Find the Slope of a Perpendicular Line -6x+9y=7

$- 6 x + 9 y = 7$

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.

$y = m x + b$

Add $6 x$ to both sides of the equation.

$9 y = 7 + 6 x$

Divide each term in $9 y = 7 + 6 x$ by $9$ and simplify.

Divide each term in $9 y = 7 + 6 x$ by $9$ .

$\frac{9 y}{9} = \frac{7}{9} + \frac{6 x}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 y}{9} = \frac{7}{9} + \frac{6 x}{9}$

Divide $y$ by $1$ .

$y = \frac{7}{9} + \frac{6 x}{9}$

Simplify the right side.

Cancel the common factor of $6$ and $9$ .

Factor $3$ out of $6 x$ .

$y = \frac{7}{9} + \frac{3 (2 x)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$y = \frac{7}{9} + \frac{3 (2 x)}{3 (3)}$

Cancel the common factor.

$y = \frac{7}{9} + \frac{3 (2 x)}{3 \cdot 3}$

Rewrite the expression.

$y = \frac{7}{9} + \frac{2 x}{3}$

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

Reorder $\frac{7}{9}$ and $\frac{2 x}{3}$ .

$y = \frac{2 x}{3} + \frac{7}{9}$

Reorder terms.

$y = \frac{2}{3} x + \frac{7}{9}$

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 $y = \frac{2}{3} x + \frac{7}{9}$ 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