Precalculus Examples

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

Step 1

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

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Step 1.1

Subtract $x$ from both sides of the equation.

$y + 2 z = 5 - x$

$2 x - y + 3 z = 4$

Step 1.2

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

$y = 5 - x - 2 z$

$2 x - y + 3 z = 4$

$y = 5 - x - 2 z$

$2 x - y + 3 z = 4$

Step 2

Solve the equation for $z$ .

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Step 2.1

Simplify $2 x - (5 - x - 2 z) + 3 z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$2 x - 1 \cdot 5 + x - (- 2 z) + 3 z = 4$

$y = 5 - x - 2 z$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $- 1$ by $5$ .

$2 x - 5 + x - (- 2 z) + 3 z = 4$

$y = 5 - x - 2 z$

Step 2.1.1.2.2

Multiply $- - x$ .

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Step 2.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$2 x - 5 + 1 x - (- 2 z) + 3 z = 4$

$y = 5 - x - 2 z$

Step 2.1.1.2.2.2

Multiply $x$ by $1$ .

$2 x - 5 + x - (- 2 z) + 3 z = 4$

$y = 5 - x - 2 z$

$2 x - 5 + x - (- 2 z) + 3 z = 4$

$y = 5 - x - 2 z$

Step 2.1.1.2.3

Multiply $- 2$ by $- 1$ .

$2 x - 5 + x + 2 z + 3 z = 4$

$y = 5 - x - 2 z$

$2 x - 5 + x + 2 z + 3 z = 4$

$y = 5 - x - 2 z$

$2 x - 5 + x + 2 z + 3 z = 4$

$y = 5 - x - 2 z$

Step 2.1.2

Simplify by adding terms.

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Step 2.1.2.1

Add $2 x$ and $x$ .

$3 x - 5 + 2 z + 3 z = 4$

$y = 5 - x - 2 z$

Step 2.1.2.2

Add $2 z$ and $3 z$ .

$3 x - 5 + 5 z = 4$

$y = 5 - x - 2 z$

$3 x - 5 + 5 z = 4$

$y = 5 - x - 2 z$

$3 x - 5 + 5 z = 4$

$y = 5 - x - 2 z$

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 $3 x$ from both sides of the equation.

$- 5 + 5 z = 4 - 3 x$

$y = 5 - x - 2 z$

Step 2.2.2

Add $5$ to both sides of the equation.

$5 z = 4 - 3 x + 5$

$y = 5 - x - 2 z$

Step 2.2.3

Add $4$ and $5$ .

$5 z = - 3 x + 9$

$y = 5 - x - 2 z$

$5 z = - 3 x + 9$

$y = 5 - x - 2 z$

Step 2.3

Divide each term in $5 z = - 3 x + 9$ by $5$ and simplify.

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Step 2.3.1

Divide each term in $5 z = - 3 x + 9$ by $5$ .

$\frac{5 z}{5} = \frac{- 3 x}{5} + \frac{9}{5}$

$y = 5 - x - 2 z$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $5$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{5 z}{5} = \frac{- 3 x}{5} + \frac{9}{5}$

$y = 5 - x - 2 z$

Step 2.3.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 3 x}{5} + \frac{9}{5}$

$y = 5 - x - 2 z$

$z = \frac{- 3 x}{5} + \frac{9}{5}$

$y = 5 - x - 2 z$

$z = \frac{- 3 x}{5} + \frac{9}{5}$

$y = 5 - x - 2 z$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

$z = - \frac{3 x}{5} + \frac{9}{5}$

$y = 5 - x - 2 z$

$z = - \frac{3 x}{5} + \frac{9}{5}$

$y = 5 - x - 2 z$

$z = - \frac{3 x}{5} + \frac{9}{5}$

$y = 5 - x - 2 z$

$z = - \frac{3 x}{5} + \frac{9}{5}$

$y = 5 - x - 2 z$

Step 3

Simplify the right side.

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Step 3.1

Simplify $5 - x - 2 (- \frac{3 x}{5} + \frac{9}{5})$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

$y = 5 - x - 2 (- \frac{3 x}{5}) - 2 (\frac{9}{5})$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.1.2

Multiply $- 2 (- \frac{3 x}{5})$ .

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Step 3.1.1.2.1

Multiply $- 1$ by $- 2$ .

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

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.1.2.2

Combine $2$ and $\frac{3 x}{5}$ .

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

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.1.2.3

Multiply $3$ by $2$ .

$y = 5 - x + \frac{6 x}{5} - 2 (\frac{9}{5})$

$z = - \frac{3 x}{5} + \frac{9}{5}$

$y = 5 - x + \frac{6 x}{5} - 2 (\frac{9}{5})$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.1.3

Multiply $- 2 (\frac{9}{5})$ .

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Step 3.1.1.3.1

Combine $- 2$ and $\frac{9}{5}$ .

$y = 5 - x + \frac{6 x}{5} + \frac{- 2 \cdot 9}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.1.3.2

Multiply $- 2$ by $9$ .

$y = 5 - x + \frac{6 x}{5} + \frac{- 18}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

$y = 5 - x + \frac{6 x}{5} + \frac{- 18}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.1.4

Move the negative in front of the fraction.

$y = 5 - x + \frac{6 x}{5} - \frac{18}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

$y = 5 - x + \frac{6 x}{5} - \frac{18}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.2

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

$y = - x + \frac{6 x}{5} + 5 \cdot \frac{5}{5} - \frac{18}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.3

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

$y = - x + \frac{6 x}{5} + \frac{5 \cdot 5}{5} - \frac{18}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.4

Combine the numerators over the common denominator.

$y = - x + \frac{6 x}{5} + \frac{5 \cdot 5 - 18}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $5$ by $5$ .

$y = - x + \frac{6 x}{5} + \frac{25 - 18}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.5.2

Subtract $18$ from $25$ .

$y = - x + \frac{6 x}{5} + \frac{7}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

$y = - x + \frac{6 x}{5} + \frac{7}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.6

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

$y = - x \cdot \frac{5}{5} + \frac{6 x}{5} + \frac{7}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.7

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

$y = \frac{- x \cdot 5}{5} + \frac{6 x}{5} + \frac{7}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.8

Combine the numerators over the common denominator.

$y = \frac{- x \cdot 5 + 6 x}{5} + \frac{7}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.9

Combine the numerators over the common denominator.

$y = \frac{- x \cdot 5 + 6 x + 7}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.10

Multiply $5$ by $- 1$ .

$y = \frac{- 5 x + 6 x + 7}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

Step 3.1.11

Add $- 5 x$ and $6 x$ .

$y = \frac{x + 7}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

$y = \frac{x + 7}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$

$y = \frac{x + 7}{5}$

$z = - \frac{3 x}{5} + \frac{9}{5}$