Find the Perpendicular Line Through (-5,1) , perpendicular to y=-9/5x+8

Through $(- 5, 1)$ , perpendicular to $y = - \frac{9}{5} x + 8$

Step 1

Find the slope when $y = - \frac{9}{5} x + 8$ .

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.

Simplify each term.

Combine $x$ and $\frac{9}{5}$ .

$y = - \frac{x \cdot 9}{5} + 8$

Move $9$ to the left of $x$ .

$y = - \frac{9 x}{5} + 8$

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

Reorder terms.

$y = - (\frac{9}{5} x) + 8$

Remove parentheses.

$y = - \frac{9}{5} x + 8$

Using the slope-intercept form, the slope is $- \frac{9}{5}$ .

$m = - \frac{9}{5}$

Step 2

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

$m_{perpendicular} = - \frac{1}{- \frac{9}{5}}$

Step 3

Simplify $- \frac{1}{- \frac{9}{5}}$ to find the slope of the perpendicular line.

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

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{9}{5}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{9}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{5}{9})$

Multiply $\frac{5}{9}$ by $1$ .

$m_{perpendicular} = \frac{5}{9}$

Multiply $- - \frac{5}{9}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{5}{9})$

Multiply $\frac{5}{9}$ by $1$ .

$m_{perpendicular} = \frac{5}{9}$

Step 4

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

Use the slope $\frac{5}{9}$ and a given point $(- 5, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = \frac{5}{9} \cdot (x - (- 5))$

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

$y - 1 = \frac{5}{9} \cdot (x + 5)$

Step 5

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

Solve for $y$ .

Simplify $\frac{5}{9} \cdot (x + 5)$ .

Rewrite.

$y - 1 = 0 + 0 + \frac{5}{9} \cdot (x + 5)$

Simplify by adding zeros.

$y - 1 = \frac{5}{9} \cdot (x + 5)$

Apply the distributive property.

$y - 1 = \frac{5}{9} x + \frac{5}{9} \cdot 5$

Combine $\frac{5}{9}$ and $x$ .

$y - 1 = \frac{5 x}{9} + \frac{5}{9} \cdot 5$

Multiply $\frac{5}{9} \cdot 5$ .

Combine $\frac{5}{9}$ and $5$ .

$y - 1 = \frac{5 x}{9} + \frac{5 \cdot 5}{9}$

Multiply $5$ by $5$ .

$y - 1 = \frac{5 x}{9} + \frac{25}{9}$

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

Add $1$ to both sides of the equation.

$y = \frac{5 x}{9} + \frac{25}{9} + 1$

Write $1$ as a fraction with a common denominator.

$y = \frac{5 x}{9} + \frac{25}{9} + \frac{9}{9}$

Combine the numerators over the common denominator.

$y = \frac{5 x}{9} + \frac{25 + 9}{9}$

Add $25$ and $9$ .

$y = \frac{5 x}{9} + \frac{34}{9}$

Reorder terms.

$y = \frac{5}{9} x + \frac{34}{9}$

Step 6

Find the Perpendicular Line Through (-5,1) , perpendicular to y=-9/5x+8

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