Precalculus Examples

$2 x + 3 y + 3 z = 2$ , $4 x + 18 y + 15 z = 55$

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.

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

$4 x + 18 y + 15 z = 55$

Step 1.1.2

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

$2 x = 2 - 3 y - 3 z$

$4 x + 18 y + 15 z = 55$

$2 x = 2 - 3 y - 3 z$

$4 x + 18 y + 15 z = 55$

Step 1.2

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

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Step 1.2.1

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

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

$4 x + 18 y + 15 z = 55$

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.

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

$4 x + 18 y + 15 z = 55$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$4 x + 18 y + 15 z = 55$

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

$4 x + 18 y + 15 z = 55$

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

$4 x + 18 y + 15 z = 55$

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 $2$ by $2$ .

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

$4 x + 18 y + 15 z = 55$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$4 x + 18 y + 15 z = 55$

Step 1.2.3.1.3

Move the negative in front of the fraction.

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

$4 x + 18 y + 15 z = 55$

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

$4 x + 18 y + 15 z = 55$

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

$4 x + 18 y + 15 z = 55$

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

$4 x + 18 y + 15 z = 55$

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

$4 x + 18 y + 15 z = 55$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify $4 (1 - \frac{3 y}{2} - \frac{3 z}{2}) + 18 y + 15 z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$4 \cdot 1 + 4 (- \frac{3 y}{2}) + 4 (- \frac{3 z}{2}) + 18 y + 15 z = 55$

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

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $4$ by $1$ .

$4 + 4 (- \frac{3 y}{2}) + 4 (- \frac{3 z}{2}) + 18 y + 15 z = 55$

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

Step 2.1.1.2.2

Cancel the common factor of $2$ .

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Step 2.1.1.2.2.1

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

$4 + 4 (\frac{- 3 y}{2}) + 4 (- \frac{3 z}{2}) + 18 y + 15 z = 55$

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

Step 2.1.1.2.2.2

Factor $2$ out of $4$ .

$4 + 2 (2) (\frac{- 3 y}{2}) + 4 (- \frac{3 z}{2}) + 18 y + 15 z = 55$

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

Step 2.1.1.2.2.3

Cancel the common factor.

$4 + 2 \cdot (2 (\frac{- 3 y}{2})) + 4 (- \frac{3 z}{2}) + 18 y + 15 z = 55$

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

Step 2.1.1.2.2.4

Rewrite the expression.

$4 + 2 (- 3 y) + 4 (- \frac{3 z}{2}) + 18 y + 15 z = 55$

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

$4 + 2 (- 3 y) + 4 (- \frac{3 z}{2}) + 18 y + 15 z = 55$

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

Step 2.1.1.2.3

Multiply $- 3$ by $2$ .

$4 - 6 y + 4 (- \frac{3 z}{2}) + 18 y + 15 z = 55$

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

Step 2.1.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.1.2.4.1

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

$4 - 6 y + 4 (\frac{- 3 z}{2}) + 18 y + 15 z = 55$

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

Step 2.1.1.2.4.2

Factor $2$ out of $4$ .

$4 - 6 y + 2 (2) (\frac{- 3 z}{2}) + 18 y + 15 z = 55$

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

Step 2.1.1.2.4.3

Cancel the common factor.

$4 - 6 y + 2 \cdot (2 (\frac{- 3 z}{2})) + 18 y + 15 z = 55$

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

Step 2.1.1.2.4.4

Rewrite the expression.

$4 - 6 y + 2 (- 3 z) + 18 y + 15 z = 55$

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

$4 - 6 y + 2 (- 3 z) + 18 y + 15 z = 55$

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

Step 2.1.1.2.5

Multiply $- 3$ by $2$ .

$4 - 6 y - 6 z + 18 y + 15 z = 55$

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

$4 - 6 y - 6 z + 18 y + 15 z = 55$

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

$4 - 6 y - 6 z + 18 y + 15 z = 55$

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

Step 2.1.2

Simplify by adding terms.

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Step 2.1.2.1

Add $- 6 y$ and $18 y$ .

$4 + 12 y - 6 z + 15 z = 55$

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

Step 2.1.2.2

Add $- 6 z$ and $15 z$ .

$4 + 12 y + 9 z = 55$

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

$4 + 12 y + 9 z = 55$

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

$4 + 12 y + 9 z = 55$

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

Step 2.2

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

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Step 2.2.1

Subtract $4$ from both sides of the equation.

$12 y + 9 z = 55 - 4$

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

Step 2.2.2

Subtract $9 z$ from both sides of the equation.

$12 y = 55 - 4 - 9 z$

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

Step 2.2.3

Subtract $4$ from $55$ .

