Find the Slope of a Perpendicular Line x-5y=-6

$x - 5 y = - 6$

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 $x$ from both sides of the equation.

$- 5 y = - 6 - x$

Divide each term in $- 5 y = - 6 - x$ by $- 5$ and simplify.

Divide each term in $- 5 y = - 6 - x$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 6}{- 5} + \frac{- x}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{- 6}{- 5} + \frac{- x}{- 5}$

Divide $y$ by $1$ .

$y = \frac{- 6}{- 5} + \frac{- x}{- 5}$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$y = \frac{6}{5} + \frac{- x}{- 5}$

Dividing two negative values results in a positive value.

$y = \frac{6}{5} + \frac{x}{5}$

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

Reorder $\frac{6}{5}$ and $\frac{x}{5}$ .

$y = \frac{x}{5} + \frac{6}{5}$

Reorder terms.

$y = \frac{1}{5} x + \frac{6}{5}$

Step 2

Using the slope-intercept form, the slope is $\frac{1}{5}$ .

$m = \frac{1}{5}$

Step 3

The equation of a perpendicular line to $y = \frac{1}{5} x + \frac{6}{5}$ must have a slope that is the negative reciprocal of the original slope.

$m_{perpendicular} = - \frac{1}{\frac{1}{5}}$

Step 4

Simplify the result.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 5)$

Multiply $- (1 \cdot 5)$ .

Multiply $5$ by $1$ .

$m_{perpendicular} = - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$m_{perpendicular} = - 5$

Step 5