Find the Perpendicular Line (-3,8) , -3x+4y=-23

$(- 3, 8)$ , $- 3 x + 4 y = - 23$

Step 1

Solve $- 3 x + 4 y = - 23$ .

Add $3 x$ to both sides of the equation.

$4 y = - 23 + 3 x$

Divide each term in $4 y = - 23 + 3 x$ by $4$ and simplify.

Divide each term in $4 y = - 23 + 3 x$ by $4$ .

$\frac{4 y}{4} = \frac{- 23}{4} + \frac{3 x}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 23}{4} + \frac{3 x}{4}$

Divide $y$ by $1$ .

$y = \frac{- 23}{4} + \frac{3 x}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{23}{4} + \frac{3 x}{4}$

Step 2

Find the slope when $y = - \frac{23}{4} + \frac{3 x}{4}$ .

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 $- \frac{23}{4}$ and $\frac{3 x}{4}$ .

$y = \frac{3 x}{4} - \frac{23}{4}$

Reorder terms.

$y = \frac{3}{4} x - \frac{23}{4}$

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

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

Step 3

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

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

Step 4

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

Step 5

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

Use the slope $- \frac{4}{3}$ and a given point $(- 3, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = - \frac{4}{3} \cdot (x - (- 3))$

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

$y - 8 = - \frac{4}{3} \cdot (x + 3)$

Step 6

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

Solve for $y$ .

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

Rewrite.

$y - 8 = 0 + 0 - \frac{4}{3} \cdot (x + 3)$

Simplify terms.

Apply the distributive property.

$y - 8 = - \frac{4}{3} x - \frac{4}{3} \cdot 3$

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

$y - 8 = - \frac{x \cdot 4}{3} - \frac{4}{3} \cdot 3$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$y - 8 = - \frac{x \cdot 4}{3} + \frac{- 4}{3} \cdot 3$

Cancel the common factor.

$y - 8 = - \frac{x \cdot 4}{3} + \frac{- 4}{3} \cdot 3$

Rewrite the expression.

$y - 8 = - \frac{x \cdot 4}{3} - 4$

Move $4$ to the left of $x$ .

$y - 8 = - \frac{4 x}{3} - 4$

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

Add $8$ to both sides of the equation.

$y = - \frac{4 x}{3} - 4 + 8$

Add $- 4$ and $8$ .

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

Reorder terms.

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

Remove parentheses.

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

Step 7

Find the Perpendicular Line (-3,8) , -3x+4y=-23

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