Precalculus Examples

$4 x - y + 5 z = - 6$ , $3 x + 3 y - 4 z = 30$ , $6 x + 2 y - 3 z = 33$

Step 1

Solve for $x$ in $4 x - y + 5 z = - 6$ .

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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 $y$ to both sides of the equation.

$4 x + 5 z = - 6 + y$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

Step 1.1.2

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

$4 x = - 6 + y - 5 z$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

$4 x = - 6 + y - 5 z$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

Step 1.2

Divide each term in $4 x = - 6 + y - 5 z$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = - 6 + y - 5 z$ by $4$ .

$\frac{4 x}{4} = \frac{- 6}{4} + \frac{y}{4} + \frac{- 5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 6}{4} + \frac{y}{4} + \frac{- 5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{4} + \frac{y}{4} + \frac{- 5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

$x = \frac{- 6}{4} + \frac{y}{4} + \frac{- 5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

$x = \frac{- 6}{4} + \frac{y}{4} + \frac{- 5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

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

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

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Step 1.2.3.1.1.1

Factor $2$ out of $- 6$ .

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

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot - 3}{2 \cdot 2} + \frac{y}{4} + \frac{- 5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 3}{2 \cdot 2} + \frac{y}{4} + \frac{- 5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 3}{2} + \frac{y}{4} + \frac{- 5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

$x = \frac{- 3}{2} + \frac{y}{4} + \frac{- 5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

$x = \frac{- 3}{2} + \frac{y}{4} + \frac{- 5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{3}{2} + \frac{y}{4} + \frac{- 5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$3 x + 3 y - 4 z = 30$

$6 x + 2 y - 3 z = 33$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $3 x + 3 y - 4 z = 30$ with $- \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$ .

$3 (- \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}) + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 (- \frac{3}{2}) + 3 (\frac{y}{4}) + 3 (- \frac{5 z}{4}) + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

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

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Step 2.2.1.1.2.1.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{3}{2}) + 3 (\frac{y}{4}) + 3 (- \frac{5 z}{4}) + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.1.2.1.2

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

$\frac{- 3 \cdot 3}{2} + 3 (\frac{y}{4}) + 3 (- \frac{5 z}{4}) + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.1.2.1.3

Multiply $- 3$ by $3$ .

$\frac{- 9}{2} + 3 (\frac{y}{4}) + 3 (- \frac{5 z}{4}) + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$\frac{- 9}{2} + 3 (\frac{y}{4}) + 3 (- \frac{5 z}{4}) + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.1.2.2

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

$\frac{- 9}{2} + \frac{3 y}{4} + 3 (- \frac{5 z}{4}) + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.1.2.3

Multiply $3 (- \frac{5 z}{4})$ .

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Step 2.2.1.1.2.3.1

Multiply $- 1$ by $3$ .

$\frac{- 9}{2} + \frac{3 y}{4} - 3 \frac{5 z}{4} + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.1.2.3.2

Combine $- 3$ and $\frac{5 z}{4}$ .

$\frac{- 9}{2} + \frac{3 y}{4} + \frac{- 3 (5 z)}{4} + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.1.2.3.3

Multiply $5$ by $- 3$ .

$\frac{- 9}{2} + \frac{3 y}{4} + \frac{- 15 z}{4} + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$\frac{- 9}{2} + \frac{3 y}{4} + \frac{- 15 z}{4} + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$\frac{- 9}{2} + \frac{3 y}{4} + \frac{- 15 z}{4} + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.1.3

Simplify each term.

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Step 2.2.1.1.3.1

Move the negative in front of the fraction.

$- \frac{9}{2} + \frac{3 y}{4} + \frac{- 15 z}{4} + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.1.3.2

Move the negative in front of the fraction.

$- \frac{9}{2} + \frac{3 y}{4} - \frac{15 z}{4} + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$- \frac{9}{2} + \frac{3 y}{4} - \frac{15 z}{4} + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$- \frac{9}{2} + \frac{3 y}{4} - \frac{15 z}{4} + 3 y - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.2

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

$- \frac{9}{2} - \frac{15 z}{4} + \frac{3 y}{4} + 3 y \cdot \frac{4}{4} - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$- \frac{9}{2} - \frac{15 z}{4} + \frac{3 y}{4} + \frac{3 y \cdot 4}{4} - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$- \frac{9}{2} - \frac{15 z}{4} + \frac{3 y + 3 y \cdot 4}{4} - 4 z = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.3.3

Combine the numerators over the common denominator.

