Precalculus Examples

$3 x + 4 y - z = 0$ , $x - 2 y + z = - 8$

Step 1

Solve the equation for $y$ .

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Step 1.1

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

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Step 1.1.1

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

$4 y - z = - 3 x$

$x - 2 y + z = - 8$

Step 1.1.2

Add $z$ to both sides of the equation.

$4 y = - 3 x + z$

$x - 2 y + z = - 8$

$4 y = - 3 x + z$

$x - 2 y + z = - 8$

Step 1.2

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

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Step 1.2.1

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

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

$x - 2 y + z = - 8$

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.

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

$x - 2 y + z = - 8$

Step 1.2.2.1.2

Divide $y$ by $1$ .

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

$x - 2 y + z = - 8$

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

$x - 2 y + z = - 8$

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

$x - 2 y + z = - 8$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

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

$x - 2 y + z = - 8$

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

$x - 2 y + z = - 8$

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

$x - 2 y + z = - 8$

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

$x - 2 y + z = - 8$

Step 2

Solve the equation for $z$ .

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Step 2.1

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

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

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

Step 2.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.2.1

Move the leading negative in $- \frac{3 x}{4}$ into the numerator.

$x - 2 \frac{- 3 x}{4} - 2 \frac{z}{4} + z = - 8$

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

Step 2.1.1.2.2

Factor $2$ out of $- 2$ .

$x + 2 (- 1) (\frac{- 3 x}{4}) - 2 \frac{z}{4} + z = - 8$

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

Step 2.1.1.2.3

Factor $2$ out of $4$ .

$x + 2 \cdot (- 1 \frac{- 3 x}{2 \cdot 2}) - 2 \frac{z}{4} + z = - 8$

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

Step 2.1.1.2.4

Cancel the common factor.

$x + 2 \cdot (- 1 \frac{- 3 x}{2 \cdot 2}) - 2 \frac{z}{4} + z = - 8$

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

Step 2.1.1.2.5

Rewrite the expression.

$x - 1 \frac{- 3 x}{2} - 2 \frac{z}{4} + z = - 8$

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

$x - 1 \frac{- 3 x}{2} - 2 \frac{z}{4} + z = - 8$

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

Step 2.1.1.3

Cancel the common factor of $2$ .

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Step 2.1.1.3.1

Factor $2$ out of $- 2$ .

$x - 1 \frac{- 3 x}{2} + 2 (- 1) (\frac{z}{4}) + z = - 8$

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

Step 2.1.1.3.2

Factor $2$ out of $4$ .

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

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

Step 2.1.1.3.3

Cancel the common factor.

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

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

Step 2.1.1.3.4

Rewrite the expression.

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

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

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

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

Step 2.1.1.4

Simplify each term.

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Step 2.1.1.4.1

Move the negative in front of the fraction.

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

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

Step 2.1.1.4.2

Multiply $- 1 (- \frac{3 x}{2})$ .

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Step 2.1.1.4.2.1

Multiply $- 1$ by $- 1$ .

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

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

Step 2.1.1.4.2.2

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

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

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

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

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

Step 2.1.1.4.3

Rewrite $- 1 \frac{z}{2}$ as $- \frac{z}{2}$ .

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

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

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

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

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

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

Step 2.1.2

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

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

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

Step 2.1.3

Simplify terms.

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Step 2.1.3.1

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

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

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

Step 2.1.3.2

Combine the numerators over the common denominator.

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

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

Step 2.1.3.3

Combine the numerators over the common denominator.

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

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

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

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

Step 2.1.4

Move $2$ to the left of $x$ .

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

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

Step 2.1.5

Add $2 x$ and $3 x$ .

$z + \frac{5 x - z}{2} = - 8$

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

Step 2.1.6

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

$z \cdot \frac{2}{2} + \frac{5 x - z}{2} = - 8$

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

Step 2.1.7

Simplify terms.

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Step 2.1.7.1

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

$\frac{z \cdot 2}{2} + \frac{5 x - z}{2} = - 8$

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

Step 2.1.7.2

Combine the numerators over the common denominator.

$\frac{z \cdot 2 + 5 x - z}{2} = - 8$

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

$\frac{z \cdot 2 + 5 x - z}{2} = - 8$

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

Step 2.1.8

Simplify the numerator.

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Step 2.1.8.1

Move $2$ to the left of $z$ .

$\frac{2 \cdot z + 5 x - z}{2} = - 8$

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

Step 2.1.8.2

Subtract $z$ from $2 z$ .

$\frac{5 x + z}{2} = - 8$

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

$\frac{5 x + z}{2} = - 8$

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

$\frac{5 x + z}{2} = - 8$

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

Step 2.2

Multiply both sides by $2$ .

$\frac{5 x + z}{2} \cdot 2 = - 8 \cdot 2$

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

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Cancel the common factor of $2$ .

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Step 2.3.1.1.1

Cancel the common factor.

$\frac{5 x + z}{2} \cdot 2 = - 8 \cdot 2$

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

Step 2.3.1.1.2

Rewrite the expression.

$5 x + z = - 8 \cdot 2$

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

$5 x + z = - 8 \cdot 2$

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

$5 x + z = - 8 \cdot 2$

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $- 8$ by $2$ .

$5 x + z = - 16$

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

$5 x + z = - 16$

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

$5 x + z = - 16$

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

Step 2.4

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

$z = - 16 - 5 x$

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

$z = - 16 - 5 x$

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

Step 3

Simplify the right side.

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Step 3.1

Simplify $- \frac{3 x}{4} + \frac{- 16 - 5 x}{4}$ .

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Step 3.1.1

Combine the numerators over the common denominator.

$y = \frac{- 3 x - 16 - 5 x}{4}$

$z = - 16 - 5 x$

Step 3.1.2

Subtract $5 x$ from $- 3 x$ .

$y = \frac{- 8 x - 16}{4}$

$z = - 16 - 5 x$

Step 3.1.3

Cancel the common factor of $- 8 x - 16$ and $4$ .

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Step 3.1.3.1

Factor $4$ out of $- 8 x$ .

$y = \frac{4 (- 2 x) - 16}{4}$

$z = - 16 - 5 x$

Step 3.1.3.2

Factor $4$ out of $- 16$ .

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

$z = - 16 - 5 x$

Step 3.1.3.3

Factor $4$ out of $4 (- 2 x) + 4 (- 4)$ .

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

$z = - 16 - 5 x$

Step 3.1.3.4

Cancel the common factors.

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Step 3.1.3.4.1

Factor $4$ out of $4$ .

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

$z = - 16 - 5 x$

Step 3.1.3.4.2

Cancel the common factor.

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

$z = - 16 - 5 x$

Step 3.1.3.4.3

Rewrite the expression.

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

$z = - 16 - 5 x$

Step 3.1.3.4.4

Divide $- 2 x - 4$ by $1$ .

$y = - 2 x - 4$

$z = - 16 - 5 x$

$y = - 2 x - 4$

$z = - 16 - 5 x$

$y = - 2 x - 4$

$z = - 16 - 5 x$

$y = - 2 x - 4$

$z = - 16 - 5 x$

$y = - 2 x - 4$

$z = - 16 - 5 x$

Step 4

Simplify the right side.

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Step 4.1

Reorder $- 16$ and $- 5 x$ .

$z = - 5 x - 16$

$y = - 2 x - 4$

$z = - 5 x - 16$

$y = - 2 x - 4$