Algebra Examples

$8 x + 7 y = 18$ , $3 x - 5 y = 22$

Step 1

Solve for $x$ in $8 x + 7 y = 18$ .

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Step 1.1

Subtract $7 y$ from both sides of the equation.

$8 x = 18 - 7 y$

$3 x - 5 y = 22$

Step 1.2

Divide each term in $8 x = 18 - 7 y$ by $8$ and simplify.

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Step 1.2.1

Divide each term in $8 x = 18 - 7 y$ by $8$ .

$\frac{8 x}{8} = \frac{18}{8} + \frac{- 7 y}{8}$

$3 x - 5 y = 22$

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.

$\frac{8 x}{8} = \frac{18}{8} + \frac{- 7 y}{8}$

$3 x - 5 y = 22$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{8} + \frac{- 7 y}{8}$

$3 x - 5 y = 22$

$x = \frac{18}{8} + \frac{- 7 y}{8}$

$3 x - 5 y = 22$

$x = \frac{18}{8} + \frac{- 7 y}{8}$

$3 x - 5 y = 22$

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 $18$ and $8$ .

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Step 1.2.3.1.1.1

Factor $2$ out of $18$ .

$x = \frac{2 (9)}{8} + \frac{- 7 y}{8}$

$3 x - 5 y = 22$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $2$ out of $8$ .

$x = \frac{2 \cdot 9}{2 \cdot 4} + \frac{- 7 y}{8}$

$3 x - 5 y = 22$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 9}{2 \cdot 4} + \frac{- 7 y}{8}$

$3 x - 5 y = 22$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{9}{4} + \frac{- 7 y}{8}$

$3 x - 5 y = 22$

$x = \frac{9}{4} + \frac{- 7 y}{8}$

$3 x - 5 y = 22$

$x = \frac{9}{4} + \frac{- 7 y}{8}$

$3 x - 5 y = 22$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = \frac{9}{4} - \frac{7 y}{8}$

$3 x - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$3 x - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$3 x - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$3 x - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$3 x - 5 y = 22$

Step 2

Replace all occurrences of $x$ with $\frac{9}{4} - \frac{7 y}{8}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x - 5 y = 22$ with $\frac{9}{4} - \frac{7 y}{8}$ .

$3 (\frac{9}{4} - \frac{7 y}{8}) - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (\frac{9}{4} - \frac{7 y}{8}) - 5 y$ .

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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.

$3 (\frac{9}{4}) + 3 (- \frac{7 y}{8}) - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.1.2

Multiply $3 (\frac{9}{4})$ .

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Step 2.2.1.1.2.1

Combine $3$ and $\frac{9}{4}$ .

$\frac{3 \cdot 9}{4} + 3 (- \frac{7 y}{8}) - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.1.2.2

Multiply $3$ by $9$ .

$\frac{27}{4} + 3 (- \frac{7 y}{8}) - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$\frac{27}{4} + 3 (- \frac{7 y}{8}) - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.1.3

Multiply $3 (- \frac{7 y}{8})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $3$ .

$\frac{27}{4} - 3 \frac{7 y}{8} - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.1.3.2

Combine $- 3$ and $\frac{7 y}{8}$ .

$\frac{27}{4} + \frac{- 3 (7 y)}{8} - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.1.3.3

Multiply $7$ by $- 3$ .

$\frac{27}{4} + \frac{- 21 y}{8} - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$\frac{27}{4} + \frac{- 21 y}{8} - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$\frac{27}{4} - \frac{21 y}{8} - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$\frac{27}{4} - \frac{21 y}{8} - 5 y = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.2

To write $- 5 y$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{27}{4} - \frac{21 y}{8} - 5 y \cdot \frac{8}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $- 5 y$ and $\frac{8}{8}$ .

$\frac{27}{4} - \frac{21 y}{8} + \frac{- 5 y \cdot 8}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{27}{4} + \frac{- 21 y - 5 y \cdot 8}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$\frac{27}{4} + \frac{- 21 y - 5 y \cdot 8}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Simplify the numerator.

