Find the Slope of a Perpendicular Line 6x-4y=9

$6 x - 4 y = 9$

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 $6 x$ from both sides of the equation.

$- 4 y = 9 - 6 x$

Divide each term in $- 4 y = 9 - 6 x$ by $- 4$ and simplify.

Divide each term in $- 4 y = 9 - 6 x$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{9}{- 4} + \frac{- 6 x}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{9}{- 4} + \frac{- 6 x}{- 4}$

Divide $y$ by $1$ .

$y = \frac{9}{- 4} + \frac{- 6 x}{- 4}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{9}{4} + \frac{- 6 x}{- 4}$

Cancel the common factor of $- 6$ and $- 4$ .

Factor $- 2$ out of $- 6 x$ .

$y = - \frac{9}{4} + \frac{- 2 (3 x)}{- 4}$

Cancel the common factors.

Factor $- 2$ out of $- 4$ .

$y = - \frac{9}{4} + \frac{- 2 (3 x)}{- 2 (2)}$

Cancel the common factor.

$y = - \frac{9}{4} + \frac{- 2 (3 x)}{- 2 \cdot 2}$

Rewrite the expression.

$y = - \frac{9}{4} + \frac{3 x}{2}$

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

Reorder $- \frac{9}{4}$ and $\frac{3 x}{2}$ .

$y = \frac{3 x}{2} - \frac{9}{4}$

Reorder terms.

$y = \frac{3}{2} x - \frac{9}{4}$

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 $y = \frac{3}{2} x - \frac{9}{4}$ 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.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

Step 5

Find the Slope of a Perpendicular Line 6x-4y=9

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