Precalculus Examples

$x + y - z = 4$ , $y + w = - 2$ , $x - y = 1$ , $x + z + w = 3$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x - z = 4 - y$

$y + w = - 2$

$x - y = 1$

$x + z + w = 3$

Step 1.2

Add $z$ to both sides of the equation.

$x = 4 - y + z$

$y + w = - 2$

$x - y = 1$

$x + z + w = 3$

$x = 4 - y + z$

$y + w = - 2$

$x - y = 1$

$x + z + w = 3$

Step 2

Replace all occurrences of $x$ with $4 - y + z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x - y = 1$ with $4 - y + z$ .

$(4 - y + z) - y = 1$

$x = 4 - y + z$

$y + w = - 2$

$x + z + w = 3$

Step 2.2

Simplify the left side.

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Step 2.2.1

Subtract $y$ from $- y$ .

$4 - 2 y + z = 1$

$x = 4 - y + z$

$y + w = - 2$

$x + z + w = 3$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$y + w = - 2$

$x + z + w = 3$

Step 2.3

Replace all occurrences of $x$ in $x + z + w = 3$ with $4 - y + z$ .

$(4 - y + z) + z + w = 3$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$y + w = - 2$

Step 2.4

Simplify the left side.

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Step 2.4.1

Add $z$ and $z$ .

$4 - y + 2 z + w = 3$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$y + w = - 2$

$4 - y + 2 z + w = 3$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$y + w = - 2$

$4 - y + 2 z + w = 3$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$y + w = - 2$

Step 3

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

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Step 3.1

Subtract $4$ from both sides of the equation.

$- y + 2 z + w = 3 - 4$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$y + w = - 2$

Step 3.2

Add $y$ to both sides of the equation.

$2 z + w = 3 - 4 + y$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$y + w = - 2$

Step 3.3

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

$w = 3 - 4 + y - 2 z$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$y + w = - 2$

Step 3.4

Subtract $4$ from $3$ .

$w = - 1 + y - 2 z$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$y + w = - 2$

$w = - 1 + y - 2 z$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$y + w = - 2$

Step 4

Replace all occurrences of $w$ with $- 1 + y - 2 z$ in each equation.

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Step 4.1

Replace all occurrences of $w$ in $y + w = - 2$ with $- 1 + y - 2 z$ .

$y - 1 + y - 2 z = - 2$

$w = - 1 + y - 2 z$

$4 - 2 y + z = 1$

$x = 4 - y + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $y - 1 + y - 2 z$ .

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Step 4.2.1.1

Remove parentheses.

$y - 1 + y - 2 z = - 2$

$w = - 1 + y - 2 z$

$4 - 2 y + z = 1$

$x = 4 - y + z$

Step 4.2.1.2

Add $y$ and $y$ .

$2 y - 1 - 2 z = - 2$

$w = - 1 + y - 2 z$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$2 y - 1 - 2 z = - 2$

$w = - 1 + y - 2 z$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$2 y - 1 - 2 z = - 2$

$w = - 1 + y - 2 z$

$4 - 2 y + z = 1$

$x = 4 - y + z$

$2 y - 1 - 2 z = - 2$

$w = - 1 + y - 2 z$

$4 - 2 y + z = 1$

$x = 4 - y + z$

Step 5

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

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Step 5.1

Subtract $4$ from both sides of the equation.

$- 2 y + z = 1 - 4$

$2 y - 1 - 2 z = - 2$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

Step 5.2

Add $2 y$ to both sides of the equation.

$z = 1 - 4 + 2 y$

$2 y - 1 - 2 z = - 2$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

Step 5.3

Subtract $4$ from $1$ .

$z = - 3 + 2 y$

$2 y - 1 - 2 z = - 2$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

$z = - 3 + 2 y$

$2 y - 1 - 2 z = - 2$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

Step 6

Replace all occurrences of $z$ with $- 3 + 2 y$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $2 y - 1 - 2 z = - 2$ with $- 3 + 2 y$ .

$2 y - 1 - 2 (- 3 + 2 y) = - 2$

$z = - 3 + 2 y$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify $2 y - 1 - 2 (- 3 + 2 y)$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Apply the distributive property.

