Precalculus Examples

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

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.

$6 y - z = 6 - 3 x$

$x + y + 2 z = 5$

Step 1.1.2

Add $z$ to both sides of the equation.

$6 y = 6 - 3 x + z$

$x + y + 2 z = 5$

$6 y = 6 - 3 x + z$

$x + y + 2 z = 5$

Step 1.2

Divide each term in $6 y = 6 - 3 x + z$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 y = 6 - 3 x + z$ by $6$ .

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

$x + y + 2 z = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$x + y + 2 z = 5$

Step 1.2.2.1.2

Divide $y$ by $1$ .

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

$x + y + 2 z = 5$

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

$x + y + 2 z = 5$

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

$x + y + 2 z = 5$

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

Divide $6$ by $6$ .

$y = 1 + \frac{- 3 x}{6} + \frac{z}{6}$

$x + y + 2 z = 5$

Step 1.2.3.1.2

Cancel the common factor of $- 3$ and $6$ .

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Step 1.2.3.1.2.1

Factor $3$ out of $- 3 x$ .

$y = 1 + \frac{3 (- x)}{6} + \frac{z}{6}$

$x + y + 2 z = 5$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $3$ out of $6$ .

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

$x + y + 2 z = 5$

Step 1.2.3.1.2.2.2

Cancel the common factor.

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

$x + y + 2 z = 5$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = 1 + \frac{- x}{2} + \frac{z}{6}$

$x + y + 2 z = 5$

$y = 1 + \frac{- x}{2} + \frac{z}{6}$

$x + y + 2 z = 5$

$y = 1 + \frac{- x}{2} + \frac{z}{6}$

$x + y + 2 z = 5$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$x + y + 2 z = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$x + y + 2 z = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$x + y + 2 z = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$x + y + 2 z = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$x + y + 2 z = 5$

Step 2

Solve the equation for $z$ .

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Step 2.1

Simplify $x + 1 - \frac{x}{2} + \frac{z}{6} + 2 z$ .

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Step 2.1.1

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

$x \cdot \frac{2}{2} - \frac{x}{2} + 1 + \frac{z}{6} + 2 z = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.2

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

$\frac{x \cdot 2}{2} - \frac{x}{2} + 1 + \frac{z}{6} + 2 z = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.3

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - x}{2} + 1 + \frac{z}{6} + 2 z = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.4

Subtract $x$ from $x \cdot 2$ .

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Step 2.1.4.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x - x}{2} + 1 + \frac{z}{6} + 2 z = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.4.2

Subtract $x$ from $2 \cdot x$ .

$\frac{x}{2} + 1 + \frac{z}{6} + 2 z = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$\frac{x}{2} + 1 + \frac{z}{6} + 2 z = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.5

To write $2 z$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{x}{2} + 1 + \frac{z}{6} + 2 z \cdot \frac{6}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.6

Simplify terms.

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Step 2.1.6.1

Combine $2 z$ and $\frac{6}{6}$ .

$\frac{x}{2} + 1 + \frac{z}{6} + \frac{2 z \cdot 6}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.6.2

Combine the numerators over the common denominator.

$\frac{x}{2} + 1 + \frac{z + 2 z \cdot 6}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$\frac{x}{2} + 1 + \frac{z + 2 z \cdot 6}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.7

Simplify each term.

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Step 2.1.7.1

Simplify the numerator.

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Step 2.1.7.1.1

Factor $z$ out of $z + 2 z \cdot 6$ .

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Step 2.1.7.1.1.1

Raise $z$ to the power of $1$ .

$\frac{x}{2} + 1 + \frac{z + 2 z \cdot 6}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.7.1.1.2

Factor $z$ out of $z^{1}$ .

$\frac{x}{2} + 1 + \frac{z \cdot 1 + 2 z \cdot 6}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.7.1.1.3

Factor $z$ out of $2 z \cdot 6$ .

$\frac{x}{2} + 1 + \frac{z \cdot 1 + z (2 \cdot 6)}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.7.1.1.4

Factor $z$ out of $z \cdot 1 + z (2 \cdot 6)$ .

$\frac{x}{2} + 1 + \frac{z (1 + 2 \cdot 6)}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$\frac{x}{2} + 1 + \frac{z (1 + 2 \cdot 6)}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.7.1.2

Multiply $2$ by $6$ .

$\frac{x}{2} + 1 + \frac{z (1 + 12)}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.7.1.3

Add $1$ and $12$ .

$\frac{x}{2} + 1 + \frac{z \cdot 13}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$\frac{x}{2} + 1 + \frac{z \cdot 13}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.1.7.2

Move $13$ to the left of $z$ .

