Algebra Examples

$\frac{1}{2} x + \frac{1}{4} y = - 7$ $- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1

Solve for $x$ in $\frac{1}{2} x + \frac{1}{4} y = - 7$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\frac{x}{2} + \frac{1}{4} y = - 7$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.1.2

Combine $\frac{1}{4}$ and $y$ .

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

$- \frac{2}{3} x + \frac{1}{2} y = 11$

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

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.2

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

$\frac{x}{2} = - 7 - \frac{y}{4}$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.3

Multiply both sides of the equation by $2$ .

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

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.4

Simplify both sides of the equation.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Cancel the common factor of $2$ .

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Step 1.4.1.1.1

Cancel the common factor.

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

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.4.1.1.2

Rewrite the expression.

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

$- \frac{2}{3} x + \frac{1}{2} y = 11$

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

$- \frac{2}{3} x + \frac{1}{2} y = 11$

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

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

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

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Step 1.4.2.1.1

Apply the distributive property.

$x = 2 \cdot - 7 + 2 (- \frac{y}{4})$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.4.2.1.2

Multiply $2$ by $- 7$ .

$x = - 14 + 2 (- \frac{y}{4})$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.4.2.1.3

Cancel the common factor of $2$ .

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Step 1.4.2.1.3.1

Move the leading negative in $- \frac{y}{4}$ into the numerator.

$x = - 14 + 2 (\frac{- y}{4})$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.4.2.1.3.2

Factor $2$ out of $4$ .

$x = - 14 + 2 (\frac{- y}{2 (2)})$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.4.2.1.3.3

Cancel the common factor.

$x = - 14 + 2 (\frac{- y}{2 \cdot 2})$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.4.2.1.3.4

Rewrite the expression.

$x = - 14 + \frac{- y}{2}$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

$x = - 14 + \frac{- y}{2}$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.4.2.1.4

Move the negative in front of the fraction.

$x = - 14 - \frac{y}{2}$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

$x = - 14 - \frac{y}{2}$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

$x = - 14 - \frac{y}{2}$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

$x = - 14 - \frac{y}{2}$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 1.5

Reorder $- 14$ and $- \frac{y}{2}$ .

$x = - \frac{y}{2} - 14$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

$- \frac{2}{3} x + \frac{1}{2} y = 11$

Step 2

Replace all occurrences of $x$ with $- \frac{y}{2} - 14$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- \frac{2}{3} x + \frac{1}{2} y = 11$ with $- \frac{y}{2} - 14$ .

$- \frac{2}{3} \cdot (- \frac{y}{2} - 14) + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- \frac{2}{3} \cdot (- \frac{y}{2} - 14) + \frac{1}{2} y$ .

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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.

$- \frac{2}{3} \cdot (- \frac{y}{2}) - \frac{2}{3} \cdot - 14 + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

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

$\frac{- 2}{3} \cdot (- \frac{y}{2}) - \frac{2}{3} \cdot - 14 + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.2.2

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

$\frac{- 2}{3} \cdot \frac{- y}{2} - \frac{2}{3} \cdot - 14 + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.2.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- y}{2} - \frac{2}{3} \cdot - 14 + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.2.4

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- y}{2} - \frac{2}{3} \cdot - 14 + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.2.5

Rewrite the expression.

$\frac{- 1}{3} \cdot (- y) - \frac{2}{3} \cdot - 14 + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

$\frac{- 1}{3} \cdot (- y) - \frac{2}{3} \cdot - 14 + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.3

Combine $\frac{- 1}{3}$ and $y$ .

$- \frac{- y}{3} - \frac{2}{3} \cdot - 14 + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.4

Multiply $- \frac{2}{3} \cdot - 14$ .

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Step 2.2.1.1.4.1

Multiply $- 14$ by $- 1$ .

$- \frac{- y}{3} + 14 (\frac{2}{3}) + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.4.2

Combine $14$ and $\frac{2}{3}$ .

$- \frac{- y}{3} + \frac{14 \cdot 2}{3} + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.4.3

Multiply $14$ by $2$ .

$- \frac{- y}{3} + \frac{28}{3} + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

$- \frac{- y}{3} + \frac{28}{3} + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.5

Simplify each term.

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Step 2.2.1.1.5.1

Move the negative in front of the fraction.

$\frac{y}{3} + \frac{28}{3} + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.5.2

Multiply $- - \frac{y}{3}$ .

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Step 2.2.1.1.5.2.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{y}{3}) + \frac{28}{3} + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.5.2.2

Multiply $\frac{y}{3}$ by $1$ .

$\frac{y}{3} + \frac{28}{3} + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

$\frac{y}{3} + \frac{28}{3} + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

$\frac{y}{3} + \frac{28}{3} + \frac{1}{2} y = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.1.6

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

$\frac{y}{3} + \frac{28}{3} + \frac{y}{2} = 11$

$x = - \frac{y}{2} - 14$

$\frac{y}{3} + \frac{28}{3} + \frac{y}{2} = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.2

To write $\frac{y}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y}{3} \cdot \frac{2}{2} + \frac{y}{2} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.3

To write $\frac{y}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{y}{3} \cdot \frac{2}{2} + \frac{y}{2} \cdot \frac{3}{3} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.4.1

Multiply $\frac{y}{3}$ by $\frac{2}{2}$ .

$\frac{y \cdot 2}{3 \cdot 2} + \frac{y}{2} \cdot \frac{3}{3} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.4.2

Multiply $3$ by $2$ .

