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

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

Step 1

Solve $5 x - 3 y = - 2$ .

Subtract $5 x$ from both sides of the equation.

$- 3 y = - 2 - 5 x$

Divide each term in $- 3 y = - 2 - 5 x$ by $- 3$ and simplify.

Divide each term in $- 3 y = - 2 - 5 x$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{- 2}{- 3} + \frac{- 5 x}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{- 2}{- 3} + \frac{- 5 x}{- 3}$

Divide $y$ by $1$ .

$y = \frac{- 2}{- 3} + \frac{- 5 x}{- 3}$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$y = \frac{2}{3} + \frac{- 5 x}{- 3}$

Dividing two negative values results in a positive value.

$y = \frac{2}{3} + \frac{5 x}{3}$

Step 2

Find the slope when $y = \frac{2}{3} + \frac{5 x}{3}$ .

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 $\frac{2}{3}$ and $\frac{5 x}{3}$ .

$y = \frac{5 x}{3} + \frac{2}{3}$

Reorder terms.

$y = \frac{5}{3} x + \frac{2}{3}$

Using the slope-intercept form, the slope is $\frac{5}{3}$ .

$m = \frac{5}{3}$

Step 3

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

$m_{perpendicular} = - \frac{1}{\frac{5}{3}}$

Step 4

Simplify $- \frac{1}{\frac{5}{3}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{3}{5}))$

Multiply $\frac{3}{5}$ by $1$ .

$m_{perpendicular} = - \frac{3}{5}$

Step 5

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

Use the slope $- \frac{3}{5}$ and a given point $(- 4, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

Step 6

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

Solve for $y$ .

Simplify $- \frac{3}{5} \cdot (x + 4)$ .

Rewrite.

$y - 5 = 0 + 0 - \frac{3}{5} \cdot (x + 4)$

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $x$ and $\frac{3}{5}$ .

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

Multiply $- \frac{3}{5} \cdot 4$ .

Multiply $4$ by $- 1$ .

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

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

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

Multiply $- 4$ by $3$ .

$y - 5 = - \frac{x \cdot 3}{5} + \frac{- 12}{5}$

Simplify each term.

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

$y - 5 = - \frac{3 \cdot x}{5} + \frac{- 12}{5}$

Move the negative in front of the fraction.

$y - 5 = - \frac{3 x}{5} - \frac{12}{5}$

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

Add $5$ to both sides of the equation.

$y = - \frac{3 x}{5} - \frac{12}{5} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{3 x}{5} - \frac{12}{5} + 5 \cdot \frac{5}{5}$

Combine $5$ and $\frac{5}{5}$ .

$y = - \frac{3 x}{5} - \frac{12}{5} + \frac{5 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = - \frac{3 x}{5} + \frac{- 12 + 5 \cdot 5}{5}$

Simplify the numerator.

Multiply $5$ by $5$ .

$y = - \frac{3 x}{5} + \frac{- 12 + 25}{5}$

Add $- 12$ and $25$ .

$y = - \frac{3 x}{5} + \frac{13}{5}$

Reorder terms.

$y = - (\frac{3}{5} x) + \frac{13}{5}$

Remove parentheses.

$y = - \frac{3}{5} x + \frac{13}{5}$

Step 7

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

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