Examples

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

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

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

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Step 3.1

Move the negative in front of the fraction.

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

Step 3.2

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

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.2

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

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

Step 4

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

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Step 4.1

Use the slope $\frac{1}{3}$ and a given point $(- 2, - 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Solve for $y$ .

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Step 5.1.1

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

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Step 5.1.1.1

Rewrite.

$y + 7 = 0 + 0 + \frac{1}{3} \cdot (x + 2)$

Step 5.1.1.2

Simplify by adding zeros.

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

Step 5.1.1.3

Apply the distributive property.

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

Step 5.1.1.4

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

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

Step 5.1.1.5

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

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

Step 5.1.2

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

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Step 5.1.2.1

Subtract $7$ from both sides of the equation.

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

Step 5.1.2.2

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

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

Step 5.1.2.3

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

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

Step 5.1.2.4

Combine the numerators over the common denominator.

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

Step 5.1.2.5

Simplify the numerator.

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Step 5.1.2.5.1

Multiply $- 7$ by $3$ .

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

Step 5.1.2.5.2

Subtract $21$ from $2$ .

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

Step 5.1.2.6

Move the negative in front of the fraction.

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

Step 5.2

Reorder terms.

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

Step 6

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