Algebra Examples

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

Step 1

Add $z$ to both sides of the equation.

$y = 5 + z$

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

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

Step 2

Replace all occurrences of $y$ with $5 + z$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $- 4 x - 4 y + z = 3$ with $5 + z$ .

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

$y = 5 + z$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 4 x - 4 (5 + z) + z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

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

$y = 5 + z$

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

Step 2.2.1.1.2

Multiply $- 4$ by $5$ .

$- 4 x - 20 - 4 z + z = 3$

$y = 5 + z$

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

$- 4 x - 20 - 4 z + z = 3$

$y = 5 + z$

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

Step 2.2.1.2

Add $- 4 z$ and $z$ .

$- 4 x - 20 - 3 z = 3$

$y = 5 + z$

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

$- 4 x - 20 - 3 z = 3$

$y = 5 + z$

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

$- 4 x - 20 - 3 z = 3$

$y = 5 + z$

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

Step 2.3

Replace all occurrences of $y$ in $- 3 x + y + 2 z = - 4$ with $5 + z$ .

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

$- 4 x - 20 - 3 z = 3$

$y = 5 + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

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

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Step 2.4.1.1

Remove parentheses.

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

$- 4 x - 20 - 3 z = 3$

$y = 5 + z$

Step 2.4.1.2

Add $z$ and $2 z$ .

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

$- 4 x - 20 - 3 z = 3$

$y = 5 + z$

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

$- 4 x - 20 - 3 z = 3$

$y = 5 + z$

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

$- 4 x - 20 - 3 z = 3$

$y = 5 + z$

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

$- 4 x - 20 - 3 z = 3$

$y = 5 + z$

Step 3

Reorder $5$ and $z$ .

$y = z + 5$

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

$- 4 x - 20 - 3 z = 3$

Step 4

Solve for $z$ in $y = z + 5$ .

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Step 4.1

Rewrite the equation as $z + 5 = y$ .

$z + 5 = y$

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

$- 4 x - 20 - 3 z = 3$

Step 4.2

Subtract $5$ from both sides of the equation.

$z = y - 5$

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

$- 4 x - 20 - 3 z = 3$

$z = y - 5$

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

$- 4 x - 20 - 3 z = 3$

Step 5

Replace all occurrences of $z$ with $y - 5$ in each equation.

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Step 5.1

Replace all occurrences of $z$ in $- 3 x + 5 + 3 z = - 4$ with $y - 5$ .

$- 3 x + 5 + 3 (y - 5) = - 4$

$z = y - 5$

$- 4 x - 20 - 3 z = 3$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $- 3 x + 5 + 3 (y - 5)$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$- 3 x + 5 + 3 y + 3 \cdot - 5 = - 4$

$z = y - 5$

$- 4 x - 20 - 3 z = 3$

Step 5.2.1.1.2

Multiply $3$ by $- 5$ .

$- 3 x + 5 + 3 y - 15 = - 4$

$z = y - 5$

$- 4 x - 20 - 3 z = 3$

$- 3 x + 5 + 3 y - 15 = - 4$

$z = y - 5$

$- 4 x - 20 - 3 z = 3$

Step 5.2.1.2

Subtract $15$ from $5$ .

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$- 4 x - 20 - 3 z = 3$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$- 4 x - 20 - 3 z = 3$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$- 4 x - 20 - 3 z = 3$

Step 5.3

Replace all occurrences of $z$ in $- 4 x - 20 - 3 z = 3$ with $y - 5$ .

$- 4 x - 20 - 3 (y - 5) = 3$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $- 4 x - 20 - 3 (y - 5)$ .

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Step 5.4.1.1

Simplify each term.

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Step 5.4.1.1.1

Apply the distributive property.

$- 4 x - 20 - 3 y - 3 \cdot - 5 = 3$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 5.4.1.1.2

Multiply $- 3$ by $- 5$ .

$- 4 x - 20 - 3 y + 15 = 3$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$- 4 x - 20 - 3 y + 15 = 3$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 5.4.1.2

Add $- 20$ and $15$ .

$- 4 x - 3 y - 5 = 3$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$- 4 x - 3 y - 5 = 3$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$- 4 x - 3 y - 5 = 3$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$- 4 x - 3 y - 5 = 3$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 6

Solve for $x$ in $- 4 x - 3 y - 5 = 3$ .

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Step 6.1

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

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Step 6.1.1

Add $3 y$ to both sides of the equation.

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

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 6.1.2

Add $5$ to both sides of the equation.

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

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 6.1.3

Add $3$ and $5$ .

$- 4 x = 3 y + 8$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$- 4 x = 3 y + 8$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 6.2

Divide each term in $- 4 x = 3 y + 8$ by $- 4$ and simplify.