$- 4 z + \frac{- 9}{2} + \frac{- 15 z + 3 y + 3 y \cdot 4}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.3.4

Multiply $4$ by $3$ .

$- 4 z + \frac{- 9}{2} + \frac{- 15 z + 3 y + 12 y}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.3.5

Add $3 y$ and $12 y$ .

$- 4 z + \frac{- 9}{2} + \frac{- 15 z + 15 y}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$- 4 z + \frac{- 9}{2} + \frac{- 15 z + 15 y}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Move the negative in front of the fraction.

$- 4 z - \frac{9}{2} + \frac{- 15 z + 15 y}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.4.2

Factor $15$ out of $- 15 z + 15 y$ .

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Step 2.2.1.4.2.1

Factor $15$ out of $- 15 z$ .

$- 4 z - \frac{9}{2} + \frac{15 (- z) + 15 y}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.4.2.2

Factor $15$ out of $15 y$ .

$- 4 z - \frac{9}{2} + \frac{15 (- z) + 15 (y)}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.4.2.3

Factor $15$ out of $15 (- z) + 15 (y)$ .

$- 4 z - \frac{9}{2} + \frac{15 (- z + y)}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$- 4 z - \frac{9}{2} + \frac{15 (- z + y)}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$- 4 z - \frac{9}{2} + \frac{15 (- z + y)}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.5

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

$- 4 z \cdot \frac{4}{4} + \frac{15 (- z + y)}{4} - \frac{9}{2} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.6

Simplify terms.

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Step 2.2.1.6.1

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

$\frac{- 4 z \cdot 4}{4} + \frac{15 (- z + y)}{4} - \frac{9}{2} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.6.2

Combine the numerators over the common denominator.

$\frac{- 4 z \cdot 4 + 15 (- z + y)}{4} - \frac{9}{2} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$\frac{- 4 z \cdot 4 + 15 (- z + y)}{4} - \frac{9}{2} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.7

Simplify the numerator.

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Step 2.2.1.7.1

Multiply $4$ by $- 4$ .

$\frac{- 16 z + 15 (- z + y)}{4} - \frac{9}{2} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.7.2

Apply the distributive property.

$\frac{- 16 z + 15 (- z) + 15 y}{4} - \frac{9}{2} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.7.3

Multiply $- 1$ by $15$ .

$\frac{- 16 z - 15 z + 15 y}{4} - \frac{9}{2} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.7.4

Subtract $15 z$ from $- 16 z$ .

$\frac{- 31 z + 15 y}{4} - \frac{9}{2} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$\frac{- 31 z + 15 y}{4} - \frac{9}{2} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.8

To write $- \frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- 31 z + 15 y}{4} - \frac{9}{2} \cdot \frac{2}{2} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.9

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.9.1

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$\frac{- 31 z + 15 y}{4} - \frac{9 \cdot 2}{2 \cdot 2} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.9.2

Multiply $2$ by $2$ .

$\frac{- 31 z + 15 y}{4} - \frac{9 \cdot 2}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$\frac{- 31 z + 15 y}{4} - \frac{9 \cdot 2}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.10

Combine the numerators over the common denominator.

$\frac{- 31 z + 15 y - 9 \cdot 2}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.11

Multiply $- 9$ by $2$ .

$\frac{- 31 z + 15 y - 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.12

Factor $- 1$ out of $- 31 z$ .

$\frac{- (31 z) + 15 y - 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.13

Factor $- 1$ out of $15 y$ .

$\frac{- (31 z) - (- 15 y) - 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.14

Factor $- 1$ out of $- (31 z) - (- 15 y)$ .

$\frac{- (31 z - 15 y) - 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.15

Rewrite $- 18$ as $- 1 (18)$ .

$\frac{- (31 z - 15 y) - 1 \cdot 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.16

Factor $- 1$ out of $- (31 z - 15 y) - 1 (18)$ .

$\frac{- (31 z - 15 y + 18)}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.17

Simplify the expression.

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Step 2.2.1.17.1

Rewrite $- (31 z - 15 y + 18)$ as $- 1 (31 z - 15 y + 18)$ .

$\frac{- 1 (31 z - 15 y + 18)}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.2.1.17.2

Move the negative in front of the fraction.

