Find the Perpendicular Line Through (5,-4) ; perpendicular to 5y=x-10

Through $(5, - 4)$ ; perpendicular to $5 y = x - 10$

Step 1

Divide each term in $5 y = x - 10$ by $5$ and simplify.

Divide each term in $5 y = x - 10$ by $5$ .

$\frac{5 y}{5} = \frac{x}{5} + \frac{- 10}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{x}{5} + \frac{- 10}{5}$

Divide $y$ by $1$ .

$y = \frac{x}{5} + \frac{- 10}{5}$

Simplify the right side.

Divide $- 10$ by $5$ .

$y = \frac{x}{5} - 2$

Step 2

Find the slope when $y = \frac{x}{5} - 2$ .

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.

$y = m x + b$

Reorder terms.

$y = \frac{1}{5} x - 2$

Using the slope-intercept form, the slope is $\frac{1}{5}$ .

$m = \frac{1}{5}$

Step 3

The equation of a perpendicular line must have a slope that is the negative reciprocal of the original slope.

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

Step 4

Simplify $- \frac{1}{\frac{1}{5}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $- (1 \cdot 5)$ .

Multiply $5$ by $1$ .

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

Multiply $- 1$ by $5$ .

$m_{perpendicular} = - 5$

Step 5

Find the equation of the perpendicular line using the point-slope formula.

Use the slope $- 5$ and a given point $(5, - 4)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 4) = - 5 \cdot (x - (5))$

Simplify the equation and keep it in point-slope form.

$y + 4 = - 5 \cdot (x - 5)$

Step 6

Solve for $y$ .

Simplify $- 5 \cdot (x - 5)$ .

Rewrite.

$y + 4 = 0 + 0 - 5 \cdot (x - 5)$

Simplify by adding zeros.

$y + 4 = - 5 \cdot (x - 5)$

Apply the distributive property.

$y + 4 = - 5 x - 5 \cdot - 5$

Multiply $- 5$ by $- 5$ .

$y + 4 = - 5 x + 25$

Move all terms not containing $y$ to the right side of the equation.

Subtract $4$ from both sides of the equation.

$y = - 5 x + 25 - 4$

Subtract $4$ from $25$ .

$y = - 5 x + 21$

Step 7

Find the Perpendicular Line Through (5,-4) ; perpendicular to 5y=x-10

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