$2 y - 1 - 2 \cdot - 3 - 2 (2 y) = - 2$

$z = - 3 + 2 y$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

Step 6.2.1.1.2

Multiply $- 2$ by $- 3$ .

$2 y - 1 + 6 - 2 (2 y) = - 2$

$z = - 3 + 2 y$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

Step 6.2.1.1.3

Multiply $2$ by $- 2$ .

$2 y - 1 + 6 - 4 y = - 2$

$z = - 3 + 2 y$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

$2 y - 1 + 6 - 4 y = - 2$

$z = - 3 + 2 y$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

Step 6.2.1.2

Simplify by adding terms.

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Step 6.2.1.2.1

Subtract $4 y$ from $2 y$ .

$- 2 y - 1 + 6 = - 2$

$z = - 3 + 2 y$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

Step 6.2.1.2.2

Add $- 1$ and $6$ .

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$w = - 1 + y - 2 z$

$x = 4 - y + z$

Step 6.3

Replace all occurrences of $z$ in $w = - 1 + y - 2 z$ with $- 3 + 2 y$ .

$w = - 1 + y - 2 (- 3 + 2 y)$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = 4 - y + z$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 1 + y - 2 (- 3 + 2 y)$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Apply the distributive property.

$w = - 1 + y - 2 \cdot - 3 - 2 (2 y)$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = 4 - y + z$

Step 6.4.1.1.2

Multiply $- 2$ by $- 3$ .

$w = - 1 + y + 6 - 2 (2 y)$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = 4 - y + z$

Step 6.4.1.1.3

Multiply $2$ by $- 2$ .

$w = - 1 + y + 6 - 4 y$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = 4 - y + z$

$w = - 1 + y + 6 - 4 y$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = 4 - y + z$

Step 6.4.1.2

Simplify by adding terms.

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Step 6.4.1.2.1

Add $- 1$ and $6$ .

$w = y + 5 - 4 y$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = 4 - y + z$

Step 6.4.1.2.2

Subtract $4 y$ from $y$ .

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = 4 - y + z$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = 4 - y + z$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = 4 - y + z$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = 4 - y + z$

Step 6.5

Replace all occurrences of $z$ in $x = 4 - y + z$ with $- 3 + 2 y$ .

$x = 4 - y - 3 + 2 y$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

Step 6.6

Simplify $x = 4 - y - 3 + 2 y$ .

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Step 6.6.1

Simplify the left side.

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Step 6.6.1.1

Remove parentheses.

$x = 4 - y - 3 + 2 y$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = 4 - y - 3 + 2 y$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

Step 6.6.2

Simplify the right side.

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Step 6.6.2.1

Simplify $4 - y - 3 + 2 y$ .

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Step 6.6.2.1.1

Subtract $3$ from $4$ .

$x = - y + 1 + 2 y$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

Step 6.6.2.1.2

Add $- y$ and $2 y$ .

$x = y + 1$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = y + 1$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = y + 1$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = y + 1$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

$x = y + 1$

$w = - 3 y + 5$

$- 2 y + 5 = - 2$

$z = - 3 + 2 y$

Step 7

Solve for $y$ in $- 2 y + 5 = - 2$ .

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Step 7.1

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

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Step 7.1.1

Subtract $5$ from both sides of the equation.

$- 2 y = - 2 - 5$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

Step 7.1.2

Subtract $5$ from $- 2$ .

$- 2 y = - 7$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

$- 2 y = - 7$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

Step 7.2

Divide each term in $- 2 y = - 7$ by $- 2$ and simplify.

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Step 7.2.1

Divide each term in $- 2 y = - 7$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- 7}{- 2}$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $- 2$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{- 7}{- 2}$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

Step 7.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 7}{- 2}$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

$y = \frac{- 7}{- 2}$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

$y = \frac{- 7}{- 2}$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Dividing two negative values results in a positive value.

$y = \frac{7}{2}$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

$y = \frac{7}{2}$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

$y = \frac{7}{2}$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

$y = \frac{7}{2}$

$x = y + 1$

$w = - 3 y + 5$

$z = - 3 + 2 y$

Step 8

Replace all occurrences of $y$ with $\frac{7}{2}$ in each equation.