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Step 6.2.1

Divide each term in $- 4 x = 3 y + 8$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{3 y}{- 4} + \frac{8}{- 4}$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $- 4$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{3 y}{- 4} + \frac{8}{- 4}$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 y}{- 4} + \frac{8}{- 4}$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$x = \frac{3 y}{- 4} + \frac{8}{- 4}$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$x = \frac{3 y}{- 4} + \frac{8}{- 4}$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Simplify each term.

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Step 6.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{3 y}{4} + \frac{8}{- 4}$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 6.2.3.1.2

Divide $8$ by $- 4$ .

$x = - \frac{3 y}{4} - 2$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$x = - \frac{3 y}{4} - 2$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$x = - \frac{3 y}{4} - 2$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$x = - \frac{3 y}{4} - 2$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

$x = - \frac{3 y}{4} - 2$

$- 3 x + 3 y - 10 = - 4$

$z = y - 5$

Step 7

Replace all occurrences of $x$ with $- \frac{3 y}{4} - 2$ in each equation.

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Step 7.1

Replace all occurrences of $x$ in $- 3 x + 3 y - 10 = - 4$ with $- \frac{3 y}{4} - 2$ .

$- 3 (- \frac{3 y}{4} - 2) + 3 y - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2

Simplify the left side.

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Step 7.2.1

Simplify $- 3 (- \frac{3 y}{4} - 2) + 3 y - 10$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Apply the distributive property.

$- 3 (- \frac{3 y}{4}) - 3 \cdot - 2 + 3 y - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.1.2

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

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Step 7.2.1.1.2.1

Multiply $- 1$ by $- 3$ .

$3 (\frac{3 y}{4}) - 3 \cdot - 2 + 3 y - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.1.2.2

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

$\frac{3 (3 y)}{4} - 3 \cdot - 2 + 3 y - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.1.2.3

Multiply $3$ by $3$ .

$\frac{9 y}{4} - 3 \cdot - 2 + 3 y - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$\frac{9 y}{4} - 3 \cdot - 2 + 3 y - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.1.3

Multiply $- 3$ by $- 2$ .

$\frac{9 y}{4} + 6 + 3 y - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$\frac{9 y}{4} + 6 + 3 y - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.2

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

$\frac{9 y}{4} + 3 y \cdot \frac{4}{4} + 6 - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.3

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

$\frac{9 y}{4} + \frac{3 y \cdot 4}{4} + 6 - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.4

Combine the numerators over the common denominator.

$\frac{9 y + 3 y \cdot 4}{4} + 6 - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.5

Find the common denominator.

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Step 7.2.1.5.1

Write $6$ as a fraction with denominator $1$ .

$\frac{9 y + 3 y \cdot 4}{4} + \frac{6}{1} - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.5.2

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$\frac{9 y + 3 y \cdot 4}{4} + \frac{6}{1} \cdot \frac{4}{4} - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.5.3

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$\frac{9 y + 3 y \cdot 4}{4} + \frac{6 \cdot 4}{4} - 10 = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.5.4

Write $- 10$ as a fraction with denominator $1$ .

$\frac{9 y + 3 y \cdot 4}{4} + \frac{6 \cdot 4}{4} + \frac{- 10}{1} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.5.5

Multiply $\frac{- 10}{1}$ by $\frac{4}{4}$ .

$\frac{9 y + 3 y \cdot 4}{4} + \frac{6 \cdot 4}{4} + \frac{- 10}{1} \cdot \frac{4}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.5.6

Multiply $\frac{- 10}{1}$ by $\frac{4}{4}$ .

$\frac{9 y + 3 y \cdot 4}{4} + \frac{6 \cdot 4}{4} + \frac{- 10 \cdot 4}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$\frac{9 y + 3 y \cdot 4}{4} + \frac{6 \cdot 4}{4} + \frac{- 10 \cdot 4}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.6

Combine the numerators over the common denominator.

$\frac{9 y + 3 y \cdot 4 + 6 \cdot 4 - 10 \cdot 4}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.7

Simplify each term.

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Step 7.2.1.7.1

Multiply $4$ by $3$ .

$\frac{9 y + 12 y + 6 \cdot 4 - 10 \cdot 4}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.7.2

Multiply $6$ by $4$ .

$\frac{9 y + 12 y + 24 - 10 \cdot 4}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.7.3

Multiply $- 10$ by $4$ .

$\frac{9 y + 12 y + 24 - 40}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$\frac{9 y + 12 y + 24 - 40}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.8

Simplify by adding terms.

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Step 7.2.1.8.1

Add $9 y$ and $12 y$ .

$\frac{21 y + 24 - 40}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 7.2.1.8.2

Subtract $40$ from $24$ .

$\frac{21 y - 16}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$\frac{21 y - 16}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$\frac{21 y - 16}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$\frac{21 y - 16}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$\frac{21 y - 16}{4} = - 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 8

Solve for $y$ in $\frac{21 y - 16}{4} = - 4$ .

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Step 8.1

Multiply both sides by $4$ .

