Precalculus Examples

$2 x + 2 y - z = 7$ , $- x - 2 y + 2 x = - 6$ , $- 2 x - 4 y + 3 z = - 9$

Step 1

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

$x = - 6 + 2 y$

$2 x + 2 y - z = 7$

$- 2 x - 4 y + 3 z = - 9$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $2 x + 2 y - z = 7$ with $- 6 + 2 y$ .

$2 (- 6 + 2 y) + 2 y - z = 7$

$x = - 6 + 2 y$

$- 2 x - 4 y + 3 z = - 9$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- 6 + 2 y) + 2 y - z$ .

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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.

$2 \cdot - 6 + 2 (2 y) + 2 y - z = 7$

$x = - 6 + 2 y$

$- 2 x - 4 y + 3 z = - 9$

Step 2.2.1.1.2

Multiply $2$ by $- 6$ .

$- 12 + 2 (2 y) + 2 y - z = 7$

$x = - 6 + 2 y$

$- 2 x - 4 y + 3 z = - 9$

Step 2.2.1.1.3

Multiply $2$ by $2$ .

$- 12 + 4 y + 2 y - z = 7$

$x = - 6 + 2 y$

$- 2 x - 4 y + 3 z = - 9$

$- 12 + 4 y + 2 y - z = 7$

$x = - 6 + 2 y$

$- 2 x - 4 y + 3 z = - 9$

Step 2.2.1.2

Add $4 y$ and $2 y$ .

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

$- 2 x - 4 y + 3 z = - 9$

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

$- 2 x - 4 y + 3 z = - 9$

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

$- 2 x - 4 y + 3 z = - 9$

Step 2.3

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

$- 2 (- 6 + 2 y) - 4 y + 3 z = - 9$

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 2 (- 6 + 2 y) - 4 y + 3 z$ .

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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.

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

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

Step 2.4.1.1.2

Multiply $- 2$ by $- 6$ .

$12 - 2 (2 y) - 4 y + 3 z = - 9$

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

Step 2.4.1.1.3

Multiply $2$ by $- 2$ .

$12 - 4 y - 4 y + 3 z = - 9$

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

$12 - 4 y - 4 y + 3 z = - 9$

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

Step 2.4.1.2

Subtract $4 y$ from $- 4 y$ .

$12 - 8 y + 3 z = - 9$

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

$12 - 8 y + 3 z = - 9$

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

$12 - 8 y + 3 z = - 9$

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

$12 - 8 y + 3 z = - 9$

$- 12 + 6 y - z = 7$

$x = - 6 + 2 y$

Step 3

Reorder $- 6$ and $2 y$ .

$x = 2 y - 6$

$12 - 8 y + 3 z = - 9$

$- 12 + 6 y - z = 7$

Step 4

Solve for $y$ in $12 - 8 y + 3 z = - 9$ .

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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 $12$ from both sides of the equation.

$- 8 y + 3 z = - 9 - 12$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

Step 4.1.2

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

$- 8 y = - 9 - 12 - 3 z$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

Step 4.1.3

Subtract $12$ from $- 9$ .

$- 8 y = - 21 - 3 z$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

$- 8 y = - 21 - 3 z$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

Step 4.2

Divide each term in $- 8 y = - 21 - 3 z$ by $- 8$ and simplify.

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Step 4.2.1

Divide each term in $- 8 y = - 21 - 3 z$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{- 21}{- 8} + \frac{- 3 z}{- 8}$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 8$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{- 21}{- 8} + \frac{- 3 z}{- 8}$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

Step 4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 21}{- 8} + \frac{- 3 z}{- 8}$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

$y = \frac{- 21}{- 8} + \frac{- 3 z}{- 8}$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

$y = \frac{- 21}{- 8} + \frac{- 3 z}{- 8}$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

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

Dividing two negative values results in a positive value.

$y = \frac{21}{8} + \frac{- 3 z}{- 8}$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

Step 4.2.3.1.2

Dividing two negative values results in a positive value.

$y = \frac{21}{8} + \frac{3 z}{8}$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

$y = \frac{21}{8} + \frac{3 z}{8}$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

$y = \frac{21}{8} + \frac{3 z}{8}$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

$y = \frac{21}{8} + \frac{3 z}{8}$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

$y = \frac{21}{8} + \frac{3 z}{8}$

$x = 2 y - 6$

$- 12 + 6 y - z = 7$

Step 5

Replace all occurrences of $y$ with $\frac{21}{8} + \frac{3 z}{8}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 2 y - 6$ with $\frac{21}{8} + \frac{3 z}{8}$ .

$x = 2 (\frac{21}{8} + \frac{3 z}{8}) - 6$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $2 (\frac{21}{8} + \frac{3 z}{8}) - 6$ .

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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.

$x = 2 (\frac{21}{8}) + 2 (\frac{3 z}{8}) - 6$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

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 $8$ .