$\frac{x}{2} + 1 + \frac{13 z}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$\frac{x}{2} + 1 + \frac{13 z}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$\frac{x}{2} + 1 + \frac{13 z}{6} = 5$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.2

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

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Step 2.2.1

Subtract $\frac{x}{2}$ from both sides of the equation.

$1 + \frac{13 z}{6} = 5 - \frac{x}{2}$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.2.2

Subtract $1$ from both sides of the equation.

$\frac{13 z}{6} = 5 - \frac{x}{2} - 1$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.2.3

Subtract $1$ from $5$ .

$\frac{13 z}{6} = - \frac{x}{2} + 4$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$\frac{13 z}{6} = - \frac{x}{2} + 4$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.3

Multiply both sides of the equation by $\frac{6}{13}$ .

$\frac{6}{13} \cdot \frac{13 z}{6} = \frac{6}{13} \cdot (- \frac{x}{2} + 4)$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify $\frac{6}{13} \cdot \frac{13 z}{6}$ .

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Step 2.4.1.1.1

Cancel the common factor of $6$ .

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Step 2.4.1.1.1.1

Cancel the common factor.

$\frac{6}{13} \cdot \frac{13 z}{6} = \frac{6}{13} \cdot (- \frac{x}{2} + 4)$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.1.1.1.2

Rewrite the expression.

$\frac{1}{13} \cdot (13 z) = \frac{6}{13} \cdot (- \frac{x}{2} + 4)$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$\frac{1}{13} \cdot (13 z) = \frac{6}{13} \cdot (- \frac{x}{2} + 4)$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.1.1.2

Cancel the common factor of $13$ .

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Step 2.4.1.1.2.1

Factor $13$ out of $13 z$ .

$\frac{1}{13} \cdot (13 (z)) = \frac{6}{13} \cdot (- \frac{x}{2} + 4)$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.1.1.2.2

Cancel the common factor.

$\frac{1}{13} \cdot (13 z) = \frac{6}{13} \cdot (- \frac{x}{2} + 4)$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.1.1.2.3

Rewrite the expression.

$z = \frac{6}{13} \cdot (- \frac{x}{2} + 4)$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$z = \frac{6}{13} \cdot (- \frac{x}{2} + 4)$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$z = \frac{6}{13} \cdot (- \frac{x}{2} + 4)$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$z = \frac{6}{13} \cdot (- \frac{x}{2} + 4)$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\frac{6}{13} (- \frac{x}{2} + 4)$ .

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Step 2.4.2.1.1

Apply the distributive property.

$z = \frac{6}{13} \cdot (- \frac{x}{2}) + \frac{6}{13} \cdot 4$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2.1.2

Cancel the common factor of $2$ .

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Step 2.4.2.1.2.1

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

$z = \frac{6}{13} \cdot \frac{- x}{2} + \frac{6}{13} \cdot 4$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2.1.2.2

Factor $2$ out of $6$ .

$z = \frac{2 (3)}{13} \cdot \frac{- x}{2} + \frac{6}{13} \cdot 4$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2.1.2.3

Cancel the common factor.

$z = \frac{2 \cdot 3}{13} \cdot \frac{- x}{2} + \frac{6}{13} \cdot 4$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2.1.2.4

Rewrite the expression.

$z = \frac{3}{13} \cdot (- x) + \frac{6}{13} \cdot 4$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$z = \frac{3}{13} \cdot (- x) + \frac{6}{13} \cdot 4$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2.1.3

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

$z = - \frac{3 x}{13} + \frac{6}{13} \cdot 4$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2.1.4

Multiply $\frac{6}{13} \cdot 4$ .

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Step 2.4.2.1.4.1

Combine $\frac{6}{13}$ and $4$ .

$z = - \frac{3 x}{13} + \frac{6 \cdot 4}{13}$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 2.4.2.1.4.2

Multiply $6$ by $4$ .

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = 1 - \frac{x}{2} + \frac{z}{6}$

Step 3

Simplify the right side.

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Step 3.1

Simplify $1 - \frac{x}{2} + \frac{- \frac{3 x}{13} + \frac{24}{13}}{6}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Simplify the numerator.

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Step 3.1.1.1.1

Combine the numerators over the common denominator.

$y = 1 - \frac{x}{2} + \frac{\frac{- 3 x + 24}{13}}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.1.1.2

Factor $3$ out of $- 3 x + 24$ .

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Step 3.1.1.1.2.1

Factor $3$ out of $- 3 x$ .

$y = 1 - \frac{x}{2} + \frac{\frac{3 (- x) + 24}{13}}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.1.1.2.2

Factor $3$ out of $24$ .

$y = 1 - \frac{x}{2} + \frac{\frac{3 (- x) + 3 (8)}{13}}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.1.1.2.3

Factor $3$ out of $3 (- x) + 3 (8)$ .

