Examples

$y = 3 x - 21$ , $(- 7, 0)$

Step 1

Use the slope-intercept form to find the slope.

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Step 1.1

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

$y = m x + b$

Step 1.2

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

$m = 3$

Step 2

The equation of a perpendicular line must have a slope that is the negative reciprocal of the original slope.

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

Step 3

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

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Step 3.1

Use the slope $- \frac{1}{3}$ and a given point $(- 7, 0)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = - \frac{1}{3} \cdot (x - (- 7))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y + 0 = - \frac{1}{3} \cdot (x + 7)$

Step 4

Write in $y = m x + b$ form.

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Step 4.1

Solve for $y$ .

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Step 4.1.1

Add $y$ and $0$ .

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

Step 4.1.2

Simplify $- \frac{1}{3} \cdot (x + 7)$ .

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Step 4.1.2.1

Apply the distributive property.

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Multiply $- \frac{1}{3} \cdot 7$ .

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Step 4.1.2.3.1

Multiply $7$ by $- 1$ .

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

Step 4.1.2.3.2

Combine $- 7$ and $\frac{1}{3}$ .

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

Step 4.1.2.4

Move the negative in front of the fraction.

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

Step 4.2

Reorder terms.

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

Step 4.3

Remove parentheses.

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

Step 5

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