Find the Slope of a Perpendicular Line -7x+6y=10

$- 7 x + 6 y = 10$

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 $7 x$ to both sides of the equation.

$6 y = 10 + 7 x$

Divide each term in $6 y = 10 + 7 x$ by $6$ and simplify.

Divide each term in $6 y = 10 + 7 x$ by $6$ .

$\frac{6 y}{6} = \frac{10}{6} + \frac{7 x}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{10}{6} + \frac{7 x}{6}$

Divide $y$ by $1$ .

$y = \frac{10}{6} + \frac{7 x}{6}$

Simplify the right side.

Cancel the common factor of $10$ and $6$ .

Factor $2$ out of $10$ .

$y = \frac{2 (5)}{6} + \frac{7 x}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$y = \frac{2 \cdot 5}{2 \cdot 3} + \frac{7 x}{6}$

Cancel the common factor.

$y = \frac{2 \cdot 5}{2 \cdot 3} + \frac{7 x}{6}$

Rewrite the expression.

$y = \frac{5}{3} + \frac{7 x}{6}$

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

Reorder $\frac{5}{3}$ and $\frac{7 x}{6}$ .

$y = \frac{7 x}{6} + \frac{5}{3}$

Reorder terms.

$y = \frac{7}{6} x + \frac{5}{3}$

Step 2

Using the slope-intercept form, the slope is $\frac{7}{6}$ .

$m = \frac{7}{6}$

Step 3

The equation of a perpendicular line to $y = \frac{7}{6} x + \frac{5}{3}$ must have a slope that is the negative reciprocal of the original slope.

$m_{perpendicular} = - \frac{1}{\frac{7}{6}}$

Step 4

Simplify the result.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{6}{7}))$

Multiply $\frac{6}{7}$ by $1$ .

$m_{perpendicular} = - \frac{6}{7}$

Step 5