Algebra Examples

$- 24 - 8 x = 12 y$ $1 + \frac{5}{9} y = - \frac{7}{18} x$

Step 1

Solve for $x$ in $- 24 - 8 x = 12 y$ .

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Step 1.1

Add $24$ to both sides of the equation.

$- 8 x = 12 y + 24$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

Step 1.2

Divide each term in $- 8 x = 12 y + 24$ by $- 8$ and simplify.

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Step 1.2.1

Divide each term in $- 8 x = 12 y + 24$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{12 y}{- 8} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 8$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{12 y}{- 8} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12 y}{- 8} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

$x = \frac{12 y}{- 8} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

$x = \frac{12 y}{- 8} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

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

Cancel the common factor of $12$ and $- 8$ .

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Step 1.2.3.1.1.1

Factor $4$ out of $12 y$ .

$x = \frac{4 (3 y)}{- 8} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $4$ out of $- 8$ .

$x = \frac{4 (3 y)}{4 (- 2)} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{4 (3 y)}{4 \cdot - 2} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{3 y}{- 2} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

$x = \frac{3 y}{- 2} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

$x = \frac{3 y}{- 2} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{3 y}{2} + \frac{24}{- 8}$

$1 + \frac{5}{9} y = - \frac{7}{18} x$

Step 1.2.3.1.3

Divide $24$ by $- 8$ .

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

$1 + \frac{5}{9} y = - \frac{7}{18} x$

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

$1 + \frac{5}{9} y = - \frac{7}{18} x$

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

$1 + \frac{5}{9} y = - \frac{7}{18} x$

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

$1 + \frac{5}{9} y = - \frac{7}{18} x$

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

$1 + \frac{5}{9} y = - \frac{7}{18} x$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $1 + \frac{5}{9} y = - \frac{7}{18} x$ with $- \frac{3 y}{2} - 3$ .

$1 + \frac{5}{9} y = - \frac{7}{18} \cdot (- \frac{3 y}{2} - 3)$

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

Step 2.2

Simplify $1 + \frac{5}{9} y = - \frac{7}{18} \cdot (- \frac{3 y}{2} - 3)$ .

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Combine $\frac{5}{9}$ and $y$ .

$1 + \frac{5 y}{9} = - \frac{7}{18} \cdot (- \frac{3 y}{2} - 3)$

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

$1 + \frac{5 y}{9} = - \frac{7}{18} \cdot (- \frac{3 y}{2} - 3)$

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

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $- \frac{7}{18} \cdot (- \frac{3 y}{2} - 3)$ .

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Step 2.2.2.1.1

Apply the distributive property.

$1 + \frac{5 y}{9} = - \frac{7}{18} \cdot (- \frac{3 y}{2}) - \frac{7}{18} \cdot - 3$

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

Step 2.2.2.1.2

Cancel the common factor of $3$ .

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Step 2.2.2.1.2.1

Move the leading negative in $- \frac{7}{18}$ into the numerator.

$1 + \frac{5 y}{9} = \frac{- 7}{18} \cdot (- \frac{3 y}{2}) - \frac{7}{18} \cdot - 3$

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

Step 2.2.2.1.2.2

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

$1 + \frac{5 y}{9} = \frac{- 7}{18} \cdot \frac{- 3 y}{2} - \frac{7}{18} \cdot - 3$

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

Step 2.2.2.1.2.3

Factor $3$ out of $18$ .

$1 + \frac{5 y}{9} = \frac{- 7}{3 (6)} \cdot \frac{- 3 y}{2} - \frac{7}{18} \cdot - 3$

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

Step 2.2.2.1.2.4

Factor $3$ out of $- 3 y$ .

$1 + \frac{5 y}{9} = \frac{- 7}{3 (6)} \cdot \frac{3 (- y)}{2} - \frac{7}{18} \cdot - 3$

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

Step 2.2.2.1.2.5

Cancel the common factor.

$1 + \frac{5 y}{9} = \frac{- 7}{3 \cdot 6} \cdot \frac{3 (- y)}{2} - \frac{7}{18} \cdot - 3$

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

Step 2.2.2.1.2.6

Rewrite the expression.

$1 + \frac{5 y}{9} = \frac{- 7}{6} \cdot \frac{- y}{2} - \frac{7}{18} \cdot - 3$

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

$1 + \frac{5 y}{9} = \frac{- 7}{6} \cdot \frac{- y}{2} - \frac{7}{18} \cdot - 3$

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

Step 2.2.2.1.3

Multiply $\frac{- 7}{6}$ by $\frac{- y}{2}$ .

$1 + \frac{5 y}{9} = \frac{- 7 (- y)}{6 \cdot 2} - \frac{7}{18} \cdot - 3$

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

Step 2.2.2.1.4

Multiply.

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Step 2.2.2.1.4.1

Multiply $- 1$ by $- 7$ .

$1 + \frac{5 y}{9} = \frac{7 y}{6 \cdot 2} - \frac{7}{18} \cdot - 3$

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

Step 2.2.2.1.4.2

Multiply $6$ by $2$ .