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$6 x + 2 y - 3 z = 33$

Step 2.3

Replace all occurrences of $x$ in $6 x + 2 y - 3 z = 33$ with $- \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$ .

$6 (- \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $6 (- \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}) + 2 y - 3 z$ .

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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.

$6 (- \frac{3}{2}) + 6 (\frac{y}{4}) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

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

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

$6 (\frac{- 3}{2}) + 6 (\frac{y}{4}) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.1.2

Factor $2$ out of $6$ .

$2 (3) (\frac{- 3}{2}) + 6 (\frac{y}{4}) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.1.3

Cancel the common factor.

$2 \cdot (3 (\frac{- 3}{2})) + 6 (\frac{y}{4}) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.1.4

Rewrite the expression.

$3 \cdot - 3 + 6 (\frac{y}{4}) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$3 \cdot - 3 + 6 (\frac{y}{4}) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.2

Multiply $3$ by $- 3$ .

$- 9 + 6 (\frac{y}{4}) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

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 $6$ .

$- 9 + 2 (3) (\frac{y}{4}) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.3.2

Factor $2$ out of $4$ .

$- 9 + 2 \cdot (3 (\frac{y}{2 \cdot 2})) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.3.3

Cancel the common factor.

$- 9 + 2 \cdot (3 (\frac{y}{2 \cdot 2})) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.3.4

Rewrite the expression.

$- 9 + 3 (\frac{y}{2}) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- 9 + 3 (\frac{y}{2}) + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.4

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

$- 9 + \frac{3 y}{2} + 6 (- \frac{5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

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 $- \frac{5 z}{4}$ into the numerator.

$- 9 + \frac{3 y}{2} + 6 (\frac{- 5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.5.2

Factor $2$ out of $6$ .

$- 9 + \frac{3 y}{2} + 2 (3) (\frac{- 5 z}{4}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.5.3

Factor $2$ out of $4$ .

$- 9 + \frac{3 y}{2} + 2 \cdot (3 (\frac{- 5 z}{2 \cdot 2})) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.5.4

Cancel the common factor.

$- 9 + \frac{3 y}{2} + 2 \cdot (3 (\frac{- 5 z}{2 \cdot 2})) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.5.5

Rewrite the expression.

$- 9 + \frac{3 y}{2} + 3 (\frac{- 5 z}{2}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- 9 + \frac{3 y}{2} + 3 (\frac{- 5 z}{2}) + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.6

Combine $3$ and $\frac{- 5 z}{2}$ .

$- 9 + \frac{3 y}{2} + \frac{3 (- 5 z)}{2} + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.2.7

Multiply $- 5$ by $3$ .

$- 9 + \frac{3 y}{2} + \frac{- 15 z}{2} + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- 9 + \frac{3 y}{2} + \frac{- 15 z}{2} + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.1.3

Move the negative in front of the fraction.

$- 9 + \frac{3 y}{2} - \frac{15 z}{2} + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- 9 + \frac{3 y}{2} - \frac{15 z}{2} + 2 y - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.2

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

$- 9 - \frac{15 z}{2} + \frac{3 y}{2} + 2 y \cdot \frac{2}{2} - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.3

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

$- 9 - \frac{15 z}{2} + \frac{3 y}{2} + \frac{2 y \cdot 2}{2} - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.4

Combine the numerators over the common denominator.

$- 9 - \frac{15 z}{2} + \frac{3 y + 2 y \cdot 2}{2} - 3 z = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.5

Combine the numerators over the common denominator.

$- 9 - 3 z + \frac{- 15 z + 3 y + 2 y \cdot 2}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.6

Multiply $2$ by $2$ .

$- 9 - 3 z + \frac{- 15 z + 3 y + 4 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.7

Add $3 y$ and $4 y$ .

$- 9 - 3 z + \frac{- 15 z + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.8

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 3 z - 9 \cdot \frac{2}{2} + \frac{- 15 z + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.9

Combine $- 9$ and $\frac{2}{2}$ .

$- 3 z + \frac{- 9 \cdot 2}{2} + \frac{- 15 z + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.10

Combine the numerators over the common denominator.

$- 3 z + \frac{- 9 \cdot 2 - 15 z + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.11

Multiply $- 9$ by $2$ .

$- 3 z + \frac{- 18 - 15 z + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.12

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

$- 3 z \cdot \frac{2}{2} + \frac{- 18 - 15 z + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.13

Simplify terms.