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Step 8.1

Replace all occurrences of $y$ in $x = y + 1$ with $\frac{7}{2}$ .

$x = (\frac{7}{2}) + 1$

$y = \frac{7}{2}$

$w = - 3 y + 5$

$z = - 3 + 2 y$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $(\frac{7}{2}) + 1$ .

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Step 8.2.1.1

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

$x = \frac{7}{2} + \frac{2}{2}$

$y = \frac{7}{2}$

$w = - 3 y + 5$

$z = - 3 + 2 y$

Step 8.2.1.2

Combine the numerators over the common denominator.

$x = \frac{7 + 2}{2}$

$y = \frac{7}{2}$

$w = - 3 y + 5$

$z = - 3 + 2 y$

Step 8.2.1.3

Add $7$ and $2$ .

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$w = - 3 y + 5$

$z = - 3 + 2 y$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$w = - 3 y + 5$

$z = - 3 + 2 y$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$w = - 3 y + 5$

$z = - 3 + 2 y$

Step 8.3

Replace all occurrences of $y$ in $w = - 3 y + 5$ with $\frac{7}{2}$ .

$w = - 3 (\frac{7}{2}) + 5$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

Step 8.4

Simplify the right side.

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Step 8.4.1

Simplify $- 3 (\frac{7}{2}) + 5$ .

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Step 8.4.1.1

Simplify each term.

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Step 8.4.1.1.1

Multiply $- 3 (\frac{7}{2})$ .

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Step 8.4.1.1.1.1

Combine $- 3$ and $\frac{7}{2}$ .

$w = \frac{- 3 \cdot 7}{2} + 5$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

Step 8.4.1.1.1.2

Multiply $- 3$ by $7$ .

$w = \frac{- 21}{2} + 5$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

$w = \frac{- 21}{2} + 5$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

Step 8.4.1.1.2

Move the negative in front of the fraction.

$w = - \frac{21}{2} + 5$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

$w = - \frac{21}{2} + 5$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

Step 8.4.1.2

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

$w = - \frac{21}{2} + 5 \cdot \frac{2}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

Step 8.4.1.3

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

$w = - \frac{21}{2} + \frac{5 \cdot 2}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

Step 8.4.1.4

Combine the numerators over the common denominator.

$w = \frac{- 21 + 5 \cdot 2}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

Step 8.4.1.5

Simplify the numerator.

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Step 8.4.1.5.1

Multiply $5$ by $2$ .

$w = \frac{- 21 + 10}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

Step 8.4.1.5.2

Add $- 21$ and $10$ .

$w = \frac{- 11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

$w = \frac{- 11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

Step 8.4.1.6

Move the negative in front of the fraction.

$w = - \frac{11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

$w = - \frac{11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

$w = - \frac{11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 2 y$

Step 8.5

Replace all occurrences of $y$ in $z = - 3 + 2 y$ with $\frac{7}{2}$ .

$z = - 3 + 2 (\frac{7}{2})$

$w = - \frac{11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

Step 8.6

Simplify the right side.

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Step 8.6.1

Simplify $- 3 + 2 (\frac{7}{2})$ .

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Step 8.6.1.1

Cancel the common factor of $2$ .

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Step 8.6.1.1.1

Cancel the common factor.

$z = - 3 + 2 (\frac{7}{2})$

$w = - \frac{11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

Step 8.6.1.1.2

Rewrite the expression.

$z = - 3 + 7$

$w = - \frac{11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = - 3 + 7$

$w = - \frac{11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

Step 8.6.1.2

Add $- 3$ and $7$ .

$z = 4$

$w = - \frac{11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = 4$

$w = - \frac{11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = 4$

$w = - \frac{11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

$z = 4$

$w = - \frac{11}{2}$

$x = \frac{9}{2}$

$y = \frac{7}{2}$

Step 9

List all of the solutions.

$z = 4, w = - \frac{11}{2}, x = \frac{9}{2}, y = \frac{7}{2}$