$1 + \frac{5 y}{9} = \frac{7 y}{12} - \frac{7}{18} \cdot - 3$

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

$1 + \frac{5 y}{9} = \frac{7 y}{12} - \frac{7}{18} \cdot - 3$

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

Step 2.2.2.1.5

Cancel the common factor of $3$ .

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Step 2.2.2.1.5.1

Move the leading negative in $- \frac{7}{18}$ into the numerator.

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{- 7}{18} \cdot - 3$

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

Step 2.2.2.1.5.2

Factor $3$ out of $18$ .

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{- 7}{3 (6)} \cdot - 3$

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

Step 2.2.2.1.5.3

Factor $3$ out of $- 3$ .

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{- 7}{3 \cdot 6} \cdot (3 \cdot - 1)$

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

Step 2.2.2.1.5.4

Cancel the common factor.

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{- 7}{3 \cdot 6} \cdot (3 \cdot - 1)$

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

Step 2.2.2.1.5.5

Rewrite the expression.

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{- 7}{6} \cdot - 1$

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

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{- 7}{6} \cdot - 1$

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

Step 2.2.2.1.6

Combine $\frac{- 7}{6}$ and $- 1$ .

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{- 7 \cdot - 1}{6}$

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

Step 2.2.2.1.7

Multiply $- 7$ by $- 1$ .

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{7}{6}$

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

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{7}{6}$

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

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{7}{6}$

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

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{7}{6}$

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

$1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{7}{6}$

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

Step 3

Solve for $y$ in $1 + \frac{5 y}{9} = \frac{7 y}{12} + \frac{7}{6}$ .

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Step 3.1

Move all terms containing $y$ to the left side of the equation.

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Step 3.1.1

Subtract $\frac{7 y}{12}$ from both sides of the equation.

$1 + \frac{5 y}{9} - \frac{7 y}{12} = \frac{7}{6}$

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

Step 3.1.2

To write $\frac{5 y}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$1 + \frac{5 y}{9} \cdot \frac{4}{4} - \frac{7 y}{12} = \frac{7}{6}$

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

Step 3.1.3

To write $- \frac{7 y}{12}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$1 + \frac{5 y}{9} \cdot \frac{4}{4} - \frac{7 y}{12} \cdot \frac{3}{3} = \frac{7}{6}$

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

Step 3.1.4

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.4.1

Multiply $\frac{5 y}{9}$ by $\frac{4}{4}$ .

$1 + \frac{5 y \cdot 4}{9 \cdot 4} - \frac{7 y}{12} \cdot \frac{3}{3} = \frac{7}{6}$

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

Step 3.1.4.2

Multiply $9$ by $4$ .

$1 + \frac{5 y \cdot 4}{36} - \frac{7 y}{12} \cdot \frac{3}{3} = \frac{7}{6}$

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

Step 3.1.4.3

Multiply $\frac{7 y}{12}$ by $\frac{3}{3}$ .

$1 + \frac{5 y \cdot 4}{36} - \frac{7 y \cdot 3}{12 \cdot 3} = \frac{7}{6}$

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

Step 3.1.4.4

Multiply $12$ by $3$ .

$1 + \frac{5 y \cdot 4}{36} - \frac{7 y \cdot 3}{36} = \frac{7}{6}$

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

$1 + \frac{5 y \cdot 4}{36} - \frac{7 y \cdot 3}{36} = \frac{7}{6}$

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

Step 3.1.5

Combine the numerators over the common denominator.

$1 + \frac{5 y \cdot 4 - 7 y \cdot 3}{36} = \frac{7}{6}$

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

Step 3.1.6

Simplify each term.

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Step 3.1.6.1

Simplify the numerator.

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Step 3.1.6.1.1

Factor $y$ out of $5 y \cdot 4 - 7 y \cdot 3$ .

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Step 3.1.6.1.1.1

Factor $y$ out of $5 y \cdot 4$ .

$1 + \frac{y (5 \cdot 4) - 7 y \cdot 3}{36} = \frac{7}{6}$

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

Step 3.1.6.1.1.2

Factor $y$ out of $- 7 y \cdot 3$ .

$1 + \frac{y (5 \cdot 4) + y (- 7 \cdot 3)}{36} = \frac{7}{6}$

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

Step 3.1.6.1.1.3

Factor $y$ out of $y (5 \cdot 4) + y (- 7 \cdot 3)$ .

$1 + \frac{y (5 \cdot 4 - 7 \cdot 3)}{36} = \frac{7}{6}$

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

$1 + \frac{y (5 \cdot 4 - 7 \cdot 3)}{36} = \frac{7}{6}$

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

Step 3.1.6.1.2

Multiply $5$ by $4$ .

$1 + \frac{y (20 - 7 \cdot 3)}{36} = \frac{7}{6}$

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

Step 3.1.6.1.3

Multiply $- 7$ by $3$ .

$1 + \frac{y (20 - 21)}{36} = \frac{7}{6}$

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

Step 3.1.6.1.4

Subtract $21$ from $20$ .