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Step 2.4.1.13.1

Combine $- 3 z$ and $\frac{2}{2}$ .

$\frac{- 3 z \cdot 2}{2} + \frac{- 18 - 15 z + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.13.2

Combine the numerators over the common denominator.

$\frac{- 3 z \cdot 2 - 18 - 15 z + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$\frac{- 3 z \cdot 2 - 18 - 15 z + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.14

Simplify the numerator.

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Step 2.4.1.14.1

Multiply $2$ by $- 3$ .

$\frac{- 6 z - 18 - 15 z + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.14.2

Subtract $15 z$ from $- 6 z$ .

$\frac{- 21 z - 18 + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$\frac{- 21 z - 18 + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.15

Simplify with factoring out.

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Step 2.4.1.15.1

Factor $- 1$ out of $- 21 z$ .

$\frac{- (21 z) - 18 + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.15.2

Rewrite $- 18$ as $- 1 (18)$ .

$\frac{- (21 z) - 1 \cdot 18 + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.15.3

Factor $- 1$ out of $- (21 z) - 1 (18)$ .

$\frac{- (21 z + 18) + 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.15.4

Factor $- 1$ out of $7 y$ .

$\frac{- (21 z + 18) - (- 7 y)}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.15.5

Factor $- 1$ out of $- (21 z + 18) - (- 7 y)$ .

$\frac{- (21 z + 18 - 7 y)}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.15.6

Simplify the expression.

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Step 2.4.1.15.6.1

Rewrite $- (21 z + 18 - 7 y)$ as $- 1 (21 z + 18 - 7 y)$ .

$\frac{- 1 (21 z + 18 - 7 y)}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 2.4.1.15.6.2

Move the negative in front of the fraction.

$- \frac{21 z + 18 - 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- \frac{21 z + 18 - 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- \frac{21 z + 18 - 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- \frac{21 z + 18 - 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- \frac{21 z + 18 - 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- \frac{21 z + 18 - 7 y}{2} = 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3

Solve for $z$ in $- \frac{21 z + 18 - 7 y}{2} = 33$ .

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Step 3.1

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{21 z + 18 - 7 y}{2}) = - 2 \cdot 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- 2 (- \frac{21 z + 18 - 7 y}{2})$ .

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Step 3.2.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{21 z + 18 - 7 y}{2}$ into the numerator.

$- 2 \frac{- (21 z + 18 - 7 y)}{2} = - 2 \cdot 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.2.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- (21 z + 18 - 7 y)}{2}) = - 2 \cdot 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.2.1.1.1.3

Cancel the common factor.

$2 \cdot (- 1 \frac{- (21 z + 18 - 7 y)}{2}) = - 2 \cdot 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.2.1.1.1.4

Rewrite the expression.

$21 z + 18 - 7 y = - 2 \cdot 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$21 z + 18 - 7 y = - 2 \cdot 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (21 z + 18 - 7 y) = - 2 \cdot 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.2.1.1.2.2

Multiply $21 z + 18 - 7 y$ by $1$ .

$21 z + 18 - 7 y = - 2 \cdot 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$21 z + 18 - 7 y = - 2 \cdot 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$21 z + 18 - 7 y = - 2 \cdot 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$21 z + 18 - 7 y = - 2 \cdot 33$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 2$ by $33$ .

$21 z + 18 - 7 y = - 66$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$21 z + 18 - 7 y = - 66$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$21 z + 18 - 7 y = - 66$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.3

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

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Step 3.3.1

Subtract $18$ from both sides of the equation.

$21 z - 7 y = - 66 - 18$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.3.2

Add $7 y$ to both sides of the equation.

$21 z = - 66 - 18 + 7 y$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.3.3

Subtract $18$ from $- 66$ .

$21 z = - 84 + 7 y$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$21 z = - 84 + 7 y$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.4

Divide each term in $21 z = - 84 + 7 y$ by $21$ and simplify.

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Step 3.4.1

Divide each term in $21 z = - 84 + 7 y$ by $21$ .

$\frac{21 z}{21} = \frac{- 84}{21} + \frac{7 y}{21}$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $21$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{21 z}{21} = \frac{- 84}{21} + \frac{7 y}{21}$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.4.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 84}{21} + \frac{7 y}{21}$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$z = \frac{- 84}{21} + \frac{7 y}{21}$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$z = \frac{- 84}{21} + \frac{7 y}{21}$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify each term.

