Find the Perpendicular Line 2x+9y=-30 ; (-8,0)

$2 x + 9 y = - 30$ ; $(- 8, 0)$

Step 1

Solve $2 x + 9 y = - 30$ .

Subtract $2 x$ from both sides of the equation.

$9 y = - 30 - 2 x$

Divide each term in $9 y = - 30 - 2 x$ by $9$ and simplify.

Divide each term in $9 y = - 30 - 2 x$ by $9$ .

$\frac{9 y}{9} = \frac{- 30}{9} + \frac{- 2 x}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 y}{9} = \frac{- 30}{9} + \frac{- 2 x}{9}$

Divide $y$ by $1$ .

$y = \frac{- 30}{9} + \frac{- 2 x}{9}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 30$ and $9$ .

Factor $3$ out of $- 30$ .

$y = \frac{3 (- 10)}{9} + \frac{- 2 x}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$y = \frac{3 \cdot - 10}{3 \cdot 3} + \frac{- 2 x}{9}$

Cancel the common factor.

$y = \frac{3 \cdot - 10}{3 \cdot 3} + \frac{- 2 x}{9}$

Rewrite the expression.

$y = \frac{- 10}{3} + \frac{- 2 x}{9}$

Move the negative in front of the fraction.

$y = - \frac{10}{3} + \frac{- 2 x}{9}$

Move the negative in front of the fraction.

$y = - \frac{10}{3} - \frac{2 x}{9}$

Step 2

Find the slope when $y = - \frac{10}{3} - \frac{2 x}{9}$ .

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 $- \frac{10}{3}$ and $- \frac{2 x}{9}$ .

$y = - \frac{2 x}{9} - \frac{10}{3}$

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

Reorder terms.

$y = - (\frac{2}{9} x) - \frac{10}{3}$

Remove parentheses.

$y = - \frac{2}{9} x - \frac{10}{3}$

Using the slope-intercept form, the slope is $- \frac{2}{9}$ .

$m = - \frac{2}{9}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{2}{9}}$

Step 4

Simplify $- \frac{1}{- \frac{2}{9}}$ to find the slope of the perpendicular line.

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

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{2}{9}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{2}{9}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{9}{2})$

Multiply $\frac{9}{2}$ by $1$ .

$m_{perpendicular} = \frac{9}{2}$

Multiply $- - \frac{9}{2}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{9}{2})$

Multiply $\frac{9}{2}$ by $1$ .

$m_{perpendicular} = \frac{9}{2}$

Step 5

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

Use the slope $\frac{9}{2}$ and a given point $(- 8, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = \frac{9}{2} \cdot (x - (- 8))$

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

$y + 0 = \frac{9}{2} \cdot (x + 8)$

Step 6

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

Solve for $y$ .

Add $y$ and $0$ .

$y = \frac{9}{2} \cdot (x + 8)$

Simplify $\frac{9}{2} \cdot (x + 8)$ .

Apply the distributive property.

$y = \frac{9}{2} x + \frac{9}{2} \cdot 8$

Combine $\frac{9}{2}$ and $x$ .

$y = \frac{9 x}{2} + \frac{9}{2} \cdot 8$

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$y = \frac{9 x}{2} + \frac{9}{2} \cdot (2 (4))$

Cancel the common factor.

$y = \frac{9 x}{2} + \frac{9}{2} \cdot (2 \cdot 4)$

Rewrite the expression.

$y = \frac{9 x}{2} + 9 \cdot 4$

Multiply $9$ by $4$ .

$y = \frac{9 x}{2} + 36$

Reorder terms.

$y = \frac{9}{2} x + 36$

Step 7

Find the Perpendicular Line 2x+9y=-30 ; (-8,0)

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