Precalculus Examples

$2 x + 3 y - z = 1$ , $3 x + 5 y + z = 2$

Step 1

Solve the equation for $x$ .

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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

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

$2 x - z = 1 - 3 y$

$3 x + 5 y + z = 2$

Step 1.1.2

Add $z$ to both sides of the equation.

$2 x = 1 - 3 y + z$

$3 x + 5 y + z = 2$

$2 x = 1 - 3 y + z$

$3 x + 5 y + z = 2$

Step 1.2

Divide each term in $2 x = 1 - 3 y + z$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 1 - 3 y + z$ by $2$ .

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

$3 x + 5 y + z = 2$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$3 x + 5 y + z = 2$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$3 x + 5 y + z = 2$

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

$3 x + 5 y + z = 2$

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

$3 x + 5 y + z = 2$

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.

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

$3 x + 5 y + z = 2$

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

$3 x + 5 y + z = 2$

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

$3 x + 5 y + z = 2$

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

$3 x + 5 y + z = 2$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify $3 (\frac{1}{2} - \frac{3 y}{2} + \frac{z}{2}) + 5 y + z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$3 (\frac{1}{2}) + 3 (- \frac{3 y}{2}) + 3 (\frac{z}{2}) + 5 y + z = 2$

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

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Combine $3$ and $\frac{1}{2}$ .

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

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

Step 2.1.1.2.2

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

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Step 2.1.1.2.2.1

Multiply $- 1$ by $3$ .

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

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

Step 2.1.1.2.2.2

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

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

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

Step 2.1.1.2.2.3

Multiply $3$ by $- 3$ .

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

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

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

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

Step 2.1.1.2.3

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

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

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

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

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

Step 2.1.1.3

Move the negative in front of the fraction.

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

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

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

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

Step 2.1.2

To write $5 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

Step 2.1.3

Combine $5 y$ and $\frac{2}{2}$ .

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

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

Step 2.1.4

Combine the numerators over the common denominator.

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

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

Step 2.1.5

Combine the numerators over the common denominator.

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

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

Step 2.1.6

Multiply $2$ by $5$ .

$z + \frac{3 + 3 z - 9 y + 10 y}{2} = 2$

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

Step 2.1.7

Add $- 9 y$ and $10 y$ .

$z + \frac{3 + 3 z + y}{2} = 2$

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

Step 2.1.8

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

$z \cdot \frac{2}{2} + \frac{3 + 3 z + y}{2} = 2$

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

Step 2.1.9

Simplify terms.

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Step 2.1.9.1

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

$\frac{z \cdot 2}{2} + \frac{3 + 3 z + y}{2} = 2$

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

Step 2.1.9.2

Combine the numerators over the common denominator.

$\frac{z \cdot 2 + 3 + 3 z + y}{2} = 2$

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

$\frac{z \cdot 2 + 3 + 3 z + y}{2} = 2$

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

Step 2.1.10

Simplify the numerator.

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Step 2.1.10.1

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

$\frac{2 \cdot z + 3 + 3 z + y}{2} = 2$

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

Step 2.1.10.2

Add $2 z$ and $3 z$ .

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

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

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

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

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

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

Step 2.2

Multiply both sides by $2$ .

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

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

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

Simplify $\frac{5 z + 3 + y}{2} \cdot 2$ .

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Step 2.3.1.1.1

Cancel the common factor of $2$ .

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Step 2.3.1.1.1.1

Cancel the common factor.

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

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

Step 2.3.1.1.1.2

Rewrite the expression.

$5 z + 3 + y = 2 \cdot 2$

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

$5 z + 3 + y = 2 \cdot 2$

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

Step 2.3.1.1.2

Simplify the expression.

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Step 2.3.1.1.2.1

Move $3$ .

$5 z + y + 3 = 2 \cdot 2$

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

Step 2.3.1.1.2.2

Reorder $5 z$ and $y$ .

$y + 5 z + 3 = 2 \cdot 2$

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

$y + 5 z + 3 = 2 \cdot 2$

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

$y + 5 z + 3 = 2 \cdot 2$

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

$y + 5 z + 3 = 2 \cdot 2$

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $2$ by $2$ .

$y + 5 z + 3 = 4$

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

$y + 5 z + 3 = 4$

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

$y + 5 z + 3 = 4$

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

Step 2.4

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

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Step 2.4.1

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

$y + 3 = 4 - 5 z$

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

Step 2.4.2

Subtract $3$ from both sides of the equation.

$y = 4 - 5 z - 3$

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

Step 2.4.3

Subtract $3$ from $4$ .

$y = - 5 z + 1$

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

$y = - 5 z + 1$

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

$y = - 5 z + 1$

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

Step 3

Simplify the right side.

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Step 3.1

Simplify $\frac{1}{2} - \frac{3 (- 5 z + 1)}{2} + \frac{z}{2}$ .

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Step 3.1.1

Combine the numerators over the common denominator.

$x = \frac{1 - 3 (- 5 z + 1) + z}{2}$

$y = - 5 z + 1$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Apply the distributive property.

$x = \frac{1 - 3 (- 5 z) - 3 \cdot 1 + z}{2}$

$y = - 5 z + 1$

Step 3.1.2.2

Multiply $- 5$ by $- 3$ .

$x = \frac{1 + 15 z - 3 \cdot 1 + z}{2}$

$y = - 5 z + 1$

Step 3.1.2.3

Multiply $- 3$ by $1$ .

$x = \frac{1 + 15 z - 3 + z}{2}$

$y = - 5 z + 1$

$x = \frac{1 + 15 z - 3 + z}{2}$

$y = - 5 z + 1$

Step 3.1.3

Simplify terms.

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Step 3.1.3.1

Subtract $3$ from $1$ .

$x = \frac{15 z - 2 + z}{2}$

$y = - 5 z + 1$

Step 3.1.3.2

Add $15 z$ and $z$ .

$x = \frac{16 z - 2}{2}$

$y = - 5 z + 1$

Step 3.1.3.3

Cancel the common factor of $16 z - 2$ and $2$ .

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Step 3.1.3.3.1

Factor $2$ out of $16 z$ .

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

$y = - 5 z + 1$

Step 3.1.3.3.2

Factor $2$ out of $- 2$ .

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

$y = - 5 z + 1$

Step 3.1.3.3.3

Factor $2$ out of $2 (8 z) + 2 (- 1)$ .

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

$y = - 5 z + 1$

Step 3.1.3.3.4

Cancel the common factors.

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Step 3.1.3.3.4.1

Factor $2$ out of $2$ .

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

$y = - 5 z + 1$

Step 3.1.3.3.4.2

Cancel the common factor.

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

$y = - 5 z + 1$

Step 3.1.3.3.4.3

Rewrite the expression.

$x = \frac{8 z - 1}{1}$

$y = - 5 z + 1$

Step 3.1.3.3.4.4

Divide $8 z - 1$ by $1$ .

$x = 8 z - 1$

$y = - 5 z + 1$

$x = 8 z - 1$

$y = - 5 z + 1$

$x = 8 z - 1$

$y = - 5 z + 1$

$x = 8 z - 1$

$y = - 5 z + 1$

$x = 8 z - 1$

$y = - 5 z + 1$

$x = 8 z - 1$

$y = - 5 z + 1$