Find the Slope of a Perpendicular Line 3x+2y=12

$3 x + 2 y = 12$

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 $3 x$ from both sides of the equation.

$2 y = 12 - 3 x$

Divide each term in $2 y = 12 - 3 x$ by $2$ and simplify.

Divide each term in $2 y = 12 - 3 x$ by $2$ .

$\frac{2 y}{2} = \frac{12}{2} + \frac{- 3 x}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{12}{2} + \frac{- 3 x}{2}$

Divide $y$ by $1$ .

$y = \frac{12}{2} + \frac{- 3 x}{2}$

Simplify the right side.

Simplify each term.

Divide $12$ by $2$ .

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

Move the negative in front of the fraction.

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

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

Reorder $6$ and $- \frac{3 x}{2}$ .

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

Reorder terms.

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

Remove parentheses.

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

Step 2

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

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

Step 3

The equation of a perpendicular line to $y = - \frac{3}{2} x + 6$ must have a slope that is the negative reciprocal of the original slope.

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

Step 4

Simplify the result.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

Multiply $- - \frac{2}{3}$ .

Multiply $- 1$ by $- 1$ .

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

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

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

Step 5

Find the Slope of a Perpendicular Line 3x+2y=12

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