Find the Perpendicular Line y=3x-2 (-9,5)

$y = 3 x - 2$ $(- 9, 5)$

Step 1

Use the slope-intercept form to find the slope.

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 $3$ .

$m = 3$

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.

Use the slope $- \frac{1}{3}$ and a given point $(- 9, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = - \frac{1}{3} \cdot (x - (- 9))$

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

$y - 5 = - \frac{1}{3} \cdot (x + 9)$

Step 4

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

Solve for $y$ .

Simplify $- \frac{1}{3} \cdot (x + 9)$ .

Rewrite.

$y - 5 = 0 + 0 - \frac{1}{3} \cdot (x + 9)$

Simplify by adding zeros.

$y - 5 = - \frac{1}{3} \cdot (x + 9)$

Apply the distributive property.

$y - 5 = - \frac{1}{3} x - \frac{1}{3} \cdot 9$

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

$y - 5 = - \frac{x}{3} - \frac{1}{3} \cdot 9$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$y - 5 = - \frac{x}{3} + \frac{- 1}{3} \cdot 9$

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

$y - 5 = - \frac{x}{3} - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$y - 5 = - \frac{x}{3} - 3$

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

Add $5$ to both sides of the equation.

$y = - \frac{x}{3} - 3 + 5$

Add $- 3$ and $5$ .

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

Reorder terms.

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

Remove parentheses.

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

Step 5

Find the Perpendicular Line y=3x-2 (-9,5)

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