$x = 2 (\frac{21}{2 (4)}) + 2 (\frac{3 z}{8}) - 6$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2.1.1.2.2

Cancel the common factor.

$x = 2 (\frac{21}{2 \cdot 4}) + 2 (\frac{3 z}{8}) - 6$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2.1.1.2.3

Rewrite the expression.

$x = \frac{21}{4} + 2 (\frac{3 z}{8}) - 6$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

$x = \frac{21}{4} + 2 (\frac{3 z}{8}) - 6$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2.1.1.3

Cancel the common factor of $2$ .

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Step 5.2.1.1.3.1

Factor $2$ out of $8$ .

$x = \frac{21}{4} + 2 (\frac{3 z}{2 (4)}) - 6$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2.1.1.3.2

Cancel the common factor.

$x = \frac{21}{4} + 2 (\frac{3 z}{2 \cdot 4}) - 6$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2.1.1.3.3

Rewrite the expression.

$x = \frac{21}{4} + \frac{3 z}{4} - 6$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

$x = \frac{21}{4} + \frac{3 z}{4} - 6$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

$x = \frac{21}{4} + \frac{3 z}{4} - 6$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2.1.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{3 z}{4} + \frac{21}{4} - 6 \cdot \frac{4}{4}$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2.1.3

Combine $- 6$ and $\frac{4}{4}$ .

$x = \frac{3 z}{4} + \frac{21}{4} + \frac{- 6 \cdot 4}{4}$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x = \frac{3 z}{4} + \frac{21 - 6 \cdot 4}{4}$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $- 6$ by $4$ .

$x = \frac{3 z}{4} + \frac{21 - 24}{4}$

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2.1.5.2

Subtract $24$ from $21$ .

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.2.1.6

Move the negative in front of the fraction.

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 6 y - z = 7$

Step 5.3

Replace all occurrences of $y$ in $- 12 + 6 y - z = 7$ with $\frac{21}{8} + \frac{3 z}{8}$ .

$- 12 + 6 (\frac{21}{8} + \frac{3 z}{8}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $- 12 + 6 (\frac{21}{8} + \frac{3 z}{8}) - z$ .

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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.

$- 12 + 6 (\frac{21}{8}) + 6 (\frac{3 z}{8}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.2

Cancel the common factor of $2$ .

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Step 5.4.1.1.2.1

Factor $2$ out of $6$ .

$- 12 + 2 (3) (\frac{21}{8}) + 6 (\frac{3 z}{8}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.2.2

Factor $2$ out of $8$ .

$- 12 + 2 \cdot (3 (\frac{21}{2 \cdot 4})) + 6 (\frac{3 z}{8}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.2.3

Cancel the common factor.

$- 12 + 2 \cdot (3 (\frac{21}{2 \cdot 4})) + 6 (\frac{3 z}{8}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.2.4

Rewrite the expression.

$- 12 + 3 (\frac{21}{4}) + 6 (\frac{3 z}{8}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + 3 (\frac{21}{4}) + 6 (\frac{3 z}{8}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.3

Combine $3$ and $\frac{21}{4}$ .

$- 12 + \frac{3 \cdot 21}{4} + 6 (\frac{3 z}{8}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.4

Multiply $3$ by $21$ .

$- 12 + \frac{63}{4} + 6 (\frac{3 z}{8}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.5

Cancel the common factor of $2$ .

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Step 5.4.1.1.5.1

Factor $2$ out of $6$ .

$- 12 + \frac{63}{4} + 2 (3) (\frac{3 z}{8}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.5.2

Factor $2$ out of $8$ .

$- 12 + \frac{63}{4} + 2 \cdot (3 (\frac{3 z}{2 \cdot 4})) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.5.3

Cancel the common factor.

$- 12 + \frac{63}{4} + 2 \cdot (3 (\frac{3 z}{2 \cdot 4})) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.5.4

Rewrite the expression.

$- 12 + \frac{63}{4} + 3 (\frac{3 z}{4}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + \frac{63}{4} + 3 (\frac{3 z}{4}) - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.6

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

$- 12 + \frac{63}{4} + \frac{3 (3 z)}{4} - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.1.7

Multiply $3$ by $3$ .

$- 12 + \frac{63}{4} + \frac{9 z}{4} - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$- 12 + \frac{63}{4} + \frac{9 z}{4} - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.2

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 12 \cdot \frac{4}{4} + \frac{63}{4} + \frac{9 z}{4} - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.3

Combine $- 12$ and $\frac{4}{4}$ .

$\frac{- 12 \cdot 4}{4} + \frac{63}{4} + \frac{9 z}{4} - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.4

Combine the numerators over the common denominator.

$\frac{- 12 \cdot 4 + 63}{4} + \frac{9 z}{4} - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.5

Simplify the numerator.