$y = 1 - \frac{x}{2} + \frac{\frac{3 (- x + 8)}{13}}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = 1 - \frac{x}{2} + \frac{\frac{3 (- x + 8)}{13}}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = 1 - \frac{x}{2} + \frac{\frac{3 (- x + 8)}{13}}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = 1 - \frac{x}{2} + \frac{3 (- x + 8)}{13} \cdot \frac{1}{6}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.1.3

Cancel the common factor of $3$ .

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Step 3.1.1.3.1

Factor $3$ out of $6$ .

$y = 1 - \frac{x}{2} + \frac{3 (- x + 8)}{13} \cdot \frac{1}{3 (2)}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.1.3.2

Cancel the common factor.

$y = 1 - \frac{x}{2} + \frac{3 (- x + 8)}{13} \cdot \frac{1}{3 \cdot 2}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.1.3.3

Rewrite the expression.

$y = 1 - \frac{x}{2} + \frac{- x + 8}{13} \cdot \frac{1}{2}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = 1 - \frac{x}{2} + \frac{- x + 8}{13} \cdot \frac{1}{2}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.1.4

Multiply $\frac{- x + 8}{13}$ by $\frac{1}{2}$ .

$y = 1 - \frac{x}{2} + \frac{- x + 8}{13 \cdot 2}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.1.5

Multiply $13$ by $2$ .

$y = 1 - \frac{x}{2} + \frac{- x + 8}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = 1 - \frac{x}{2} + \frac{- x + 8}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.2

Combine into one fraction.

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Step 3.1.2.1

Write $1$ as a fraction with a common denominator.

$y = - \frac{x}{2} + \frac{26}{26} + \frac{- x + 8}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.2.2

Combine the numerators over the common denominator.

$y = - \frac{x}{2} + \frac{26 - x + 8}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.2.3

Add $26$ and $8$ .

$y = - \frac{x}{2} + \frac{- x + 34}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = - \frac{x}{2} + \frac{- x + 34}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.3

To write $- \frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$y = - \frac{x}{2} \cdot \frac{13}{13} + \frac{- x + 34}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.4

Write each expression with a common denominator of $26$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.4.1

Multiply $\frac{x}{2}$ by $\frac{13}{13}$ .

$y = - \frac{x \cdot 13}{2 \cdot 13} + \frac{- x + 34}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.4.2

Multiply $2$ by $13$ .

$y = - \frac{x \cdot 13}{26} + \frac{- x + 34}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = - \frac{x \cdot 13}{26} + \frac{- x + 34}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.5

Combine the numerators over the common denominator.

$y = \frac{- x \cdot 13 - x + 34}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.6

Simplify the numerator.

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Step 3.1.6.1

Multiply $13$ by $- 1$ .

$y = \frac{- 13 x - x + 34}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.6.2

Subtract $x$ from $- 13 x$ .

$y = \frac{- 14 x + 34}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.6.3

Factor $2$ out of $- 14 x + 34$ .

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Step 3.1.6.3.1

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

$y = \frac{2 (- 7 x) + 34}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.6.3.2

Factor $2$ out of $34$ .

$y = \frac{2 (- 7 x) + 2 (17)}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.6.3.3

Factor $2$ out of $2 (- 7 x) + 2 (17)$ .

$y = \frac{2 (- 7 x + 17)}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = \frac{2 (- 7 x + 17)}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = \frac{2 (- 7 x + 17)}{26}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.7

Cancel the common factors.

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Step 3.1.7.1

Factor $2$ out of $26$ .

$y = \frac{2 (- 7 x + 17)}{2 \cdot 13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.7.2

Cancel the common factor.

$y = \frac{2 (- 7 x + 17)}{2 \cdot 13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.7.3

Rewrite the expression.

$y = \frac{- 7 x + 17}{13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = \frac{- 7 x + 17}{13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.8

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

$y = \frac{- (7 x) + 17}{13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.9

Rewrite $17$ as $- 1 (- 17)$ .

$y = \frac{- (7 x) - 1 \cdot - 17}{13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.10

Factor $- 1$ out of $- (7 x) - 1 (- 17)$ .

$y = \frac{- (7 x - 17)}{13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.11

Simplify the expression.

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Step 3.1.11.1

Rewrite $- (7 x - 17)$ as $- 1 (7 x - 17)$ .

$y = \frac{- 1 (7 x - 17)}{13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

Step 3.1.11.2

Move the negative in front of the fraction.

$y = - \frac{7 x - 17}{13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = - \frac{7 x - 17}{13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = - \frac{7 x - 17}{13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$

$y = - \frac{7 x - 17}{13}$

$z = - \frac{3 x}{13} + \frac{24}{13}$