Precalculus Examples

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

Step 1

Solve for $x$ in $\frac{x}{2} + \frac{y}{3} = 4$ .

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Step 1.1

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

$\frac{x}{2} = 4 - \frac{y}{3}$

$\frac{x}{4} + \frac{y}{6} = 2$

Step 1.2

Multiply both sides of the equation by $2$ .

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

$\frac{x}{4} + \frac{y}{6} = 2$

Step 1.3

Simplify both sides of the equation.

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Step 1.3.1

Simplify the left side.

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Step 1.3.1.1

Cancel the common factor of $2$ .

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Step 1.3.1.1.1

Cancel the common factor.

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

$\frac{x}{4} + \frac{y}{6} = 2$

Step 1.3.1.1.2

Rewrite the expression.

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

$\frac{x}{4} + \frac{y}{6} = 2$

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

$\frac{x}{4} + \frac{y}{6} = 2$

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

$\frac{x}{4} + \frac{y}{6} = 2$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Simplify $2 (4 - \frac{y}{3})$ .

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Step 1.3.2.1.1

Apply the distributive property.

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

$\frac{x}{4} + \frac{y}{6} = 2$

Step 1.3.2.1.2

Multiply $2$ by $4$ .

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

$\frac{x}{4} + \frac{y}{6} = 2$

Step 1.3.2.1.3

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

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Step 1.3.2.1.3.1

Multiply $- 1$ by $2$ .

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

$\frac{x}{4} + \frac{y}{6} = 2$

Step 1.3.2.1.3.2

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

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

$\frac{x}{4} + \frac{y}{6} = 2$

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

$\frac{x}{4} + \frac{y}{6} = 2$

Step 1.3.2.1.4

Move the negative in front of the fraction.

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

$\frac{x}{4} + \frac{y}{6} = 2$

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

$\frac{x}{4} + \frac{y}{6} = 2$

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

$\frac{x}{4} + \frac{y}{6} = 2$

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

$\frac{x}{4} + \frac{y}{6} = 2$

Step 1.4

Reorder $8$ and $- \frac{2 y}{3}$ .

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

$\frac{x}{4} + \frac{y}{6} = 2$

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

$\frac{x}{4} + \frac{y}{6} = 2$

Step 2

Replace all occurrences of $x$ with $- \frac{2 y}{3} + 8$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{x}{4} + \frac{y}{6} = 2$ with $- \frac{2 y}{3} + 8$ .

$\frac{- \frac{2 y}{3} + 8}{4} + \frac{y}{6} = 2$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{- \frac{2 y}{3} + 8}{4} + \frac{y}{6}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Cancel the common factor of $- \frac{2 y}{3} + 8$ and $4$ .

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Step 2.2.1.1.1.1

Factor $2$ out of $- \frac{2 y}{3}$ .

$\frac{2 (- \frac{y}{3}) + 8}{4} + \frac{y}{6} = 2$

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

Step 2.2.1.1.1.2

Factor $2$ out of $8$ .

$\frac{2 (- \frac{y}{3}) + 2 (4)}{4} + \frac{y}{6} = 2$

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

Step 2.2.1.1.1.3

Factor $2$ out of $2 (- \frac{y}{3}) + 2 (4)$ .

$\frac{2 (- \frac{y}{3} + 4)}{4} + \frac{y}{6} = 2$

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

Step 2.2.1.1.1.4

Cancel the common factors.

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Step 2.2.1.1.1.4.1

Factor $2$ out of $4$ .

$\frac{2 (- \frac{y}{3} + 4)}{2 (2)} + \frac{y}{6} = 2$

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

Step 2.2.1.1.1.4.2

Cancel the common factor.

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

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

Step 2.2.1.1.1.4.3

Rewrite the expression.

$\frac{- \frac{y}{3} + 4}{2} + \frac{y}{6} = 2$

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

$\frac{- \frac{y}{3} + 4}{2} + \frac{y}{6} = 2$

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

$\frac{- \frac{y}{3} + 4}{2} + \frac{y}{6} = 2$

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

Step 2.2.1.1.2

Simplify the numerator.

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Step 2.2.1.1.2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

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

Step 2.2.1.1.2.2

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

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

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

Step 2.2.1.1.2.3

Combine the numerators over the common denominator.

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

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

Step 2.2.1.1.2.4

Multiply $4$ by $3$ .

$\frac{\frac{- y + 12}{3}}{2} + \frac{y}{6} = 2$

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

$\frac{\frac{- y + 12}{3}}{2} + \frac{y}{6} = 2$

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

Step 2.2.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{- y + 12}{3} \cdot \frac{1}{2} + \frac{y}{6} = 2$

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

Step 2.2.1.1.4

Multiply $\frac{- y + 12}{3} \cdot \frac{1}{2}$ .

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Step 2.2.1.1.4.1

Multiply $\frac{- y + 12}{3}$ by $\frac{1}{2}$ .

$\frac{- y + 12}{3 \cdot 2} + \frac{y}{6} = 2$

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

Step 2.2.1.1.4.2

Multiply $3$ by $2$ .

$\frac{- y + 12}{6} + \frac{y}{6} = 2$

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

$\frac{- y + 12}{6} + \frac{y}{6} = 2$

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

$\frac{- y + 12}{6} + \frac{y}{6} = 2$

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

Step 2.2.1.2

Simplify terms.

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Step 2.2.1.2.1

Combine the numerators over the common denominator.

$\frac{- y + 12 + y}{6} = 2$

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

Step 2.2.1.2.2

Combine the opposite terms in $- y + 12 + y$ .

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Step 2.2.1.2.2.1

Add $- y$ and $y$ .

$\frac{0 + 12}{6} = 2$

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

Step 2.2.1.2.2.2

Add $0$ and $12$ .

$\frac{12}{6} = 2$

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

$\frac{12}{6} = 2$

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

Step 2.2.1.2.3

Divide $12$ by $6$ .

$2 = 2$

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

$2 = 2$

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

$2 = 2$

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

$2 = 2$

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

$2 = 2$

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

Step 3

Remove any equations from the system that are always true.

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

Step 4