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Step 5.4.1.5.1

Multiply $- 12$ by $4$ .

$\frac{- 48 + 63}{4} + \frac{9 z}{4} - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.5.2

Add $- 48$ and $63$ .

$\frac{15}{4} + \frac{9 z}{4} - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$\frac{15}{4} + \frac{9 z}{4} - z = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.6

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

$\frac{15}{4} + \frac{9 z}{4} - z \cdot \frac{4}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.7

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

$\frac{15}{4} + \frac{9 z}{4} + \frac{- z \cdot 4}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.8

Combine the numerators over the common denominator.

$\frac{15}{4} + \frac{9 z - z \cdot 4}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.9

Combine the numerators over the common denominator.

$\frac{15 + 9 z - z \cdot 4}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.10

Multiply $4$ by $- 1$ .

$\frac{15 + 9 z - 4 z}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.11

Subtract $4 z$ from $9 z$ .

$\frac{15 + 5 z}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.12

Factor $5$ out of $15 + 5 z$ .

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Step 5.4.1.12.1

Factor $5$ out of $15$ .

$\frac{5 \cdot 3 + 5 z}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 5.4.1.12.2

Factor $5$ out of $5 \cdot 3 + 5 z$ .

$\frac{5 (3 + z)}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$\frac{5 (3 + z)}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$\frac{5 (3 + z)}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$\frac{5 (3 + z)}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$\frac{5 (3 + z)}{4} = 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6

Solve for $z$ in $\frac{5 (3 + z)}{4} = 7$ .

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Step 6.1

Multiply both sides of the equation by $\frac{4}{5}$ .

$\frac{4}{5} \cdot \frac{5 (3 + z)}{4} = \frac{4}{5} \cdot 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.2

Simplify both sides of the equation.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify $\frac{4}{5} \cdot \frac{5 (3 + z)}{4}$ .

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Step 6.2.1.1.1

Cancel the common factor of $4$ .

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Step 6.2.1.1.1.1

Cancel the common factor.

$\frac{4}{5} \cdot \frac{5 (3 + z)}{4} = \frac{4}{5} \cdot 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.2.1.1.1.2

Rewrite the expression.

$\frac{1}{5} \cdot (5 (3 + z)) = \frac{4}{5} \cdot 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$\frac{1}{5} \cdot (5 (3 + z)) = \frac{4}{5} \cdot 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.2.1.1.2

Cancel the common factor of $5$ .

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Step 6.2.1.1.2.1

Cancel the common factor.

$\frac{1}{5} \cdot (5 (3 + z)) = \frac{4}{5} \cdot 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.2.1.1.2.2

Rewrite the expression.

$3 + z = \frac{4}{5} \cdot 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$3 + z = \frac{4}{5} \cdot 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$3 + z = \frac{4}{5} \cdot 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$3 + z = \frac{4}{5} \cdot 7$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Multiply $\frac{4}{5} \cdot 7$ .

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Step 6.2.2.1.1

Combine $\frac{4}{5}$ and $7$ .

$3 + z = \frac{4 \cdot 7}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.2.2.1.2

Multiply $4$ by $7$ .

$3 + z = \frac{28}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$3 + z = \frac{28}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$3 + z = \frac{28}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$3 + z = \frac{28}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.3

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

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Step 6.3.1

Subtract $3$ from both sides of the equation.

$z = \frac{28}{5} - 3$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.3.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$z = \frac{28}{5} - 3 \cdot \frac{5}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.3.3

Combine $- 3$ and $\frac{5}{5}$ .

$z = \frac{28}{5} + \frac{- 3 \cdot 5}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.3.4

Combine the numerators over the common denominator.

$z = \frac{28 - 3 \cdot 5}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.3.5

Simplify the numerator.

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Step 6.3.5.1

Multiply $- 3$ by $5$ .

$z = \frac{28 - 15}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 6.3.5.2

Subtract $15$ from $28$ .

$z = \frac{13}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$z = \frac{13}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$z = \frac{13}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

$z = \frac{13}{5}$

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

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7

Replace all occurrences of $z$ with $\frac{13}{5}$ in each equation.

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Step 7.1

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

$x = \frac{3 (\frac{13}{5})}{4} - \frac{3}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{3 (\frac{13}{5})}{4} - \frac{3}{4}$ .

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Step 7.2.1.1

Combine the numerators over the common denominator.

$x = \frac{3 (\frac{13}{5}) - 3}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2.1.2

Multiply $3 (\frac{13}{5})$ .

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Step 7.2.1.2.1

Combine $3$ and $\frac{13}{5}$ .

$x = \frac{\frac{3 \cdot 13}{5} - 3}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2.1.2.2

Multiply $3$ by $13$ .