$\frac{y \cdot 2}{6} + \frac{y}{2} \cdot \frac{3}{3} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.4.3

Multiply $\frac{y}{2}$ by $\frac{3}{3}$ .

$\frac{y \cdot 2}{6} + \frac{y \cdot 3}{2 \cdot 3} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.4.4

Multiply $2$ by $3$ .

$\frac{y \cdot 2}{6} + \frac{y \cdot 3}{6} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

$\frac{y \cdot 2}{6} + \frac{y \cdot 3}{6} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{y \cdot 2 + y \cdot 3}{6} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.6

Reorder $y$ and $3$ .

$\frac{2 \cdot y + 3 \cdot y}{6} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

Step 2.2.1.7

Add $2 \cdot y$ and $3 \cdot y$ .

$\frac{5 y}{6} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

$\frac{5 y}{6} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

$\frac{5 y}{6} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

$\frac{5 y}{6} + \frac{28}{3} = 11$

$x = - \frac{y}{2} - 14$

Step 3

Solve for $y$ in $\frac{5 y}{6} + \frac{28}{3} = 11$ .

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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 $\frac{28}{3}$ from both sides of the equation.

$\frac{5 y}{6} = 11 - \frac{28}{3}$

$x = - \frac{y}{2} - 14$

Step 3.1.2

To write $11$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5 y}{6} = 11 \cdot \frac{3}{3} - \frac{28}{3}$

$x = - \frac{y}{2} - 14$

Step 3.1.3

Combine $11$ and $\frac{3}{3}$ .

$\frac{5 y}{6} = \frac{11 \cdot 3}{3} - \frac{28}{3}$

$x = - \frac{y}{2} - 14$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{5 y}{6} = \frac{11 \cdot 3 - 28}{3}$

$x = - \frac{y}{2} - 14$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $11$ by $3$ .

$\frac{5 y}{6} = \frac{33 - 28}{3}$

$x = - \frac{y}{2} - 14$

Step 3.1.5.2

Subtract $28$ from $33$ .

$\frac{5 y}{6} = \frac{5}{3}$

$x = - \frac{y}{2} - 14$

$\frac{5 y}{6} = \frac{5}{3}$

$x = - \frac{y}{2} - 14$

$\frac{5 y}{6} = \frac{5}{3}$

$x = - \frac{y}{2} - 14$

Step 3.2

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} \cdot \frac{5 y}{6} = \frac{6}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

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 $\frac{6}{5} \cdot \frac{5 y}{6}$ .

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Step 3.3.1.1.1

Cancel the common factor of $6$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 y}{6} = \frac{6}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{5} \cdot (5 y) = \frac{6}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

$\frac{1}{5} \cdot (5 y) = \frac{6}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

Step 3.3.1.1.2

Cancel the common factor of $5$ .

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Step 3.3.1.1.2.1

Factor $5$ out of $5 y$ .

$\frac{1}{5} \cdot (5 (y)) = \frac{6}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{5} \cdot (5 y) = \frac{6}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{6}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

$y = \frac{6}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

$y = \frac{6}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

$y = \frac{6}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{6}{5} \cdot \frac{5}{3}$ .

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Step 3.3.2.1.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1.1

Factor $3$ out of $6$ .

$y = \frac{3 (2)}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

Step 3.3.2.1.1.2

Cancel the common factor.

$y = \frac{3 \cdot 2}{5} \cdot \frac{5}{3}$

$x = - \frac{y}{2} - 14$

Step 3.3.2.1.1.3

Rewrite the expression.

$y = \frac{2}{5} \cdot 5$

$x = - \frac{y}{2} - 14$

$y = \frac{2}{5} \cdot 5$

$x = - \frac{y}{2} - 14$

Step 3.3.2.1.2

Cancel the common factor of $5$ .

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Step 3.3.2.1.2.1

Cancel the common factor.

$y = \frac{2}{5} \cdot 5$

$x = - \frac{y}{2} - 14$

Step 3.3.2.1.2.2

Rewrite the expression.

$y = 2$

$x = - \frac{y}{2} - 14$

$y = 2$

$x = - \frac{y}{2} - 14$

$y = 2$

$x = - \frac{y}{2} - 14$

$y = 2$

$x = - \frac{y}{2} - 14$

$y = 2$

$x = - \frac{y}{2} - 14$

$y = 2$

$x = - \frac{y}{2} - 14$

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 $x = - \frac{y}{2} - 14$ with $2$ .

$x = - \frac{2}{2} - 14$

$y = 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{2}{2} - 14$ .

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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 $2$ .

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Step 4.2.1.1.1.1

Cancel the common factor.

$x = - \frac{2}{2} - 14$

$y = 2$

Step 4.2.1.1.1.2

Rewrite the expression.

$x = - 1 \cdot 1 - 14$

$y = 2$

$x = - 1 \cdot 1 - 14$

$y = 2$

Step 4.2.1.1.2

Multiply $- 1$ by $1$ .

$x = - 1 - 14$

$y = 2$

$x = - 1 - 14$

$y = 2$

Step 4.2.1.2

Subtract $14$ from $- 1$ .

$x = - 15$

$y = 2$

$x = - 15$

$y = 2$

$x = - 15$

$y = 2$

$x = - 15$

$y = 2$

Step 5

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

$(- 15, 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 15, 2)$

Equation Form:

$x = - 15, y = 2$

Step 7