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Step 3.4.3.1.1

Divide $- 84$ by $21$ .

$z = - 4 + \frac{7 y}{21}$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.4.3.1.2

Cancel the common factor of $7$ and $21$ .

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Step 3.4.3.1.2.1

Factor $7$ out of $7 y$ .

$z = - 4 + \frac{7 (y)}{21}$

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.4.3.1.2.2

Cancel the common factors.

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Step 3.4.3.1.2.2.1

Factor $7$ out of $21$ .

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

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.4.3.1.2.2.2

Cancel the common factor.

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

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 3.4.3.1.2.2.3

Rewrite the expression.

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

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

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

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

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

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

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

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

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

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

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

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

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

$- \frac{31 z - 15 y + 18}{4} = 30$

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4

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

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Step 4.1

Replace all occurrences of $z$ in $- \frac{31 z - 15 y + 18}{4} = 30$ with $- 4 + \frac{y}{3}$ .

$- \frac{31 (- 4 + \frac{y}{3}) - 15 y + 18}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- \frac{31 (- 4 + \frac{y}{3}) - 15 y + 18}{4}$ .

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Step 4.2.1.1

Simplify the numerator.

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Step 4.2.1.1.1

Apply the distributive property.

$- \frac{31 \cdot - 4 + 31 (\frac{y}{3}) - 15 y + 18}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.2

Multiply $31$ by $- 4$ .

$- \frac{- 124 + 31 (\frac{y}{3}) - 15 y + 18}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.3

Combine $31$ and $\frac{y}{3}$ .

$- \frac{- 124 + \frac{31 y}{3} - 15 y + 18}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.4

Add $- 124$ and $18$ .

$- \frac{\frac{31 y}{3} - 15 y - 106}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.5

To write $- 15 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{\frac{31 y}{3} - 15 y \cdot \frac{3}{3} - 106}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.6

Combine $- 15 y$ and $\frac{3}{3}$ .

$- \frac{\frac{31 y}{3} + \frac{- 15 y \cdot 3}{3} - 106}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.7

Combine the numerators over the common denominator.

$- \frac{\frac{31 y - 15 y \cdot 3}{3} - 106}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.8

To write $- 106$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{\frac{31 y - 15 y \cdot 3}{3} - 106 \cdot \frac{3}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.9

Combine $- 106$ and $\frac{3}{3}$ .

$- \frac{\frac{31 y - 15 y \cdot 3}{3} + \frac{- 106 \cdot 3}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.10

Combine the numerators over the common denominator.

$- \frac{\frac{31 y - 15 y \cdot 3 - 106 \cdot 3}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.11

Reorder terms.

$- \frac{\frac{- 15 \cdot (3 y) + 31 y - 106 \cdot 3}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.12

Rewrite $\frac{- 15 \cdot 3 y + 31 y - 106 \cdot 3}{3}$ in a factored form.

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Step 4.2.1.1.12.1

Multiply $- 15$ by $3$ .

$- \frac{\frac{- 45 y + 31 y - 106 \cdot 3}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.12.2

Multiply $- 106$ by $3$ .

$- \frac{\frac{- 45 y + 31 y - 318}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.12.3

Add $- 45 y$ and $31 y$ .

$- \frac{\frac{- 14 y - 318}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.12.4

Factor $2$ out of $- 14 y - 318$ .

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Step 4.2.1.1.12.4.1

Factor $2$ out of $- 14 y$ .

$- \frac{\frac{2 (- 7 y) - 318}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.12.4.2

Factor $2$ out of $- 318$ .

$- \frac{\frac{2 (- 7 y) + 2 (- 159)}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.1.12.4.3

Factor $2$ out of $2 (- 7 y) + 2 (- 159)$ .

$- \frac{\frac{2 (- 7 y - 159)}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- \frac{\frac{2 (- 7 y - 159)}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- \frac{\frac{2 (- 7 y - 159)}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- \frac{\frac{2 (- 7 y - 159)}{3}}{4} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$- (\frac{2 (- 7 y - 159)}{3} \cdot \frac{1}{4}) = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.3

Cancel the common factor of $2$ .

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Step 4.2.1.3.1

Factor $2$ out of $4$ .

$- (\frac{2 (- 7 y - 159)}{3} \cdot \frac{1}{2 (2)}) = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.3.2

Cancel the common factor.

