Find the Slope of a Perpendicular Line 5x+7y=6

$5 x + 7 y = 6$

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 $5 x$ from both sides of the equation.

$7 y = 6 - 5 x$

Divide each term in $7 y = 6 - 5 x$ by $7$ and simplify.

Divide each term in $7 y = 6 - 5 x$ by $7$ .

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

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{6}{7} - \frac{5 x}{7}$

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

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

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

Reorder terms.

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

Remove parentheses.

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

Step 2

Using the slope-intercept form, the slope is $- \frac{5}{7}$ .

$m = - \frac{5}{7}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{5}{7}}$

Step 4

Simplify the result.

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

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{5}{7}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{5}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{7}{5})$

Multiply $\frac{7}{5}$ by $1$ .

$m_{perpendicular} = \frac{7}{5}$

Multiply $- - \frac{7}{5}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{7}{5})$

Multiply $\frac{7}{5}$ by $1$ .

$m_{perpendicular} = \frac{7}{5}$

Step 5

Find the Slope of a Perpendicular Line 5x+7y=6

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