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Step 2.2.1.4.1.1

Factor $y$ out of $- 21 y - 5 y \cdot 8$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $- 21 y$ .

$\frac{27}{4} + \frac{y \cdot - 21 - 5 y \cdot 8}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.4.1.1.2

Factor $y$ out of $- 5 y \cdot 8$ .

$\frac{27}{4} + \frac{y \cdot - 21 + y (- 5 \cdot 8)}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot - 21 + y (- 5 \cdot 8)$ .

$\frac{27}{4} + \frac{y (- 21 - 5 \cdot 8)}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$\frac{27}{4} + \frac{y (- 21 - 5 \cdot 8)}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.4.1.2

Multiply $- 5$ by $8$ .

$\frac{27}{4} + \frac{y (- 21 - 40)}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.4.1.3

Subtract $40$ from $- 21$ .

$\frac{27}{4} + \frac{y \cdot - 61}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$\frac{27}{4} + \frac{y \cdot - 61}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.4.2

Move $- 61$ to the left of $y$ .

$\frac{27}{4} + \frac{- 61 \cdot y}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 2.2.1.4.3

Move the negative in front of the fraction.

$\frac{27}{4} - \frac{61 y}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$\frac{27}{4} - \frac{61 y}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$\frac{27}{4} - \frac{61 y}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$\frac{27}{4} - \frac{61 y}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

$\frac{27}{4} - \frac{61 y}{8} = 22$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3

Solve for $y$ in $\frac{27}{4} - \frac{61 y}{8} = 22$ .

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Step 3.1

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

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Step 3.1.1

Subtract $\frac{27}{4}$ from both sides of the equation.

$- \frac{61 y}{8} = 22 - \frac{27}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.1.2

To write $22$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{61 y}{8} = 22 \cdot \frac{4}{4} - \frac{27}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.1.3

Combine $22$ and $\frac{4}{4}$ .

$- \frac{61 y}{8} = \frac{22 \cdot 4}{4} - \frac{27}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.1.4

Combine the numerators over the common denominator.

$- \frac{61 y}{8} = \frac{22 \cdot 4 - 27}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $22$ by $4$ .

$- \frac{61 y}{8} = \frac{88 - 27}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.1.5.2

Subtract $27$ from $88$ .

$- \frac{61 y}{8} = \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

$- \frac{61 y}{8} = \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

$- \frac{61 y}{8} = \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.2

Multiply both sides of the equation by $- \frac{8}{61}$ .

$- \frac{8}{61} \cdot (- \frac{61 y}{8}) = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $- \frac{8}{61} (- \frac{61 y}{8})$ .

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Step 3.3.1.1.1

Cancel the common factor of $8$ .

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Step 3.3.1.1.1.1

Move the leading negative in $- \frac{8}{61}$ into the numerator.

$\frac{- 8}{61} \cdot (- \frac{61 y}{8}) = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{61 y}{8}$ into the numerator.

$\frac{- 8}{61} \cdot \frac{- 61 y}{8} = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.1.1.1.3

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{61} \cdot \frac{- 61 y}{8} = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{8 \cdot - 1}{61} \cdot \frac{- 61 y}{8} = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{61} \cdot (- 61 y) = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

$\frac{- 1}{61} \cdot (- 61 y) = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.1.1.2

Cancel the common factor of $61$ .

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Step 3.3.1.1.2.1

Factor $61$ out of $- 61 y$ .

$\frac{- 1}{61} \cdot (61 (- y)) = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{61} \cdot (61 (- y)) = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

$y = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.1.1.3

Multiply.

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Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

$y = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

$y = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

$y = - \frac{8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{8}{61} \cdot \frac{61}{4}$ .

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Step 3.3.2.1.1

Cancel the common factor of $4$ .

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Step 3.3.2.1.1.1

Move the leading negative in $- \frac{8}{61}$ into the numerator.

$y = \frac{- 8}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.2.1.1.2

Factor $4$ out of $- 8$ .

