Find the Slope of a Perpendicular Line 5x-6y=1

$5 x - 6 y = 1$

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 $5 x$ from both sides of the equation.

$- 6 y = 1 - 5 x$

Divide each term in $- 6 y = 1 - 5 x$ by $- 6$ and simplify.

Divide each term in $- 6 y = 1 - 5 x$ by $- 6$ .

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

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Dividing two negative values results in a positive value.

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

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

Reorder $- \frac{1}{6}$ and $\frac{5 x}{6}$ .

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

Reorder terms.

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

Step 2

Using the slope-intercept form, the slope is $\frac{5}{6}$ .

$m = \frac{5}{6}$

Step 3

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

$m_{perpendicular} = - \frac{1}{\frac{5}{6}}$

Step 4

Simplify the result.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{6}{5}))$

Multiply $\frac{6}{5}$ by $1$ .

$m_{perpendicular} = - \frac{6}{5}$

Step 5