Find the Slope of a Perpendicular Line 3x-2y=5

$3 x - 2 y = 5$

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 $3 x$ from both sides of the equation.

$- 2 y = 5 - 3 x$

Divide each term in $- 2 y = 5 - 3 x$ by $- 2$ and simplify.

Divide each term in $- 2 y = 5 - 3 x$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{5}{- 2} + \frac{- 3 x}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{5}{- 2} + \frac{- 3 x}{- 2}$

Divide $y$ by $1$ .

$y = \frac{5}{- 2} + \frac{- 3 x}{- 2}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{5}{2} + \frac{- 3 x}{- 2}$

Dividing two negative values results in a positive value.

$y = - \frac{5}{2} + \frac{3 x}{2}$

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

Reorder $- \frac{5}{2}$ and $\frac{3 x}{2}$ .

$y = \frac{3 x}{2} - \frac{5}{2}$

Reorder terms.

$y = \frac{3}{2} x - \frac{5}{2}$

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

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

The equation of a perpendicular line to $y = \frac{3}{2} x - \frac{5}{2}$ must have a slope that is the negative reciprocal of the original slope.

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

Simplify the result.

Multiply the numerator by the reciprocal of the denominator.

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

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

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