Find the Slope of the Perpendicular Line to the Line Through the Two Points (6,-9) , (0,4)

$(6, - 9)$ , $(0, 4)$

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{4 - (- 9)}{0 - (6)}$

Simplify.

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Simplify the numerator.

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Multiply $- 1$ by $- 9$ .

$m = \frac{4 + 9}{0 - (6)}$

Add $4$ and $9$ .

$m = \frac{13}{0 - (6)}$

Simplify the denominator.

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Multiply $- 1$ by $6$ .

$m = \frac{13}{0 - 6}$

Subtract $6$ from $0$ .

$m = \frac{13}{- 6}$

Move the negative in front of the fraction.

$m = - \frac{13}{6}$

The slope of a perpendicular line is the negative reciprocal of the slope of the line that passes through the two given points.

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

Simplify $- \frac{1}{- \frac{13}{6}}$ .

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

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{13}{6}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{13}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{6}{13})$

Multiply $\frac{6}{13}$ by $1$ .

$m_{perpendicular} = \frac{6}{13}$

Multiply $- - \frac{6}{13}$ .

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Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{6}{13})$

Multiply $\frac{6}{13}$ by $1$ .

$m_{perpendicular} = \frac{6}{13}$

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