Find the Perpendicular Line Find the equation of the line through (2,-2) which is perpendicular to the line y=x/4+7

Find the equation of the line through $(2, - 2)$ which is perpendicular to the line $y = \frac{x}{4} + 7$

Step 1

Find the slope when $y = \frac{x}{4} + 7$ .

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.

$y = \frac{1}{4} x + 7$

Using the slope-intercept form, the slope is $\frac{1}{4}$ .

$m = \frac{1}{4}$

Step 2

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

$m_{perpendicular} = - \frac{1}{\frac{1}{4}}$

Step 3

Simplify $- \frac{1}{\frac{1}{4}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 4)$

Multiply $- (1 \cdot 4)$ .

Multiply $4$ by $1$ .

$m_{perpendicular} = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$m_{perpendicular} = - 4$

Step 4

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

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

$y - (- 2) = - 4 \cdot (x - (2))$

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

$y + 2 = - 4 \cdot (x - 2)$

Step 5

Solve for $y$ .

Simplify $- 4 \cdot (x - 2)$ .

Rewrite.

$y + 2 = 0 + 0 - 4 \cdot (x - 2)$

Simplify by adding zeros.

$y + 2 = - 4 \cdot (x - 2)$

Apply the distributive property.

$y + 2 = - 4 x - 4 \cdot - 2$

Multiply $- 4$ by $- 2$ .

$y + 2 = - 4 x + 8$

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

Subtract $2$ from both sides of the equation.

$y = - 4 x + 8 - 2$

Subtract $2$ from $8$ .

$y = - 4 x + 6$

Step 6

Find the Perpendicular Line Find the equation of the line through (2,-2) which is perpendicular to the line y=x/4+7

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