$12 y = 51 - 9 z$

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

$12 y = 51 - 9 z$

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

Step 2.3

Divide each term in $12 y = 51 - 9 z$ by $12$ and simplify.

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Step 2.3.1

Divide each term in $12 y = 51 - 9 z$ by $12$ .

$\frac{12 y}{12} = \frac{51}{12} + \frac{- 9 z}{12}$

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $12$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{12 y}{12} = \frac{51}{12} + \frac{- 9 z}{12}$

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

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{51}{12} + \frac{- 9 z}{12}$

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

$y = \frac{51}{12} + \frac{- 9 z}{12}$

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

$y = \frac{51}{12} + \frac{- 9 z}{12}$

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Cancel the common factor of $51$ and $12$ .

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Step 2.3.3.1.1.1

Factor $3$ out of $51$ .

$y = \frac{3 (17)}{12} + \frac{- 9 z}{12}$

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

Step 2.3.3.1.1.2

Cancel the common factors.

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Step 2.3.3.1.1.2.1

Factor $3$ out of $12$ .

$y = \frac{3 \cdot 17}{3 \cdot 4} + \frac{- 9 z}{12}$

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

Step 2.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot 17}{3 \cdot 4} + \frac{- 9 z}{12}$

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

Step 2.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{17}{4} + \frac{- 9 z}{12}$

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

$y = \frac{17}{4} + \frac{- 9 z}{12}$

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

$y = \frac{17}{4} + \frac{- 9 z}{12}$

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

Step 2.3.3.1.2

Cancel the common factor of $- 9$ and $12$ .

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Step 2.3.3.1.2.1

Factor $3$ out of $- 9 z$ .

$y = \frac{17}{4} + \frac{3 (- 3 z)}{12}$

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

Step 2.3.3.1.2.2

Cancel the common factors.

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Step 2.3.3.1.2.2.1

Factor $3$ out of $12$ .

$y = \frac{17}{4} + \frac{3 (- 3 z)}{3 (4)}$

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

Step 2.3.3.1.2.2.2

Cancel the common factor.

$y = \frac{17}{4} + \frac{3 (- 3 z)}{3 \cdot 4}$

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

Step 2.3.3.1.2.2.3

Rewrite the expression.

$y = \frac{17}{4} + \frac{- 3 z}{4}$

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

$y = \frac{17}{4} + \frac{- 3 z}{4}$

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

$y = \frac{17}{4} + \frac{- 3 z}{4}$

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

Step 2.3.3.1.3

Move the negative in front of the fraction.

$y = \frac{17}{4} - \frac{3 z}{4}$

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

$y = \frac{17}{4} - \frac{3 z}{4}$

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

$y = \frac{17}{4} - \frac{3 z}{4}$

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

$y = \frac{17}{4} - \frac{3 z}{4}$

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

$y = \frac{17}{4} - \frac{3 z}{4}$

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

Step 3

Simplify the right side.

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Step 3.1

Simplify $1 - \frac{3 (\frac{17}{4} - \frac{3 z}{4})}{2} - \frac{3 z}{2}$ .

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Step 3.1.1

Combine the numerators over the common denominator.

$x = 1 + \frac{- 3 (\frac{17}{4} - \frac{3 z}{4}) - 3 z}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Apply the distributive property.

$x = 1 + \frac{- 3 (\frac{17}{4}) - 3 (- \frac{3 z}{4}) - 3 z}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.2.2

Multiply $- 3 (\frac{17}{4})$ .

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Step 3.1.2.2.1

Combine $- 3$ and $\frac{17}{4}$ .

$x = 1 + \frac{\frac{- 3 \cdot 17}{4} - 3 (- \frac{3 z}{4}) - 3 z}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.2.2.2

Multiply $- 3$ by $17$ .

$x = 1 + \frac{\frac{- 51}{4} - 3 (- \frac{3 z}{4}) - 3 z}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = 1 + \frac{\frac{- 51}{4} - 3 (- \frac{3 z}{4}) - 3 z}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.2.3

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

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Step 3.1.2.3.1

Multiply $- 1$ by $- 3$ .

$x = 1 + \frac{\frac{- 51}{4} + 3 (\frac{3 z}{4}) - 3 z}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.2.3.2

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

$x = 1 + \frac{\frac{- 51}{4} + \frac{3 (3 z)}{4} - 3 z}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.2.3.3

Multiply $3$ by $3$ .

$x = 1 + \frac{\frac{- 51}{4} + \frac{9 z}{4} - 3 z}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = 1 + \frac{\frac{- 51}{4} + \frac{9 z}{4} - 3 z}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.2.4

Move the negative in front of the fraction.

