Find the Perpendicular Line Find the equation of a line perpendicular to y+1=-1/2x that passes through the point (-8,7)

Find the equation of a line perpendicular to $y + 1 = - \frac{1}{2} x$ that passes through the point $(- 8, 7)$

Step 1

Solve $y + 1 = - \frac{1}{2} x$ .

Simplify $- \frac{1}{2} x$ .

Rewrite.

$y + 1 = 0 + 0 - \frac{1}{2} x$

Simplify by adding zeros.

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

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

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

Subtract $1$ from both sides of the equation.

$y = - \frac{x}{2} - 1$

Step 2

Find the slope when $y = - \frac{x}{2} - 1$ .

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.

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

Remove parentheses.

$y = - \frac{1}{2} x - 1$

Using the slope-intercept form, the slope is $- \frac{1}{2}$ .

$m = - \frac{1}{2}$

Step 3

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

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

Step 4

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

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 2$

Multiply $- - (1 \cdot 2)$ .

Multiply $2$ by $1$ .

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

Multiply $- 1$ by $2$ .

$m_{perpendicular} = 2$

Multiply $- 1$ by $- 2$ .

$m_{perpendicular} = 2$

Step 5

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

Use the slope $2$ and a given point $(- 8, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = 2 \cdot (x - (- 8))$

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

$y - 7 = 2 \cdot (x + 8)$

Step 6

Solve for $y$ .

Simplify $2 \cdot (x + 8)$ .

Rewrite.

$y - 7 = 0 + 0 + 2 \cdot (x + 8)$

Simplify by adding zeros.

$y - 7 = 2 \cdot (x + 8)$

Apply the distributive property.

$y - 7 = 2 x + 2 \cdot 8$

Multiply $2$ by $8$ .

$y - 7 = 2 x + 16$

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

Add $7$ to both sides of the equation.

$y = 2 x + 16 + 7$

Add $16$ and $7$ .

$y = 2 x + 23$

Step 7

Find the Perpendicular Line Find the equation of a line perpendicular to y+1=-1/2x that passes through the point (-8,7)

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