$- (\frac{2 (- 7 y - 159)}{3} \cdot \frac{1}{2 \cdot 2}) = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.3.3

Rewrite the expression.

$- (\frac{- 7 y - 159}{3} \cdot \frac{1}{2}) = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$- (\frac{- 7 y - 159}{3} \cdot \frac{1}{2}) = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.4

Multiply $\frac{- 7 y - 159}{3}$ by $\frac{1}{2}$ .

$- \frac{- 7 y - 159}{3 \cdot 2} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.5

Multiply $3$ by $2$ .

$- \frac{- 7 y - 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.6

Factor $- 1$ out of $- 7 y$ .

$- \frac{- (7 y) - 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.7

Rewrite $- 159$ as $- 1 (159)$ .

$- \frac{- (7 y) - 1 \cdot 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.8

Factor $- 1$ out of $- (7 y) - 1 (159)$ .

$- \frac{- (7 y + 159)}{6} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.9

Simplify the expression.

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Step 4.2.1.9.1

Rewrite $- (7 y + 159)$ as $- 1 (7 y + 159)$ .

$- \frac{- 1 (7 y + 159)}{6} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.9.2

Move the negative in front of the fraction.

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.9.3

Multiply $- 1$ by $- 1$ .

$1 (\frac{7 y + 159}{6}) = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.2.1.9.4

Multiply $\frac{7 y + 159}{6}$ by $1$ .

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{y}{4} - \frac{5 z}{4}$

Step 4.3

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

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

$\frac{7 y + 159}{6} = 30$

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

Step 4.4

Simplify the right side.

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Step 4.4.1

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

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Step 4.4.1.1

Combine the numerators over the common denominator.

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

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.2

Simplify each term.

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Step 4.4.1.2.1

Apply the distributive property.

$x = \frac{- 3}{2} + \frac{y - 5 \cdot - 4 - 5 \frac{y}{3}}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.2.2

Multiply $- 5$ by $- 4$ .

$x = \frac{- 3}{2} + \frac{y + 20 - 5 \frac{y}{3}}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.2.3

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

$x = \frac{- 3}{2} + \frac{y + 20 + \frac{- 5 y}{3}}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.2.4

Move the negative in front of the fraction.

$x = \frac{- 3}{2} + \frac{y + 20 - \frac{5 y}{3}}{4}$

$\frac{7 y + 159}{6} = 30$

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

$x = \frac{- 3}{2} + \frac{y + 20 - \frac{5 y}{3}}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.3

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

$x = \frac{- 3}{2} + \frac{y \cdot \frac{3}{3} - \frac{5 y}{3} + 20}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.4

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

$x = \frac{- 3}{2} + \frac{\frac{y \cdot 3}{3} - \frac{5 y}{3} + 20}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.5

Combine the numerators over the common denominator.

$x = \frac{- 3}{2} + \frac{\frac{y \cdot 3 - 5 y}{3} + 20}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.6

Subtract $5 y$ from $y \cdot 3$ .

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Step 4.4.1.6.1

Reorder $y$ and $3$ .

$x = \frac{- 3}{2} + \frac{\frac{3 \cdot y - 5 y}{3} + 20}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.6.2

Subtract $5 y$ from $3 \cdot y$ .

$x = \frac{- 3}{2} + \frac{\frac{- 2 \cdot y}{3} + 20}{4}$

$\frac{7 y + 159}{6} = 30$

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

$x = \frac{- 3}{2} + \frac{\frac{- 2 \cdot y}{3} + 20}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.7

Move the negative in front of the fraction.

$x = \frac{- 3}{2} + \frac{- \frac{2 y}{3} + 20}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8

Simplify each term.

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Step 4.4.1.8.1

Move the negative in front of the fraction.

$x = - \frac{3}{2} + \frac{- \frac{2 y}{3} + 20}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.2

Cancel the common factor of $- \frac{2 y}{3} + 20$ and $4$ .

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Step 4.4.1.8.2.1

Factor $2$ out of $- \frac{2 y}{3}$ .

$x = - \frac{3}{2} + \frac{2 (- \frac{y}{3}) + 20}{4}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.2.2

Factor $2$ out of $20$ .

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

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.2.3

Factor $2$ out of $2 (- \frac{y}{3}) + 2 (10)$ .