$y = \frac{4 (- 2)}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.2.1.1.3

Cancel the common factor.

$y = \frac{4 \cdot - 2}{61} \cdot \frac{61}{4}$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.2.1.1.4

Rewrite the expression.

$y = \frac{- 2}{61} \cdot 61$

$x = \frac{9}{4} - \frac{7 y}{8}$

$y = \frac{- 2}{61} \cdot 61$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.2.1.2

Cancel the common factor of $61$ .

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Step 3.3.2.1.2.1

Cancel the common factor.

$y = \frac{- 2}{61} \cdot 61$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 3.3.2.1.2.2

Rewrite the expression.

$y = - 2$

$x = \frac{9}{4} - \frac{7 y}{8}$

$y = - 2$

$x = \frac{9}{4} - \frac{7 y}{8}$

$y = - 2$

$x = \frac{9}{4} - \frac{7 y}{8}$

$y = - 2$

$x = \frac{9}{4} - \frac{7 y}{8}$

$y = - 2$

$x = \frac{9}{4} - \frac{7 y}{8}$

$y = - 2$

$x = \frac{9}{4} - \frac{7 y}{8}$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{9}{4} - \frac{7 y}{8}$ with $- 2$ .

$x = \frac{9}{4} - \frac{7 (- 2)}{8}$

$y = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{9}{4} - \frac{7 (- 2)}{8}$ .

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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 $- 2$ and $8$ .

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Step 4.2.1.1.1.1

Factor $2$ out of $7 (- 2)$ .

$x = \frac{9}{4} - \frac{2 (7 \cdot (- 1))}{8}$

$y = - 2$

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 $8$ .

$x = \frac{9}{4} - \frac{2 (7 \cdot (- 1))}{2 (4)}$

$y = - 2$

Step 4.2.1.1.1.2.2

Cancel the common factor.

$x = \frac{9}{4} - \frac{2 (7 \cdot (- 1))}{2 \cdot 4}$

$y = - 2$

Step 4.2.1.1.1.2.3

Rewrite the expression.

$x = \frac{9}{4} - \frac{7 \cdot (- 1)}{4}$

$y = - 2$

$x = \frac{9}{4} - \frac{7 \cdot (- 1)}{4}$

$y = - 2$

$x = \frac{9}{4} - \frac{7 \cdot (- 1)}{4}$

$y = - 2$

Step 4.2.1.1.2

Multiply $7$ by $- 1$ .

$x = \frac{9}{4} - \frac{- 7}{4}$

$y = - 2$

Step 4.2.1.1.3

Move the negative in front of the fraction.

$x = \frac{9}{4} + \frac{7}{4}$

$y = - 2$

Step 4.2.1.1.4

Multiply $- - \frac{7}{4}$ .

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Step 4.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$x = \frac{9}{4} + 1 (\frac{7}{4})$

$y = - 2$

Step 4.2.1.1.4.2

Multiply $\frac{7}{4}$ by $1$ .

$x = \frac{9}{4} + \frac{7}{4}$

$y = - 2$

$x = \frac{9}{4} + \frac{7}{4}$

$y = - 2$

$x = \frac{9}{4} + \frac{7}{4}$

$y = - 2$

Step 4.2.1.2

Combine fractions.

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Step 4.2.1.2.1

Combine the numerators over the common denominator.

$x = \frac{9 + 7}{4}$

$y = - 2$

Step 4.2.1.2.2

Simplify the expression.

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Step 4.2.1.2.2.1

Add $9$ and $7$ .

$x = \frac{16}{4}$

$y = - 2$

Step 4.2.1.2.2.2

Divide $16$ by $4$ .

$x = 4$

$y = - 2$

$x = 4$

$y = - 2$

$x = 4$

$y = - 2$

$x = 4$

$y = - 2$

$x = 4$

$y = - 2$

$x = 4$

$y = - 2$

Step 5

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

$(4, - 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(4, - 2)$

Equation Form:

$x = 4, y = - 2$

Step 7