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

$(2, 2)$ , $y = - \frac{1}{5} x + 5$

Step 1

Find the slope when $y = - \frac{1}{5} x + 5$ .

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.

$y = m x + b$

Simplify the right side.

Combine $x$ and $\frac{1}{5}$ .

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

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

Reorder terms.

$y = - (\frac{1}{5} x) + 5$

Remove parentheses.

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

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

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

Step 2

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 3

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

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

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

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

Move the negative in front of the fraction.

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

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 $5$ .

$m_{perpendicular} = 5$

Multiply $- 1$ by $- 5$ .

$m_{perpendicular} = 5$

Step 4

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

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

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

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

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

Step 5

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Multiply $5$ by $- 2$ .

$y - 2 = 5 x - 10$

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

Add $2$ to both sides of the equation.

$y = 5 x - 10 + 2$

Add $- 10$ and $2$ .

$y = 5 x - 8$

Step 6

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

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