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

$x + 6 y = 5$

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.

$6 y = 5 - x$

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

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

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

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

Reorder terms.

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

Remove parentheses.

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

Using the slope-intercept form, the slope is $- \frac{1}{6}$ .

$m = - \frac{1}{6}$

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

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

Simplify the result.

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 6$

Multiply $6$ by $1$ .

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

Multiply $- 1$ by $6$ .

$m_{perpendicular} = 6$

Multiply $- 1$ by $- 6$ .

$m_{perpendicular} = 6$