Algebra Examples

$- x - 2 y = - 4 z + 12$ $3 x - 6 y + z = 15$ $2 x + 5 y + 1 = 0$

Step 1

Rewrite the equation as $- 4 z + 12 = - x - 2 (- \frac{2 x}{5} - \frac{1}{5})$ .

$- 4 z + 12 = - x - 2 (- \frac{2 x}{5} - \frac{1}{5})$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2

Simplify $- x - 2 (- \frac{2 x}{5} - \frac{1}{5})$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$- 4 z + 12 = - x - 2 (- \frac{2 x}{5}) - 2 (- \frac{1}{5})$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.1.2

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

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Step 2.1.2.1

Multiply $- 1$ by $- 2$ .

$- 4 z + 12 = - x + 2 (\frac{2 x}{5}) - 2 (- \frac{1}{5})$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.1.2.2

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

$- 4 z + 12 = - x + \frac{2 (2 x)}{5} - 2 (- \frac{1}{5})$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.1.2.3

Multiply $2$ by $2$ .

$- 4 z + 12 = - x + \frac{4 x}{5} - 2 (- \frac{1}{5})$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$- 4 z + 12 = - x + \frac{4 x}{5} - 2 (- \frac{1}{5})$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.1.3

Multiply $- 2 (- \frac{1}{5})$ .

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Step 2.1.3.1

Multiply $- 1$ by $- 2$ .

$- 4 z + 12 = - x + \frac{4 x}{5} + 2 (\frac{1}{5})$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.1.3.2

Combine $2$ and $\frac{1}{5}$ .

$- 4 z + 12 = - x + \frac{4 x}{5} + \frac{2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$- 4 z + 12 = - x + \frac{4 x}{5} + \frac{2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$- 4 z + 12 = - x + \frac{4 x}{5} + \frac{2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.2

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

$- 4 z + 12 = - x \cdot \frac{5}{5} + \frac{4 x}{5} + \frac{2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.3

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

$- 4 z + 12 = \frac{- x \cdot 5}{5} + \frac{4 x}{5} + \frac{2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.4

Combine the numerators over the common denominator.

$- 4 z + 12 = \frac{- x \cdot 5 + 4 x}{5} + \frac{2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.5

Combine the numerators over the common denominator.

$- 4 z + 12 = \frac{- x \cdot 5 + 4 x + 2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.6

Multiply $5$ by $- 1$ .

$- 4 z + 12 = \frac{- 5 x + 4 x + 2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.7

Add $- 5 x$ and $4 x$ .

$- 4 z + 12 = \frac{- x + 2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.8

Factor $- 1$ out of $- x$ .

$- 4 z + 12 = \frac{- (x) + 2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.9

Rewrite $2$ as $- 1 (- 2)$ .

$- 4 z + 12 = \frac{- (x) - 1 \cdot - 2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.10

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$- 4 z + 12 = \frac{- (x - 2)}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$- 4 z + 12 = \frac{- 1 (x - 2)}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 2.11.2

Move the negative in front of the fraction.

$- 4 z + 12 = - \frac{x - 2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$- 4 z + 12 = - \frac{x - 2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$- 4 z + 12 = - \frac{x - 2}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3

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

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Step 3.1

Subtract $12$ from both sides of the equation.

$- 4 z = - \frac{x - 2}{5} - 12$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3.2

Simplify each term.

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Step 3.2.1

Split the fraction $\frac{x - 2}{5}$ into two fractions.

$- 4 z = - (\frac{x}{5} + \frac{- 2}{5}) - 12$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3.2.2

Move the negative in front of the fraction.

$- 4 z = - (\frac{x}{5} - \frac{2}{5}) - 12$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3.2.3

Apply the distributive property.

$- 4 z = - \frac{x}{5} + \frac{2}{5} - 12$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3.2.4

Multiply $- - \frac{2}{5}$ .

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Step 3.2.4.1

Multiply $- 1$ by $- 1$ .

$- 4 z = - \frac{x}{5} + 1 (\frac{2}{5}) - 12$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3.2.4.2

Multiply $\frac{2}{5}$ by $1$ .

$- 4 z = - \frac{x}{5} + \frac{2}{5} - 12$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$- 4 z = - \frac{x}{5} + \frac{2}{5} - 12$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$- 4 z = - \frac{x}{5} + \frac{2}{5} - 12$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3.3

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 4 z = - \frac{x}{5} + \frac{2}{5} - 12 \cdot \frac{5}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3.4

Combine $- 12$ and $\frac{5}{5}$ .