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

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.2.4

Cancel the common factors.

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Step 4.4.1.8.2.4.1

Factor $2$ out of $4$ .

$x = - \frac{3}{2} + \frac{2 (- \frac{y}{3} + 10)}{2 (2)}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.2.4.2

Cancel the common factor.

$x = - \frac{3}{2} + \frac{2 (- \frac{y}{3} + 10)}{2 \cdot 2}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.2.4.3

Rewrite the expression.

$x = - \frac{3}{2} + \frac{- \frac{y}{3} + 10}{2}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{- \frac{y}{3} + 10}{2}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{- \frac{y}{3} + 10}{2}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.3

Simplify the numerator.

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Step 4.4.1.8.3.1

To write $10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{3}{2} + \frac{- \frac{y}{3} + 10 \cdot \frac{3}{3}}{2}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.3.2

Combine $10$ and $\frac{3}{3}$ .

$x = - \frac{3}{2} + \frac{- \frac{y}{3} + \frac{10 \cdot 3}{3}}{2}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.3.3

Combine the numerators over the common denominator.

$x = - \frac{3}{2} + \frac{\frac{- y + 10 \cdot 3}{3}}{2}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.3.4

Multiply $10$ by $3$ .

$x = - \frac{3}{2} + \frac{\frac{- y + 30}{3}}{2}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{\frac{- y + 30}{3}}{2}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.4

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{3}{2} + \frac{- y + 30}{3} \cdot \frac{1}{2}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.5

Multiply $\frac{- y + 30}{3} \cdot \frac{1}{2}$ .

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Step 4.4.1.8.5.1

Multiply $\frac{- y + 30}{3}$ by $\frac{1}{2}$ .

$x = - \frac{3}{2} + \frac{- y + 30}{3 \cdot 2}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.8.5.2

Multiply $3$ by $2$ .

$x = - \frac{3}{2} + \frac{- y + 30}{6}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{- y + 30}{6}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{3}{2} + \frac{- y + 30}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.9

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

$x = - \frac{3}{2} \cdot \frac{3}{3} + \frac{- y + 30}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.10

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1.10.1

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

$x = - \frac{3 \cdot 3}{2 \cdot 3} + \frac{- y + 30}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.10.2

Multiply $2$ by $3$ .

$x = - \frac{3 \cdot 3}{6} + \frac{- y + 30}{6}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{3 \cdot 3}{6} + \frac{- y + 30}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.11

Combine the numerators over the common denominator.

$x = \frac{- 3 \cdot 3 - y + 30}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.12

Simplify the numerator.

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Step 4.4.1.12.1

Multiply $- 3$ by $3$ .

$x = \frac{- 9 - y + 30}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.12.2

Add $- 9$ and $30$ .

$x = \frac{- y + 21}{6}$

$\frac{7 y + 159}{6} = 30$

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

$x = \frac{- y + 21}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.13

Simplify with factoring out.

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Step 4.4.1.13.1

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

$x = \frac{- (y) + 21}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.13.2

Rewrite $21$ as $- 1 (- 21)$ .

$x = \frac{- (y) - 1 \cdot - 21}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.13.3

Factor $- 1$ out of $- (y) - 1 (- 21)$ .

$x = \frac{- (y - 21)}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.13.4

Simplify the expression.

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Step 4.4.1.13.4.1

Rewrite $- (y - 21)$ as $- 1 (y - 21)$ .

$x = \frac{- 1 (y - 21)}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 4.4.1.13.4.2

Move the negative in front of the fraction.

$x = - \frac{y - 21}{6}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{y - 21}{6}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{y - 21}{6}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{y - 21}{6}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{y - 21}{6}$

$\frac{7 y + 159}{6} = 30$

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

$x = - \frac{y - 21}{6}$

$\frac{7 y + 159}{6} = 30$

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

Step 5

Solve for $y$ in $\frac{7 y + 159}{6} = 30$ .

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Step 5.1

Multiply both sides by $6$ .

$\frac{7 y + 159}{6} \cdot 6 = 30 \cdot 6$

$x = - \frac{y - 21}{6}$

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

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $6$ .

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Step 5.2.1.1.1

Cancel the common factor.

$\frac{7 y + 159}{6} \cdot 6 = 30 \cdot 6$

$x = - \frac{y - 21}{6}$

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

Step 5.2.1.1.2

Rewrite the expression.

