Find the Perpendicular Line y=-2x-6 ; P=(4,6)

$y = - 2 x - 6$ ; $P = (4, 6)$

Step 1

Use the slope-intercept form to find the slope.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Using the slope-intercept form, the slope is $- 2$ .

$m = - 2$

Step 2

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

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

Step 3

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

Move the negative in front of the fraction.

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

Multiply $- - \frac{1}{2}$ .

Multiply $- 1$ by $- 1$ .

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

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

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

Step 4

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

Use the slope $\frac{1}{2}$ and a given point $P = (4, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = \frac{1}{2} \cdot (x - (4))$

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

$y - 6 = \frac{1}{2} \cdot (x - 4)$

Step 5

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

Solve for $y$ .

Simplify $\frac{1}{2} \cdot (x - 4)$ .

Rewrite.

$y - 6 = 0 + 0 + \frac{1}{2} \cdot (x - 4)$

Simplify by adding zeros.

$y - 6 = \frac{1}{2} \cdot (x - 4)$

Apply the distributive property.

$y - 6 = \frac{1}{2} x + \frac{1}{2} \cdot - 4$

Combine $\frac{1}{2}$ and $x$ .

$y - 6 = \frac{x}{2} + \frac{1}{2} \cdot - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Add $6$ to both sides of the equation.

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

Add $- 2$ and $6$ .

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

Reorder terms.

$y = \frac{1}{2} x + 4$

Step 6

Find the Perpendicular Line y=-2x-6 ; P=(4,6)

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