$- 4 z = - \frac{x}{5} + \frac{2}{5} + \frac{- 12 \cdot 5}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3.5

Combine the numerators over the common denominator.

$- 4 z = - \frac{x}{5} + \frac{2 - 12 \cdot 5}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 12$ by $5$ .

$- 4 z = - \frac{x}{5} + \frac{2 - 60}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3.6.2

Subtract $60$ from $2$ .

$- 4 z = - \frac{x}{5} + \frac{- 58}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$- 4 z = - \frac{x}{5} + \frac{- 58}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 3.7

Move the negative in front of the fraction.

$- 4 z = - \frac{x}{5} - \frac{58}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$- 4 z = - \frac{x}{5} - \frac{58}{5}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4

Divide each term in $- 4 z = - \frac{x}{5} - \frac{58}{5}$ by $- 4$ and simplify.

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Step 4.1

Divide each term in $- 4 z = - \frac{x}{5} - \frac{58}{5}$ by $- 4$ .

$\frac{- 4 z}{- 4} = \frac{- \frac{x}{5}}{- 4} + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 4 z}{- 4} = \frac{- \frac{x}{5}}{- 4} + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.2.1.2

Divide $z$ by $1$ .

$z = \frac{- \frac{x}{5}}{- 4} + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$z = \frac{- \frac{x}{5}}{- 4} + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$z = \frac{- \frac{x}{5}}{- 4} + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$z = - \frac{x}{5} \cdot \frac{1}{- 4} + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.2

Move the negative in front of the fraction.

$z = - \frac{x}{5} \cdot (- \frac{1}{4}) + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.3

Multiply $- \frac{x}{5} (- \frac{1}{4})$ .

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Step 4.3.1.3.1

Multiply $- 1$ by $- 1$ .

$z = 1 (\frac{x}{5}) \cdot \frac{1}{4} + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.3.2

Multiply $\frac{x}{5}$ by $1$ .

$z = \frac{x}{5} \cdot \frac{1}{4} + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.3.3

Multiply $\frac{x}{5}$ by $\frac{1}{4}$ .

$z = \frac{x}{5 \cdot 4} + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.3.4

Multiply $5$ by $4$ .

$z = \frac{x}{20} + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$z = \frac{x}{20} + \frac{- \frac{58}{5}}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.4

Multiply the numerator by the reciprocal of the denominator.

$z = \frac{x}{20} - \frac{58}{5} \cdot \frac{1}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.5

Cancel the common factor of $2$ .

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Step 4.3.1.5.1

Move the leading negative in $- \frac{58}{5}$ into the numerator.

$z = \frac{x}{20} + \frac{- 58}{5} \cdot \frac{1}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.5.2

Factor $2$ out of $- 58$ .

$z = \frac{x}{20} + \frac{2 (- 29)}{5} \cdot \frac{1}{- 4}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.5.3

Factor $2$ out of $- 4$ .

$z = \frac{x}{20} + \frac{2 \cdot - 29}{5} \cdot \frac{1}{2 \cdot - 2}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.5.4

Cancel the common factor.

$z = \frac{x}{20} + \frac{2 \cdot - 29}{5} \cdot \frac{1}{2 \cdot - 2}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.5.5

Rewrite the expression.

$z = \frac{x}{20} + \frac{- 29}{5} \cdot \frac{1}{- 2}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$z = \frac{x}{20} + \frac{- 29}{5} \cdot \frac{1}{- 2}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.6

Multiply $\frac{- 29}{5}$ by $\frac{1}{- 2}$ .

$z = \frac{x}{20} + \frac{- 29}{5 \cdot - 2}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.7

Multiply $5$ by $- 2$ .

$z = \frac{x}{20} + \frac{- 29}{- 10}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

Step 4.3.1.8

Dividing two negative values results in a positive value.

$z = \frac{x}{20} + \frac{29}{10}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$z = \frac{x}{20} + \frac{29}{10}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$z = \frac{x}{20} + \frac{29}{10}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$

$z = \frac{x}{20} + \frac{29}{10}$

$3 x - 6 y + z = 15$

$y = - \frac{2 x}{5} - \frac{1}{5}$