Find the Perpendicular Line line 4x-3y=5 , point (-3,2)

line $4 x - 3 y = 5$ , point $(- 3, 2)$

Step 1

Solve $4 x - 3 y = 5$ .

Subtract $4 x$ from both sides of the equation.

$- 3 y = 5 - 4 x$

Divide each term in $- 3 y = 5 - 4 x$ by $- 3$ and simplify.

Divide each term in $- 3 y = 5 - 4 x$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{5}{- 3} + \frac{- 4 x}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{5}{- 3} + \frac{- 4 x}{- 3}$

Divide $y$ by $1$ .

$y = \frac{5}{- 3} + \frac{- 4 x}{- 3}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{5}{3} + \frac{- 4 x}{- 3}$

Dividing two negative values results in a positive value.

$y = - \frac{5}{3} + \frac{4 x}{3}$

Step 2

Find the slope when $y = - \frac{5}{3} + \frac{4 x}{3}$ .

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 $- \frac{5}{3}$ and $\frac{4 x}{3}$ .

$y = \frac{4 x}{3} - \frac{5}{3}$

Reorder terms.

$y = \frac{4}{3} x - \frac{5}{3}$

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

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

Step 3

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

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

Step 4

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

Step 5

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

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

$y - (2) = - \frac{3}{4} \cdot (x - (- 3))$

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

$y - 2 = - \frac{3}{4} \cdot (x + 3)$

Step 6

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

Solve for $y$ .

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

Rewrite.

$y - 2 = 0 + 0 - \frac{3}{4} \cdot (x + 3)$

Simplify by adding zeros.

$y - 2 = - \frac{3}{4} \cdot (x + 3)$

Apply the distributive property.

$y - 2 = - \frac{3}{4} x - \frac{3}{4} \cdot 3$

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

$y - 2 = - \frac{x \cdot 3}{4} - \frac{3}{4} \cdot 3$

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

Multiply $3$ by $- 1$ .

$y - 2 = - \frac{x \cdot 3}{4} - 3 (\frac{3}{4})$

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

$y - 2 = - \frac{x \cdot 3}{4} + \frac{- 3 \cdot 3}{4}$

Multiply $- 3$ by $3$ .

$y - 2 = - \frac{x \cdot 3}{4} + \frac{- 9}{4}$

Simplify each term.

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

$y - 2 = - \frac{3 \cdot x}{4} + \frac{- 9}{4}$

Move the negative in front of the fraction.

$y - 2 = - \frac{3 x}{4} - \frac{9}{4}$

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

Add $2$ to both sides of the equation.

$y = - \frac{3 x}{4} - \frac{9}{4} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - \frac{3 x}{4} - \frac{9}{4} + 2 \cdot \frac{4}{4}$

Combine $2$ and $\frac{4}{4}$ .

$y = - \frac{3 x}{4} - \frac{9}{4} + \frac{2 \cdot 4}{4}$

Combine the numerators over the common denominator.

$y = - \frac{3 x}{4} + \frac{- 9 + 2 \cdot 4}{4}$

Simplify the numerator.

Multiply $2$ by $4$ .

$y = - \frac{3 x}{4} + \frac{- 9 + 8}{4}$

Add $- 9$ and $8$ .

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

Move the negative in front of the fraction.

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

Reorder terms.

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

Remove parentheses.

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

Step 7

Find the Perpendicular Line line 4x-3y=5 , point (-3,2)

Related Questions

Support LCMGCF.com by disabling your adblocker.