$x = 1 + \frac{- \frac{51}{4} + \frac{9 z}{4} - 3 z}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = 1 + \frac{- \frac{51}{4} + \frac{9 z}{4} - 3 z}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.3

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

$x = 1 + \frac{- \frac{51}{4} + \frac{9 z}{4} - 3 z \cdot \frac{4}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4

Simplify terms.

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Step 3.1.4.1

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

$x = 1 + \frac{- \frac{51}{4} + \frac{9 z}{4} + \frac{- 3 z \cdot 4}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.2

Simplify terms.

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Step 3.1.4.2.1

Combine the numerators over the common denominator.

$x = 1 + \frac{- \frac{51}{4} + \frac{9 z - 3 z \cdot 4}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.2.2

Combine the numerators over the common denominator.

$x = 1 + \frac{\frac{- 51 + 9 z - 3 z \cdot 4}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.2.3

Multiply $4$ by $- 3$ .

$x = 1 + \frac{\frac{- 51 + 9 z - 12 z}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.2.4

Subtract $12 z$ from $9 z$ .

$x = 1 + \frac{\frac{- 51 - 3 z}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.2.5

Factor $3$ out of $- 51 - 3 z$ .

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Step 3.1.4.2.5.1

Factor $3$ out of $- 51$ .

$x = 1 + \frac{\frac{3 (- 17) - 3 z}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.2.5.2

Factor $3$ out of $- 3 z$ .

$x = 1 + \frac{\frac{3 (- 17) + 3 (- z)}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.2.5.3

Factor $3$ out of $3 (- 17) + 3 (- z)$ .

$x = 1 + \frac{\frac{3 (- 17 - z)}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = 1 + \frac{\frac{3 (- 17 - z)}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.2.6

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

$x = 1 + \frac{\frac{3 (- 1 \cdot 17 - z)}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.2.7

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

$x = 1 + \frac{\frac{3 (- 1 \cdot 17 - (z))}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.2.8

Factor $- 1$ out of $- 1 (17) - (z)$ .

$x = 1 + \frac{\frac{3 (- 1 (17 + z))}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.2.9

Move the negative in front of the fraction.

$x = 1 + \frac{- \frac{3 (17 + z)}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = 1 + \frac{- \frac{3 (17 + z)}{4}}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.3

Simplify each term.

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Step 3.1.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = 1 - \frac{3 (17 + z)}{4} \cdot \frac{1}{2}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.3.2

Multiply $- \frac{3 (17 + z)}{4} \cdot \frac{1}{2}$ .

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Step 3.1.4.3.2.1

Multiply $\frac{1}{2}$ by $\frac{3 (17 + z)}{4}$ .

$x = 1 - \frac{3 (17 + z)}{2 \cdot 4}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.3.2.2

Multiply $2$ by $4$ .

$x = 1 - \frac{3 (17 + z)}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = 1 - \frac{3 (17 + z)}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = 1 - \frac{3 (17 + z)}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.4

Combine into one fraction.

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Step 3.1.4.4.1

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

$x = \frac{8}{8} - \frac{3 (17 + z)}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.4.4.2

Combine the numerators over the common denominator.

$x = \frac{8 - 3 (17 + z)}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = \frac{8 - 3 (17 + z)}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = \frac{8 - 3 (17 + z)}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Apply the distributive property.

$x = \frac{8 - 3 \cdot 17 - 3 z}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.5.2

Multiply $- 3$ by $17$ .

$x = \frac{8 - 51 - 3 z}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.5.3

Subtract $51$ from $8$ .

$x = \frac{- 43 - 3 z}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = \frac{- 43 - 3 z}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.6

Simplify with factoring out.

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Step 3.1.6.1

Rewrite $- 43$ as $- 1 (43)$ .

$x = \frac{- 1 \cdot 43 - 3 z}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.6.2

Factor $- 1$ out of $- 3 z$ .

$x = \frac{- 1 \cdot 43 - (3 z)}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.6.3

Factor $- 1$ out of $- 1 (43) - (3 z)$ .

$x = \frac{- 1 (43 + 3 z)}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 3.1.6.4

Move the negative in front of the fraction.

$x = - \frac{43 + 3 z}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = - \frac{43 + 3 z}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = - \frac{43 + 3 z}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

$x = - \frac{43 + 3 z}{8}$

$y = \frac{17}{4} - \frac{3 z}{4}$

Step 4

Simplify the right side.

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Step 4.1

Reorder $\frac{17}{4}$ and $- \frac{3 z}{4}$ .

$y = - \frac{3 z}{4} + \frac{17}{4}$

$x = - \frac{43 + 3 z}{8}$

$y = - \frac{3 z}{4} + \frac{17}{4}$

$x = - \frac{43 + 3 z}{8}$