Precalculus Examples

$\frac{1}{2} x + \frac{1}{3} y = 16$ , $\frac{1}{6} x + \frac{1}{4} y = 9$

Step 1

Solve for $x$ in $\frac{1}{2} x + \frac{1}{3} y = 16$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\frac{x}{2} + \frac{1}{3} y = 16$

$\frac{1}{6} x + \frac{1}{4} y = 9$

Step 1.1.2

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

$\frac{x}{2} + \frac{y}{3} = 16$

$\frac{1}{6} x + \frac{1}{4} y = 9$

$\frac{x}{2} + \frac{y}{3} = 16$

$\frac{1}{6} x + \frac{1}{4} y = 9$

Step 1.2

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

$\frac{x}{2} = 16 - \frac{y}{3}$

$\frac{1}{6} x + \frac{1}{4} y = 9$

Step 1.3

Multiply both sides of the equation by $2$ .

$2 (\frac{x}{2}) = 2 (16 - \frac{y}{3})$

$\frac{1}{6} x + \frac{1}{4} y = 9$

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.

$2 (\frac{x}{2}) = 2 (16 - \frac{y}{3})$

$\frac{1}{6} x + \frac{1}{4} y = 9$

Step 1.4.1.1.2

Rewrite the expression.

$x = 2 (16 - \frac{y}{3})$

$\frac{1}{6} x + \frac{1}{4} y = 9$

$x = 2 (16 - \frac{y}{3})$

$\frac{1}{6} x + \frac{1}{4} y = 9$

$x = 2 (16 - \frac{y}{3})$

$\frac{1}{6} x + \frac{1}{4} y = 9$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify $2 (16 - \frac{y}{3})$ .

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Step 1.4.2.1.1

Apply the distributive property.

$x = 2 \cdot 16 + 2 (- \frac{y}{3})$

$\frac{1}{6} x + \frac{1}{4} y = 9$

Step 1.4.2.1.2

Multiply $2$ by $16$ .

$x = 32 + 2 (- \frac{y}{3})$

$\frac{1}{6} x + \frac{1}{4} y = 9$

Step 1.4.2.1.3

Multiply $2 (- \frac{y}{3})$ .

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Step 1.4.2.1.3.1

Multiply $- 1$ by $2$ .

$x = 32 - 2 \frac{y}{3}$

$\frac{1}{6} x + \frac{1}{4} y = 9$

Step 1.4.2.1.3.2

Combine $- 2$ and $\frac{y}{3}$ .

$x = 32 + \frac{- 2 y}{3}$

$\frac{1}{6} x + \frac{1}{4} y = 9$

$x = 32 + \frac{- 2 y}{3}$

$\frac{1}{6} x + \frac{1}{4} y = 9$

Step 1.4.2.1.4

Move the negative in front of the fraction.

$x = 32 - \frac{2 y}{3}$

$\frac{1}{6} x + \frac{1}{4} y = 9$

$x = 32 - \frac{2 y}{3}$

$\frac{1}{6} x + \frac{1}{4} y = 9$

$x = 32 - \frac{2 y}{3}$

$\frac{1}{6} x + \frac{1}{4} y = 9$

$x = 32 - \frac{2 y}{3}$

$\frac{1}{6} x + \frac{1}{4} y = 9$

Step 1.5

Reorder $32$ and $- \frac{2 y}{3}$ .

$x = - \frac{2 y}{3} + 32$

$\frac{1}{6} x + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

$\frac{1}{6} x + \frac{1}{4} y = 9$

Step 2

Replace all occurrences of $x$ with $- \frac{2 y}{3} + 32$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{1}{6} x + \frac{1}{4} y = 9$ with $- \frac{2 y}{3} + 32$ .

$\frac{1}{6} \cdot (- \frac{2 y}{3} + 32) + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{1}{6} \cdot (- \frac{2 y}{3} + 32) + \frac{1}{4} y$ .

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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.

$\frac{1}{6} \cdot (- \frac{2 y}{3}) + \frac{1}{6} \cdot 32 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

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 $- \frac{2 y}{3}$ into the numerator.

$\frac{1}{6} \cdot \frac{- 2 y}{3} + \frac{1}{6} \cdot 32 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.2.2

Factor $2$ out of $6$ .

$\frac{1}{2 (3)} \cdot \frac{- 2 y}{3} + \frac{1}{6} \cdot 32 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.2.3

Factor $2$ out of $- 2 y$ .

$\frac{1}{2 (3)} \cdot \frac{2 (- y)}{3} + \frac{1}{6} \cdot 32 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.2.4

Cancel the common factor.

$\frac{1}{2 \cdot 3} \cdot \frac{2 (- y)}{3} + \frac{1}{6} \cdot 32 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.2.5

Rewrite the expression.