$x = \frac{\frac{39}{5} - 3}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

$x = \frac{\frac{39}{5} - 3}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2.1.3

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = \frac{\frac{39}{5} - 3 \cdot \frac{5}{5}}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2.1.4

Combine $- 3$ and $\frac{5}{5}$ .

$x = \frac{\frac{39}{5} + \frac{- 3 \cdot 5}{5}}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{39 - 3 \cdot 5}{5}}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2.1.6

Simplify the numerator.

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Step 7.2.1.6.1

Multiply $- 3$ by $5$ .

$x = \frac{\frac{39 - 15}{5}}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2.1.6.2

Subtract $15$ from $39$ .

$x = \frac{\frac{24}{5}}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

$x = \frac{\frac{24}{5}}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2.1.7

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{24}{5} \cdot \frac{1}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2.1.8

Cancel the common factor of $4$ .

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Step 7.2.1.8.1

Factor $4$ out of $24$ .

$x = \frac{4 (6)}{5} \cdot \frac{1}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2.1.8.2

Cancel the common factor.

$x = \frac{4 \cdot 6}{5} \cdot \frac{1}{4}$

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.2.1.8.3

Rewrite the expression.

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

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

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

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

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

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

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

$z = \frac{13}{5}$

$y = \frac{21}{8} + \frac{3 z}{8}$

Step 7.3

Replace all occurrences of $z$ in $y = \frac{21}{8} + \frac{3 z}{8}$ with $\frac{13}{5}$ .

$y = \frac{21}{8} + \frac{3 (\frac{13}{5})}{8}$

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

$z = \frac{13}{5}$

Step 7.4

Simplify the right side.

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Step 7.4.1

Simplify $\frac{21}{8} + \frac{3 (\frac{13}{5})}{8}$ .

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Step 7.4.1.1

Combine the numerators over the common denominator.

$y = \frac{21 + 3 (\frac{13}{5})}{8}$

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

$z = \frac{13}{5}$

Step 7.4.1.2

Multiply $3 (\frac{13}{5})$ .

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Step 7.4.1.2.1

Combine $3$ and $\frac{13}{5}$ .

$y = \frac{21 + \frac{3 \cdot 13}{5}}{8}$

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

$z = \frac{13}{5}$

Step 7.4.1.2.2

Multiply $3$ by $13$ .

$y = \frac{21 + \frac{39}{5}}{8}$

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

$z = \frac{13}{5}$

$y = \frac{21 + \frac{39}{5}}{8}$

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

$z = \frac{13}{5}$

Step 7.4.1.3

To write $21$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{21 \cdot \frac{5}{5} + \frac{39}{5}}{8}$

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

$z = \frac{13}{5}$

Step 7.4.1.4

Combine $21$ and $\frac{5}{5}$ .

$y = \frac{\frac{21 \cdot 5}{5} + \frac{39}{5}}{8}$

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

$z = \frac{13}{5}$

Step 7.4.1.5

Combine the numerators over the common denominator.

$y = \frac{\frac{21 \cdot 5 + 39}{5}}{8}$

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

$z = \frac{13}{5}$

Step 7.4.1.6

Simplify the numerator.

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Step 7.4.1.6.1

Multiply $21$ by $5$ .

$y = \frac{\frac{105 + 39}{5}}{8}$

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

$z = \frac{13}{5}$

Step 7.4.1.6.2

Add $105$ and $39$ .

$y = \frac{\frac{144}{5}}{8}$

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

$z = \frac{13}{5}$

$y = \frac{\frac{144}{5}}{8}$

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

$z = \frac{13}{5}$

Step 7.4.1.7

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{144}{5} \cdot \frac{1}{8}$

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

$z = \frac{13}{5}$

Step 7.4.1.8

Cancel the common factor of $8$ .

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Step 7.4.1.8.1

Factor $8$ out of $144$ .

$y = \frac{8 (18)}{5} \cdot \frac{1}{8}$

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

$z = \frac{13}{5}$

Step 7.4.1.8.2

Cancel the common factor.

$y = \frac{8 \cdot 18}{5} \cdot \frac{1}{8}$

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

$z = \frac{13}{5}$

Step 7.4.1.8.3

Rewrite the expression.

$y = \frac{18}{5}$

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

$z = \frac{13}{5}$

$y = \frac{18}{5}$

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

$z = \frac{13}{5}$

$y = \frac{18}{5}$

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

$z = \frac{13}{5}$

$y = \frac{18}{5}$

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

$z = \frac{13}{5}$

$y = \frac{18}{5}$

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

$z = \frac{13}{5}$

Step 8

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

$(\frac{6}{5}, \frac{18}{5}, \frac{13}{5})$

Step 9

The result can be shown in multiple forms.

Point Form:

$(\frac{6}{5}, \frac{18}{5}, \frac{13}{5})$

Equation Form:

$x = \frac{6}{5}, y = \frac{18}{5}, z = \frac{13}{5}$