$7 y + 159 = 30 \cdot 6$

$x = - \frac{y - 21}{6}$

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

$7 y + 159 = 30 \cdot 6$

$x = - \frac{y - 21}{6}$

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

$7 y + 159 = 30 \cdot 6$

$x = - \frac{y - 21}{6}$

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $30$ by $6$ .

$7 y + 159 = 180$

$x = - \frac{y - 21}{6}$

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

$7 y + 159 = 180$

$x = - \frac{y - 21}{6}$

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

$7 y + 159 = 180$

$x = - \frac{y - 21}{6}$

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

Step 5.3

Solve for $y$ .

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Step 5.3.1

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

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Step 5.3.1.1

Subtract $159$ from both sides of the equation.

$7 y = 180 - 159$

$x = - \frac{y - 21}{6}$

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

Step 5.3.1.2

Subtract $159$ from $180$ .

$7 y = 21$

$x = - \frac{y - 21}{6}$

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

$7 y = 21$

$x = - \frac{y - 21}{6}$

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

Step 5.3.2

Divide each term in $7 y = 21$ by $7$ and simplify.

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Step 5.3.2.1

Divide each term in $7 y = 21$ by $7$ .

$\frac{7 y}{7} = \frac{21}{7}$

$x = - \frac{y - 21}{6}$

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

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $7$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{21}{7}$

$x = - \frac{y - 21}{6}$

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

Step 5.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{21}{7}$

$x = - \frac{y - 21}{6}$

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

$y = \frac{21}{7}$

$x = - \frac{y - 21}{6}$

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

$y = \frac{21}{7}$

$x = - \frac{y - 21}{6}$

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

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $21$ by $7$ .

$y = 3$

$x = - \frac{y - 21}{6}$

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

$y = 3$

$x = - \frac{y - 21}{6}$

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

$y = 3$

$x = - \frac{y - 21}{6}$

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

$y = 3$

$x = - \frac{y - 21}{6}$

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

$y = 3$

$x = - \frac{y - 21}{6}$

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

Step 6

Replace all occurrences of $y$ with $3$ in each equation.

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Step 6.1

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

$x = - \frac{(3) - 21}{6}$

$y = 3$

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

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{(3) - 21}{6}$ .

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Step 6.2.1.1

Cancel the common factor of $(3) - 21$ and $6$ .

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Step 6.2.1.1.1

Factor $3$ out of $3$ .

$x = - \frac{3 (1) - 21}{6}$

$y = 3$

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

Step 6.2.1.1.2

Factor $3$ out of $- 21$ .

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

$y = 3$

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

Step 6.2.1.1.3

Factor $3$ out of $3 \cdot 1 + 3 \cdot - 7$ .

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

$y = 3$

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

Step 6.2.1.1.4

Cancel the common factors.

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Step 6.2.1.1.4.1

Factor $3$ out of $6$ .

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

$y = 3$

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

Step 6.2.1.1.4.2

Cancel the common factor.

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

$y = 3$

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

Step 6.2.1.1.4.3

Rewrite the expression.

$x = - \frac{1 - 7}{2}$

$y = 3$

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

$x = - \frac{1 - 7}{2}$

$y = 3$

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

$x = - \frac{1 - 7}{2}$

$y = 3$

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

Step 6.2.1.2

Simplify the expression.

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Step 6.2.1.2.1

Subtract $7$ from $1$ .

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

$y = 3$

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

Step 6.2.1.2.2

Divide $- 6$ by $2$ .

$x = 3$

$y = 3$

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

Step 6.2.1.2.3

Multiply $- 1$ by $- 3$ .

$x = 3$

$y = 3$

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

$x = 3$

$y = 3$

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

$x = 3$

$y = 3$

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

$x = 3$

$y = 3$

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

Step 6.3

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

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

$x = 3$

$y = 3$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 4 + \frac{3}{3}$ .

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Step 6.4.1.1

Divide $3$ by $3$ .

$z = - 4 + 1$

$x = 3$

$y = 3$

Step 6.4.1.2

Add $- 4$ and $1$ .

$z = - 3$

$x = 3$

$y = 3$

$z = - 3$

$x = 3$

$y = 3$

$z = - 3$

$x = 3$

$y = 3$

$z = - 3$

$x = 3$

$y = 3$

Step 7

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

$(3, 3, - 3)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(3, 3, - 3)$

Equation Form:

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