Find the Perpendicular Line line y=2/3x-4 , point (2,-6)

line $y = \frac{2}{3} x - 4$ , point $(2, - 6)$

Step 1

Find the slope when $y = \frac{2}{3} x - 4$ .

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.

$y = m x + b$

Simplify the right side.

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

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

Reorder terms.

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

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

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

Step 2

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

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

Step 3

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

Step 4

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

Use the slope $- \frac{3}{2}$ and a given point $(2, - 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

Step 5

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

Solve for $y$ .

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

Rewrite.

$y + 6 = 0 + 0 - \frac{3}{2} \cdot (x - 2)$

Simplify terms.

Apply the distributive property.

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

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

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Factor $2$ out of $- 2$ .

$y + 6 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot (2 (- 1))$

Cancel the common factor.

$y + 6 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot (2 \cdot - 1)$

Rewrite the expression.

$y + 6 = - \frac{x \cdot 3}{2} - 3 \cdot - 1$

Multiply $- 3$ by $- 1$ .

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

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

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

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

Subtract $6$ from both sides of the equation.

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

Subtract $6$ from $3$ .

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

Reorder terms.

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

Remove parentheses.

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

Step 6

Find the Perpendicular Line line y=2/3x-4 , point (2,-6)

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