$\frac{1}{3} \cdot \frac{- y}{3} + \frac{1}{6} \cdot 32 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

$\frac{1}{3} \cdot \frac{- y}{3} + \frac{1}{6} \cdot 32 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.3

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

$\frac{- y}{3 \cdot 3} + \frac{1}{6} \cdot 32 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.4

Multiply $3$ by $3$ .

$\frac{- y}{9} + \frac{1}{6} \cdot 32 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.5

Cancel the common factor of $2$ .

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Step 2.2.1.1.5.1

Factor $2$ out of $6$ .

$\frac{- y}{9} + \frac{1}{2 (3)} \cdot 32 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.5.2

Factor $2$ out of $32$ .

$\frac{- y}{9} + \frac{1}{2 \cdot 3} \cdot (2 \cdot 16) + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.5.3

Cancel the common factor.

$\frac{- y}{9} + \frac{1}{2 \cdot 3} \cdot (2 \cdot 16) + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.5.4

Rewrite the expression.

$\frac{- y}{9} + \frac{1}{3} \cdot 16 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

$\frac{- y}{9} + \frac{1}{3} \cdot 16 + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.6

Combine $\frac{1}{3}$ and $16$ .

$\frac{- y}{9} + \frac{16}{3} + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.7

Move the negative in front of the fraction.

$- \frac{y}{9} + \frac{16}{3} + \frac{1}{4} y = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.1.8

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

$- \frac{y}{9} + \frac{16}{3} + \frac{y}{4} = 9$

$x = - \frac{2 y}{3} + 32$

$- \frac{y}{9} + \frac{16}{3} + \frac{y}{4} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.2

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

$- \frac{y}{9} \cdot \frac{4}{4} + \frac{y}{4} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.3

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

$- \frac{y}{9} \cdot \frac{4}{4} + \frac{y}{4} \cdot \frac{9}{9} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.4

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

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Step 2.2.1.4.1

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

$- \frac{y \cdot 4}{9 \cdot 4} + \frac{y}{4} \cdot \frac{9}{9} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.4.2

Multiply $9$ by $4$ .

$- \frac{y \cdot 4}{36} + \frac{y}{4} \cdot \frac{9}{9} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.4.3

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

$- \frac{y \cdot 4}{36} + \frac{y \cdot 9}{4 \cdot 9} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.4.4

Multiply $4$ by $9$ .

$- \frac{y \cdot 4}{36} + \frac{y \cdot 9}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

$- \frac{y \cdot 4}{36} + \frac{y \cdot 9}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- y \cdot 4 + y \cdot 9}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.6

Simplify each term.

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Step 2.2.1.6.1

Simplify the numerator.

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Step 2.2.1.6.1.1

Factor $y$ out of $- y \cdot 4 + y \cdot 9$ .

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Step 2.2.1.6.1.1.1

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

$\frac{y (- 1 \cdot 4) + y \cdot 9}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.6.1.1.2

Factor $y$ out of $y \cdot 9$ .

$\frac{y (- 1 \cdot 4) + y (9)}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.6.1.1.3

Factor $y$ out of $y (- 1 \cdot 4) + y (9)$ .

$\frac{y (- 1 \cdot 4 + 9)}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

$\frac{y (- 1 \cdot 4 + 9)}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.6.1.2

Multiply $- 1$ by $4$ .

$\frac{y (- 4 + 9)}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.6.1.3

Add $- 4$ and $9$ .

$\frac{y \cdot 5}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

$\frac{y \cdot 5}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 2.2.1.6.2

Move $5$ to the left of $y$ .

$\frac{5 y}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

$\frac{5 y}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

$\frac{5 y}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

$\frac{5 y}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

$\frac{5 y}{36} + \frac{16}{3} = 9$

$x = - \frac{2 y}{3} + 32$

Step 3

Solve for $y$ in $\frac{5 y}{36} + \frac{16}{3} = 9$ .

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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 $\frac{16}{3}$ from both sides of the equation.

$\frac{5 y}{36} = 9 - \frac{16}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.1.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5 y}{36} = 9 \cdot \frac{3}{3} - \frac{16}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.1.3

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

$\frac{5 y}{36} = \frac{9 \cdot 3}{3} - \frac{16}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{5 y}{36} = \frac{9 \cdot 3 - 16}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $9$ by $3$ .

$\frac{5 y}{36} = \frac{27 - 16}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.1.5.2

Subtract $16$ from $27$ .

$\frac{5 y}{36} = \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

$\frac{5 y}{36} = \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

$\frac{5 y}{36} = \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.2

Multiply both sides of the equation by $\frac{36}{5}$ .

