Find the Perpendicular Line y+4=-2/3(x-2) ; (4,-2)

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

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

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

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

Multiply $- 2$ by $- 1$ .

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

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

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

Multiply $2$ by $2$ .

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

Move $2$ to the left of $x$ .

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

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

Subtract $4$ from both sides of the equation.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 4$ by $3$ .

$y = - \frac{2 x}{3} + \frac{4 - 12}{3}$

Subtract $12$ from $4$ .

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

Move the negative in front of the fraction.

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

Find the slope when $y = - \frac{2 x}{3} - \frac{8}{3}$ .

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.

$y = m x + b$

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

Reorder terms.

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

Remove parentheses.

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

Using the slope-intercept form, the slope is $- \frac{2}{3}$ .

$m = - \frac{2}{3}$

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

$m_{perpendicular} = - \frac{1}{- \frac{2}{3}}$

Simplify $- \frac{1}{- \frac{2}{3}}$ to find the slope of the perpendicular line.

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

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{2}{3}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{3}{2})$

Multiply $\frac{3}{2}$ by $1$ .

$m_{perpendicular} = \frac{3}{2}$

Multiply $- - \frac{3}{2}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{3}{2})$

Multiply $\frac{3}{2}$ by $1$ .

$m_{perpendicular} = \frac{3}{2}$

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

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

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

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

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

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

Solve for $y$ .

Simplify $\frac{3}{2} \cdot (x - 4)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $- 2$ .

$y + 2 = \frac{3 x}{2} - 6$

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

Subtract $2$ from both sides of the equation.

$y = \frac{3 x}{2} - 6 - 2$

Subtract $2$ from $- 6$ .

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

Reorder terms.

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

Find the Perpendicular Line y+4=-2/3(x-2) ; (4,-2)

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