Find the Perpendicular Line through (2,-9) perpendicular to 9y=x-18

through $(2, - 9)$ perpendicular to $9 y = x - 18$

Step 1

Divide each term in $9 y = x - 18$ by $9$ and simplify.

Divide each term in $9 y = x - 18$ by $9$ .

$\frac{9 y}{9} = \frac{x}{9} + \frac{- 18}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 y}{9} = \frac{x}{9} + \frac{- 18}{9}$

Divide $y$ by $1$ .

$y = \frac{x}{9} + \frac{- 18}{9}$

Simplify the right side.

Divide $- 18$ by $9$ .

$y = \frac{x}{9} - 2$

Step 2

Find the slope when $y = \frac{x}{9} - 2$ .

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.

$y = \frac{1}{9} x - 2$

Using the slope-intercept form, the slope is $\frac{1}{9}$ .

$m = \frac{1}{9}$

Step 3

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

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

Step 4

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 9)$

Multiply $- (1 \cdot 9)$ .

Multiply $9$ by $1$ .

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

Multiply $- 1$ by $9$ .

$m_{perpendicular} = - 9$

Step 5

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

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

$y - (- 9) = - 9 \cdot (x - (2))$

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

$y + 9 = - 9 \cdot (x - 2)$

Step 6

Solve for $y$ .

Simplify $- 9 \cdot (x - 2)$ .

Rewrite.

$y + 9 = 0 + 0 - 9 \cdot (x - 2)$

Simplify by adding zeros.

$y + 9 = - 9 \cdot (x - 2)$

Apply the distributive property.

$y + 9 = - 9 x - 9 \cdot - 2$

Multiply $- 9$ by $- 2$ .

$y + 9 = - 9 x + 18$

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

Subtract $9$ from both sides of the equation.

$y = - 9 x + 18 - 9$

Subtract $9$ from $18$ .

$y = - 9 x + 9$

Step 7

Find the Perpendicular Line through (2,-9) perpendicular to 9y=x-18

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