - 9$

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

$x = \frac{7 - 2 z}{6}$

$- 6 - 2 z = - 9$

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

$x = \frac{7 - 2 z}{6}$

$- 6 - 2 z = - 9$

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

$x = \frac{7 - 2 z}{6}$

$- 6 - 2 z = - 9$

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

$x = \frac{7 - 2 z}{6}$

$- 6 - 2 z = - 9$

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

Step 5

Solve for $z$ in $- 6 - 2 z = - 9$ .

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Step 5.1

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

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Step 5.1.1

Add $6$ to both sides of the equation.

$- 2 z = - 9 + 6$

$x = \frac{7 - 2 z}{6}$

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

Step 5.1.2

Add $- 9$ and $6$ .

$- 2 z = - 3$

$x = \frac{7 - 2 z}{6}$

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

$- 2 z = - 3$

$x = \frac{7 - 2 z}{6}$

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

Step 5.2

Divide each term in $- 2 z = - 3$ by $- 2$ and simplify.

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Step 5.2.1

Divide each term in $- 2 z = - 3$ by $- 2$ .

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

$x = \frac{7 - 2 z}{6}$

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

$x = \frac{7 - 2 z}{6}$

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

Step 5.2.2.1.2

Divide $z$ by $1$ .

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

$x = \frac{7 - 2 z}{6}$

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

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

$x = \frac{7 - 2 z}{6}$

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

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

$x = \frac{7 - 2 z}{6}$

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Dividing two negative values results in a positive value.

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

$x = \frac{7 - 2 z}{6}$

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

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

$x = \frac{7 - 2 z}{6}$

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

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

$x = \frac{7 - 2 z}{6}$

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

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

$x = \frac{7 - 2 z}{6}$

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

Step 6

Replace all occurrences of $z$ with $\frac{3}{2}$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = \frac{7 - 2 z}{6}$ with $\frac{3}{2}$ .

$x = \frac{7 - 2 (\frac{3}{2})}{6}$

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

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

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{7 - 2 (\frac{3}{2})}{6}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1.1

Factor $2$ out of $- 2$ .

$x = \frac{7 + 2 (- 1) (\frac{3}{2})}{6}$

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

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

Step 6.2.1.1.1.2

Cancel the common factor.

$x = \frac{7 + 2 \cdot (- 1 (\frac{3}{2}))}{6}$

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

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

Step 6.2.1.1.1.3

Rewrite the expression.

$x = \frac{7 - 1 \cdot 3}{6}$

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

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

$x = \frac{7 - 1 \cdot 3}{6}$

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

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

Step 6.2.1.1.2

Multiply $- 1$ by $3$ .

$x = \frac{7 - 3}{6}$

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

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

Step 6.2.1.1.3

Subtract $3$ from $7$ .

$x = \frac{4}{6}$

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

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

$x = \frac{4}{6}$

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

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

Step 6.2.1.2

Cancel the common factor of $4$ and $6$ .

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Step 6.2.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{6}$

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

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

Step 6.2.1.2.2

Cancel the common factors.

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Step 6.2.1.2.2.1

Factor $2$ out of $6$ .

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

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

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

Step 6.2.1.2.2.2

Cancel the common factor.

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

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

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

Step 6.2.1.2.2.3

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Step 6.3

Replace all occurrences of $z$ in $y = \frac{1}{4} - \frac{z}{2}$ with $\frac{3}{2}$ .

$y = \frac{1}{4} - \frac{\frac{3}{2}}{2}$

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

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

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $\frac{1}{4} - \frac{\frac{3}{2}}{2}$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{1}{4} - (\frac{3}{2} \cdot \frac{1}{2})$

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

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

Step 6.4.1.1.2

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

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Step 6.4.1.1.2.1

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$y = \frac{1}{4} - \frac{3}{2 \cdot 2}$

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

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

Step 6.4.1.1.2.2

Multiply $2$ by $2$ .

$y = \frac{1}{4} - \frac{3}{4}$

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

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

$y = \frac{1}{4} - \frac{3}{4}$

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

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

$y = \frac{1}{4} - \frac{3}{4}$

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

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

Step 6.4.1.2

Simplify terms.

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Step 6.4.1.2.1

Combine the numerators over the common denominator.

$y = \frac{1 - 3}{4}$

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

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

Step 6.4.1.2.2

Subtract $3$ from $1$ .

$y = \frac{- 2}{4}$

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

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

Step 6.4.1.2.3

Cancel the common factor of $- 2$ and $4$ .

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Step 6.4.1.2.3.1

Factor $2$ out of $- 2$ .

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

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

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

Step 6.4.1.2.3.2

Cancel the common factors.

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Step 6.4.1.2.3.2.1

Factor $2$ out of $4$ .

$y = \frac{2 \cdot - 1}{2 \cdot 2}$

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

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

Step 6.4.1.2.3.2.2

Cancel the common factor.

$y = \frac{2 \cdot - 1}{2 \cdot 2}$

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

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

Step 6.4.1.2.3.2.3

Rewrite the expression.

$y = \frac{- 1}{2}$

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

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

$y = \frac{- 1}{2}$

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

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

$y = \frac{- 1}{2}$

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

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

Step 6.4.1.2.4

Move the negative in front of the fraction.

$y = - \frac{1}{2}$

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

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

$y = - \frac{1}{2}$

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

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

$y = - \frac{1}{2}$

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

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

$y = - \frac{1}{2}$

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

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

$y = - \frac{1}{2}$

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

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

Step 7

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

$(\frac{2}{3}, - \frac{1}{2}, \frac{3}{2})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{2}{3}, - \frac{1}{2}, \frac{3}{2})$

Equation Form:

$x = \frac{2}{3}, y = - \frac{1}{2}, z = \frac{3}{2}$