Find the Perpendicular Line 5y=x-4 (-2,1)

$5 y = x - 4$ $(- 2, 1)$

Divide each term in $5 y = x - 4$ by $5$ and simplify.

Divide each term in $5 y = x - 4$ by $5$ .

$\frac{5 y}{5} = \frac{x}{5} + \frac{- 4}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{x}{5} + \frac{- 4}{5}$

Divide $y$ by $1$ .

$y = \frac{x}{5} + \frac{- 4}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{x}{5} - \frac{4}{5}$

Find the slope when $y = \frac{x}{5} - \frac{4}{5}$ .

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.

$y = m x + b$

Reorder terms.

$y = \frac{1}{5} x - \frac{4}{5}$

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

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

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

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

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 $5$ by $1$ .

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

Multiply $- 1$ by $5$ .

$m_{perpendicular} = - 5$

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

Use the slope $- 5$ and a given point $(- 2, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - 5 \cdot (x - (- 2))$

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

$y - 1 = - 5 \cdot (x + 2)$

Solve for $y$ .

Simplify $- 5 \cdot (x + 2)$ .

Rewrite.

$y - 1 = 0 + 0 - 5 \cdot (x + 2)$

Simplify by adding zeros.

$y - 1 = - 5 \cdot (x + 2)$

Apply the distributive property.

$y - 1 = - 5 x - 5 \cdot 2$

Multiply $- 5$ by $2$ .

$y - 1 = - 5 x - 10$

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

Add $1$ to both sides of the equation.

$y = - 5 x - 10 + 1$

Add $- 10$ and $1$ .

$y = - 5 x - 9$

Find the Perpendicular Line 5y=x-4 (-2,1)

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