$\frac{21 y - 16}{4} \cdot 4 = - 4 \cdot 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 8.2

Simplify.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

Cancel the common factor of $4$ .

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Step 8.2.1.1.1

Cancel the common factor.

$\frac{21 y - 16}{4} \cdot 4 = - 4 \cdot 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 8.2.1.1.2

Rewrite the expression.

$21 y - 16 = - 4 \cdot 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$21 y - 16 = - 4 \cdot 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$21 y - 16 = - 4 \cdot 4$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Multiply $- 4$ by $4$ .

$21 y - 16 = - 16$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$21 y - 16 = - 16$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$21 y - 16 = - 16$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 8.3

Solve for $y$ .

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Step 8.3.1

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

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Step 8.3.1.1

Add $16$ to both sides of the equation.

$21 y = - 16 + 16$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 8.3.1.2

Add $- 16$ and $16$ .

$21 y = 0$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$21 y = 0$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 8.3.2

Divide each term in $21 y = 0$ by $21$ and simplify.

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Step 8.3.2.1

Divide each term in $21 y = 0$ by $21$ .

$\frac{21 y}{21} = \frac{0}{21}$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 8.3.2.2

Simplify the left side.

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Step 8.3.2.2.1

Cancel the common factor of $21$ .

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Step 8.3.2.2.1.1

Cancel the common factor.

$\frac{21 y}{21} = \frac{0}{21}$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 8.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{21}$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$y = \frac{0}{21}$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$y = \frac{0}{21}$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 8.3.2.3

Simplify the right side.

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Step 8.3.2.3.1

Divide $0$ by $21$ .

$y = 0$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$y = 0$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$y = 0$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$y = 0$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

$y = 0$

$x = - \frac{3 y}{4} - 2$

$z = y - 5$

Step 9

Replace all occurrences of $y$ with $0$ in each equation.

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Step 9.1

Replace all occurrences of $y$ in $x = - \frac{3 y}{4} - 2$ with $0$ .

$x = - \frac{3 (0)}{4} - 2$

$y = 0$

$z = y - 5$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $- \frac{3 (0)}{4} - 2$ .

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Step 9.2.1.1

Simplify each term.

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Step 9.2.1.1.1

Cancel the common factor of $0$ and $4$ .

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Step 9.2.1.1.1.1

Factor $4$ out of $3 (0)$ .

$x = - \frac{4 (3 \cdot (0))}{4} - 2$

$y = 0$

$z = y - 5$

Step 9.2.1.1.1.2

Cancel the common factors.

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Step 9.2.1.1.1.2.1

Factor $4$ out of $4$ .

$x = - \frac{4 (3 \cdot (0))}{4 (1)} - 2$

$y = 0$

$z = y - 5$

Step 9.2.1.1.1.2.2

Cancel the common factor.

$x = - \frac{4 (3 \cdot (0))}{4 \cdot 1} - 2$

$y = 0$

$z = y - 5$

Step 9.2.1.1.1.2.3

Rewrite the expression.

$x = - \frac{3 \cdot (0)}{1} - 2$

$y = 0$

$z = y - 5$

Step 9.2.1.1.1.2.4

Divide $3 \cdot (0)$ by $1$ .

$x = - (3 \cdot (0)) - 2$

$y = 0$

$z = y - 5$

$x = - (3 \cdot (0)) - 2$

$y = 0$

$z = y - 5$

$x = - (3 \cdot (0)) - 2$

$y = 0$

$z = y - 5$

Step 9.2.1.1.2

Multiply $- (3 \cdot (0))$ .

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Step 9.2.1.1.2.1

Multiply $3$ by $0$ .

$x = - 0 - 2$

$y = 0$

$z = y - 5$

Step 9.2.1.1.2.2

Multiply $- 1$ by $0$ .

$x = 0 - 2$

$y = 0$

$z = y - 5$

$x = 0 - 2$

$y = 0$

$z = y - 5$

$x = 0 - 2$

$y = 0$

$z = y - 5$

Step 9.2.1.2

Subtract $2$ from $0$ .

$x = - 2$

$y = 0$

$z = y - 5$

$x = - 2$

$y = 0$

$z = y - 5$

$x = - 2$

$y = 0$

$z = y - 5$

Step 9.3

Replace all occurrences of $y$ in $z = y - 5$ with $0$ .

$z = (0) - 5$

$x = - 2$

$y = 0$

Step 9.4

Simplify the right side.

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Step 9.4.1

Subtract $5$ from $0$ .

$z = - 5$

$x = - 2$

$y = 0$

$z = - 5$

$x = - 2$

$y = 0$

$z = - 5$

$x = - 2$

$y = 0$

Step 10

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 2, 0, - 5)$

Step 11

The result can be shown in multiple forms.

Point Form:

$(- 2, 0, - 5)$

Equation Form:

$x = - 2, y = 0, z = - 5$