$\frac{36}{5} \cdot \frac{5 y}{36} = \frac{36}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

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 $\frac{36}{5} \cdot \frac{5 y}{36}$ .

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Step 3.3.1.1.1

Cancel the common factor of $36$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{36}{5} \cdot \frac{5 y}{36} = \frac{36}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{5} \cdot (5 y) = \frac{36}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

$\frac{1}{5} \cdot (5 y) = \frac{36}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

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 $5 y$ .

$\frac{1}{5} \cdot (5 (y)) = \frac{36}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{5} \cdot (5 y) = \frac{36}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{36}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

$y = \frac{36}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

$y = \frac{36}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

$y = \frac{36}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{36}{5} \cdot \frac{11}{3}$ .

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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 $36$ .

$y = \frac{3 (12)}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.3.2.1.1.2

Cancel the common factor.

$y = \frac{3 \cdot 12}{5} \cdot \frac{11}{3}$

$x = - \frac{2 y}{3} + 32$

Step 3.3.2.1.1.3

Rewrite the expression.

$y = \frac{12}{5} \cdot 11$

$x = - \frac{2 y}{3} + 32$

$y = \frac{12}{5} \cdot 11$

$x = - \frac{2 y}{3} + 32$

Step 3.3.2.1.2

Combine $\frac{12}{5}$ and $11$ .

$y = \frac{12 \cdot 11}{5}$

$x = - \frac{2 y}{3} + 32$

Step 3.3.2.1.3

Multiply $12$ by $11$ .

$y = \frac{132}{5}$

$x = - \frac{2 y}{3} + 32$

$y = \frac{132}{5}$

$x = - \frac{2 y}{3} + 32$

$y = \frac{132}{5}$

$x = - \frac{2 y}{3} + 32$

$y = \frac{132}{5}$

$x = - \frac{2 y}{3} + 32$

$y = \frac{132}{5}$

$x = - \frac{2 y}{3} + 32$

Step 4

Replace all occurrences of $y$ with $\frac{132}{5}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{2 y}{3} + 32$ with $\frac{132}{5}$ .

$x = - \frac{2 (\frac{132}{5})}{3} + 32$

$y = \frac{132}{5}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{2 (\frac{132}{5})}{3} + 32$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Combine $2$ and $\frac{132}{5}$ .

$x = - \frac{\frac{2 \cdot 132}{5}}{3} + 32$

$y = \frac{132}{5}$

Step 4.2.1.1.2

Multiply $2$ by $132$ .

$x = - \frac{\frac{264}{5}}{3} + 32$

$y = \frac{132}{5}$

Step 4.2.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = - (\frac{264}{5} \cdot \frac{1}{3}) + 32$

$y = \frac{132}{5}$

Step 4.2.1.1.4

Cancel the common factor of $3$ .

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Step 4.2.1.1.4.1

Factor $3$ out of $264$ .

$x = - (\frac{3 (88)}{5} \cdot \frac{1}{3}) + 32$

$y = \frac{132}{5}$

Step 4.2.1.1.4.2

Cancel the common factor.

$x = - (\frac{3 \cdot 88}{5} \cdot \frac{1}{3}) + 32$

$y = \frac{132}{5}$

Step 4.2.1.1.4.3

Rewrite the expression.

$x = - \frac{88}{5} + 32$

$y = \frac{132}{5}$

$x = - \frac{88}{5} + 32$

$y = \frac{132}{5}$

$x = - \frac{88}{5} + 32$

$y = \frac{132}{5}$

Step 4.2.1.2

To write $32$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = - \frac{88}{5} + 32 \cdot \frac{5}{5}$

$y = \frac{132}{5}$

Step 4.2.1.3

Combine $32$ and $\frac{5}{5}$ .

$x = - \frac{88}{5} + \frac{32 \cdot 5}{5}$

$y = \frac{132}{5}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 88 + 32 \cdot 5}{5}$

$y = \frac{132}{5}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $32$ by $5$ .

$x = \frac{- 88 + 160}{5}$

$y = \frac{132}{5}$

Step 4.2.1.5.2

Add $- 88$ and $160$ .

$x = \frac{72}{5}$

$y = \frac{132}{5}$

$x = \frac{72}{5}$

$y = \frac{132}{5}$

$x = \frac{72}{5}$

$y = \frac{132}{5}$

$x = \frac{72}{5}$

$y = \frac{132}{5}$

$x = \frac{72}{5}$

$y = \frac{132}{5}$

Step 5

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

$(\frac{72}{5}, \frac{132}{5})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{72}{5}, \frac{132}{5})$

Equation Form:

$x = \frac{72}{5}, y = \frac{132}{5}$

Step 7