Find the Perpendicular Line y=-3/4x+1 and (9,12)

$y = - \frac{3}{4} x + 1$ and $(9, 12)$

Step 1

Find the slope when $y = - \frac{3}{4} x + 1$ .

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 $\frac{3}{4}$ .

$y = - \frac{x \cdot 3}{4} + 1$

Move $3$ to the left of $x$ .

$y = - \frac{3 x}{4} + 1$

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

Reorder terms.

$y = - (\frac{3}{4} x) + 1$

Remove parentheses.

$y = - \frac{3}{4} x + 1$

Using the slope-intercept form, the slope is $- \frac{3}{4}$ .

$m = - \frac{3}{4}$

Step 2

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

$m_{perpendicular} = - \frac{1}{- \frac{3}{4}}$

Step 3

Simplify $- \frac{1}{- \frac{3}{4}}$ to find the slope of the perpendicular line.

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

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{3}{4}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{3}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{4}{3})$

Multiply $\frac{4}{3}$ by $1$ .

$m_{perpendicular} = \frac{4}{3}$

Multiply $- - \frac{4}{3}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{4}{3})$

Multiply $\frac{4}{3}$ by $1$ .

$m_{perpendicular} = \frac{4}{3}$

Step 4

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

Use the slope $\frac{4}{3}$ and a given point $(9, 12)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (12) = \frac{4}{3} \cdot (x - (9))$

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

$y - 12 = \frac{4}{3} \cdot (x - 9)$

Step 5

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

Solve for $y$ .

Simplify $\frac{4}{3} \cdot (x - 9)$ .

Rewrite.

$y - 12 = 0 + 0 + \frac{4}{3} \cdot (x - 9)$

Simplify by adding zeros.

$y - 12 = \frac{4}{3} \cdot (x - 9)$

Apply the distributive property.

$y - 12 = \frac{4}{3} x + \frac{4}{3} \cdot - 9$

Combine $\frac{4}{3}$ and $x$ .

$y - 12 = \frac{4 x}{3} + \frac{4}{3} \cdot - 9$

Cancel the common factor of $3$ .

Factor $3$ out of $- 9$ .

$y - 12 = \frac{4 x}{3} + \frac{4}{3} \cdot (3 (- 3))$

Cancel the common factor.

$y - 12 = \frac{4 x}{3} + \frac{4}{3} \cdot (3 \cdot - 3)$

Rewrite the expression.

$y - 12 = \frac{4 x}{3} + 4 \cdot - 3$

Multiply $4$ by $- 3$ .

$y - 12 = \frac{4 x}{3} - 12$

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

Add $12$ to both sides of the equation.

$y = \frac{4 x}{3} - 12 + 12$

Combine the opposite terms in $\frac{4 x}{3} - 12 + 12$ .

Add $- 12$ and $12$ .

$y = \frac{4 x}{3} + 0$

Add $\frac{4 x}{3}$ and $0$ .

$y = \frac{4 x}{3}$

Reorder terms.

$y = \frac{4}{3} x$

Step 6

Find the Perpendicular Line y=-3/4x+1 and (9,12)

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