Find the Perpendicular Line y-3=8/3(x+2) , (-2,3)

$y - 3 = \frac{8}{3} (x + 2)$ , $(- 2, 3)$

Step 1

Solve $y - 3 = \frac{8}{3} \cdot (x + 2)$ .

Simplify $\frac{8}{3} (x + 2)$ .

Rewrite.

$y - 3 = 0 + 0 + \frac{8}{3} (x + 2)$

Simplify by adding zeros.

$y - 3 = \frac{8}{3} (x + 2)$

Apply the distributive property.

$y - 3 = \frac{8}{3} x + \frac{8}{3} \cdot 2$

Combine $\frac{8}{3}$ and $x$ .

$y - 3 = \frac{8 x}{3} + \frac{8}{3} \cdot 2$

Multiply $\frac{8}{3} \cdot 2$ .

Combine $\frac{8}{3}$ and $2$ .

$y - 3 = \frac{8 x}{3} + \frac{8 \cdot 2}{3}$

Multiply $8$ by $2$ .

$y - 3 = \frac{8 x}{3} + \frac{16}{3}$

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

Add $3$ to both sides of the equation.

$y = \frac{8 x}{3} + \frac{16}{3} + 3$

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{8 x}{3} + \frac{16}{3} + 3 \cdot \frac{3}{3}$

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

$y = \frac{8 x}{3} + \frac{16}{3} + \frac{3 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{8 x}{3} + \frac{16 + 3 \cdot 3}{3}$

Simplify the numerator.

Multiply $3$ by $3$ .

$y = \frac{8 x}{3} + \frac{16 + 9}{3}$

Add $16$ and $9$ .

$y = \frac{8 x}{3} + \frac{25}{3}$

Step 2

Find the slope when $y = \frac{8 x}{3} + \frac{25}{3}$ .

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 terms.

$y = \frac{8}{3} x + \frac{25}{3}$

Using the slope-intercept form, the slope is $\frac{8}{3}$ .

$m = \frac{8}{3}$

Step 3

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

$m_{perpendicular} = - \frac{1}{\frac{8}{3}}$

Step 4

Simplify $- \frac{1}{\frac{8}{3}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{3}{8}))$

Multiply $\frac{3}{8}$ by $1$ .

$m_{perpendicular} = - \frac{3}{8}$

Step 5

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

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

$y - (3) = - \frac{3}{8} \cdot (x - (- 2))$

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

$y - 3 = - \frac{3}{8} \cdot (x + 2)$

Step 6

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

Solve for $y$ .

Simplify $- \frac{3}{8} \cdot (x + 2)$ .

Rewrite.

$y - 3 = 0 + 0 - \frac{3}{8} \cdot (x + 2)$

Simplify terms.

Apply the distributive property.

$y - 3 = - \frac{3}{8} x - \frac{3}{8} \cdot 2$

Combine $x$ and $\frac{3}{8}$ .

$y - 3 = - \frac{x \cdot 3}{8} - \frac{3}{8} \cdot 2$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{8}$ into the numerator.

$y - 3 = - \frac{x \cdot 3}{8} + \frac{- 3}{8} \cdot 2$

Factor $2$ out of $8$ .

$y - 3 = - \frac{x \cdot 3}{8} + \frac{- 3}{2 (4)} \cdot 2$

Cancel the common factor.

$y - 3 = - \frac{x \cdot 3}{8} + \frac{- 3}{2 \cdot 4} \cdot 2$

Rewrite the expression.

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

Simplify each term.

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

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

Move the negative in front of the fraction.

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

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

Add $3$ to both sides of the equation.

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

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $4$ .

$y = - \frac{3 x}{8} + \frac{- 3 + 12}{4}$

Add $- 3$ and $12$ .

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

Reorder terms.

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

Remove parentheses.

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

Step 7

Find the Perpendicular Line y-3=8/3(x+2) , (-2,3)

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