Find the Slope of a Perpendicular Line 2x-3y=6

$2 x - 3 y = 6$

Step 1

Rewrite in slope-intercept form.

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

$y = m x + b$

Subtract $2 x$ from both sides of the equation.

$- 3 y = 6 - 2 x$

Divide each term in $- 3 y = 6 - 2 x$ by $- 3$ and simplify.

Divide each term in $- 3 y = 6 - 2 x$ by $- 3$ .

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

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Divide $6$ by $- 3$ .

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

Dividing two negative values results in a positive value.

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

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

Reorder $- 2$ and $\frac{2 x}{3}$ .

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

Reorder terms.

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

Step 2

Using the slope-intercept form, the slope is $\frac{2}{3}$ .

$m = \frac{2}{3}$

Step 3

The equation of a perpendicular line to $y = \frac{2}{3} x - 2$ must have a slope that is the negative reciprocal of the original slope.

$m_{perpendicular} = - \frac{1}{\frac{2}{3}}$

Step 4

Simplify the result.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{3}{2}))$

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

$m_{perpendicular} = - \frac{3}{2}$

Step 5