Precalculus Examples

$(- 2, - 7)$ , $y = - 3 x$

Use the slope-intercept form to find the slope.

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The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Using the slope-intercept form, the slope is $- 3$ .

$m = - 3$

The equation of a perpendicular line to $y = - 3 x$ must have a slope that is the negative reciprocal of the original slope.

$m_{perpendicular} = - \frac{1}{- 3}$

Simplify $- \frac{1}{- 3}$ to find the slope of the perpendicular line.

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Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{3}$

Simplify $- - \frac{1}{3}$ .

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Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{3})$

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

$m_{perpendicular} = \frac{1}{3}$

Find the equation of the perpendicular line using the point-slope formula.

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Using the point-slope form $y - y_{1} = m (x - x_{1})$ , plug in $m = \frac{1}{3}$ , $x_{1} = - 2$ , and $y_{1} = - 7$ .

$y - (- 7) = (\frac{1}{3}) (x - (- 2))$

Solve for $y$ .

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Multiply $- 1$ by $- 7$ .

$y + 7 = (\frac{1}{3}) (x - (- 2))$

Simplify $(\frac{1}{3}) (x - (- 2))$ .

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Multiply $- 1$ by $- 2$ .

$y + 7 = \frac{1}{3} \cdot (x + 2)$

Apply the distributive property.

$y + 7 = \frac{1}{3} x + \frac{1}{3} \cdot 2$

Simplify $\frac{1}{3} x$ .

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Write $x$ as a fraction with denominator $1$ .

$y + 7 = \frac{1}{3} \cdot \frac{x}{1} + \frac{1}{3} \cdot 2$

Multiply $\frac{1}{3}$ and $\frac{x}{1}$ .

$y + 7 = \frac{x}{3} + \frac{1}{3} \cdot 2$

Simplify $\frac{1}{3} \cdot 2$ .

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Write $2$ as a fraction with denominator $1$ .

$y + 7 = \frac{x}{3} + \frac{1}{3} \cdot \frac{2}{1}$

Multiply $\frac{1}{3}$ and $\frac{2}{1}$ .

$y + 7 = \frac{x}{3} + \frac{2}{3}$

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

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Subtract $7$ from both sides of the equation.

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

Simplify the right side of the equation.

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To write $- \frac{- 7}{1}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Write each expression with a common denominator of $3$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Multiply $3$ by $1$ .

$y = \frac{x}{3} + (\frac{2}{3} + \frac{- 7 \cdot 3}{3})$

Combine the numerators over the common denominator.

$y = \frac{x}{3} + \frac{2 - 7 \cdot 3}{3}$

Simplify the numerator.

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Multiply $- 7$ by $3$ .

$y = \frac{x}{3} + \frac{2 - 21}{3}$

Subtract $21$ from $2$ .

$y = \frac{x}{3} + \frac{- 19}{3}$

Move the negative in front of the fraction.

$y = \frac{x}{3} - \frac{19}{3}$

Reorder terms.

$y = \frac{1}{3} x - \frac{19}{3}$

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