Find the Perpendicular Line Through (7,-8) ; perpendicular to 7y=x-14

Through $(7, - 8)$ ; perpendicular to $7 y = x - 14$

Step 1

Divide each term in $7 y = x - 14$ by $7$ and simplify.

Divide each term in $7 y = x - 14$ by $7$ .

$\frac{7 y}{7} = \frac{x}{7} + \frac{- 14}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y}{7} = \frac{x}{7} + \frac{- 14}{7}$

Divide $y$ by $1$ .

$y = \frac{x}{7} + \frac{- 14}{7}$

Simplify the right side.

Divide $- 14$ by $7$ .

$y = \frac{x}{7} - 2$

Step 2

Find the slope when $y = \frac{x}{7} - 2$ .

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.

$y = \frac{1}{7} x - 2$

Using the slope-intercept form, the slope is $\frac{1}{7}$ .

$m = \frac{1}{7}$

Step 3

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

$m_{perpendicular} = - \frac{1}{\frac{1}{7}}$

Step 4

Simplify $- \frac{1}{\frac{1}{7}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 7)$

Multiply $- (1 \cdot 7)$ .

Multiply $7$ by $1$ .

$m_{perpendicular} = - 1 \cdot 7$

Multiply $- 1$ by $7$ .

$m_{perpendicular} = - 7$

Step 5

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

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

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

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

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

Step 6

Solve for $y$ .

Simplify $- 7 \cdot (x - 7)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Multiply $- 7$ by $- 7$ .

$y + 8 = - 7 x + 49$

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

Subtract $8$ from both sides of the equation.

$y = - 7 x + 49 - 8$

Subtract $8$ from $49$ .

$y = - 7 x + 41$

Step 7

Find the Perpendicular Line Through (7,-8) ; perpendicular to 7y=x-14

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