$1 + \frac{y \cdot - 1}{36} = \frac{7}{6}$

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

$1 + \frac{y \cdot - 1}{36} = \frac{7}{6}$

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

Step 3.1.6.2

Move $- 1$ to the left of $y$ .

$1 + \frac{- 1 \cdot y}{36} = \frac{7}{6}$

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

Step 3.1.6.3

Move the negative in front of the fraction.

$1 - \frac{y}{36} = \frac{7}{6}$

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

$1 - \frac{y}{36} = \frac{7}{6}$

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

$1 - \frac{y}{36} = \frac{7}{6}$

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

Step 3.2

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

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$- \frac{y}{36} = \frac{7}{6} - 1$

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

Step 3.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{y}{36} = \frac{7}{6} - 1 \cdot \frac{6}{6}$

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

Step 3.2.3

Combine $- 1$ and $\frac{6}{6}$ .

$- \frac{y}{36} = \frac{7}{6} + \frac{- 1 \cdot 6}{6}$

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

Step 3.2.4

Combine the numerators over the common denominator.

$- \frac{y}{36} = \frac{7 - 1 \cdot 6}{6}$

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

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 1$ by $6$ .

$- \frac{y}{36} = \frac{7 - 6}{6}$

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

Step 3.2.5.2

Subtract $6$ from $7$ .

$- \frac{y}{36} = \frac{1}{6}$

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

$- \frac{y}{36} = \frac{1}{6}$

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

$- \frac{y}{36} = \frac{1}{6}$

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

Step 3.3

Multiply both sides of the equation by $- 36$ .

$- 36 (- \frac{y}{36}) = - 36 (\frac{1}{6})$

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

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify $- 36 (- \frac{y}{36})$ .

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Step 3.4.1.1.1

Cancel the common factor of $36$ .

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Step 3.4.1.1.1.1

Move the leading negative in $- \frac{y}{36}$ into the numerator.

$- 36 \frac{- y}{36} = - 36 (\frac{1}{6})$

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

Step 3.4.1.1.1.2

Factor $36$ out of $- 36$ .

$36 (- 1) (\frac{- y}{36}) = - 36 (\frac{1}{6})$

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

Step 3.4.1.1.1.3

Cancel the common factor.

$36 \cdot (- 1 \frac{- y}{36}) = - 36 (\frac{1}{6})$

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

Step 3.4.1.1.1.4

Rewrite the expression.

$y = - 36 (\frac{1}{6})$

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

$y = - 36 (\frac{1}{6})$

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

Step 3.4.1.1.2

Multiply.

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Step 3.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - 36 (\frac{1}{6})$

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

Step 3.4.1.1.2.2

Multiply $y$ by $1$ .

$y = - 36 (\frac{1}{6})$

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

$y = - 36 (\frac{1}{6})$

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

$y = - 36 (\frac{1}{6})$

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

$y = - 36 (\frac{1}{6})$

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

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Cancel the common factor of $6$ .

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Step 3.4.2.1.1

Factor $6$ out of $- 36$ .

$y = 6 (- 6) (\frac{1}{6})$

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

Step 3.4.2.1.2

Cancel the common factor.

$y = 6 \cdot (- 6 (\frac{1}{6}))$

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

Step 3.4.2.1.3

Rewrite the expression.

$y = - 6$

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

$y = - 6$

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

$y = - 6$

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

$y = - 6$

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

$y = - 6$

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

Step 4

Replace all occurrences of $y$ with $- 6$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{3 y}{2} - 3$ with $- 6$ .

$x = - \frac{3 (- 6)}{2} - 3$

$y = - 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{3 (- 6)}{2} - 3$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $- 6$ and $2$ .

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Step 4.2.1.1.1.1

Factor $2$ out of $3 (- 6)$ .

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

$y = - 6$

Step 4.2.1.1.1.2

Cancel the common factors.

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Step 4.2.1.1.1.2.1

Factor $2$ out of $2$ .

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

$y = - 6$

Step 4.2.1.1.1.2.2

Cancel the common factor.

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

$y = - 6$

Step 4.2.1.1.1.2.3

Rewrite the expression.

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

$y = - 6$

Step 4.2.1.1.1.2.4

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

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

$y = - 6$

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

$y = - 6$

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

$y = - 6$

Step 4.2.1.1.2

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

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Step 4.2.1.1.2.1

Multiply $3$ by $- 3$ .

$x = 9 - 3$

$y = - 6$

Step 4.2.1.1.2.2

Multiply $- 1$ by $- 9$ .

$x = 9 - 3$

$y = - 6$

$x = 9 - 3$

$y = - 6$

$x = 9 - 3$

$y = - 6$

Step 4.2.1.2

Subtract $3$ from $9$ .

$x = 6$

$y = - 6$

$x = 6$

$y = - 6$

$x = 6$

$y = - 6$

$x = 6$

$y = - 6$

Step 5

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

$(6, - 6)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(6, - 6)$

Equation Form:

$x = 6, y = - 6$

Step 7