Find the Perpendicular Line passing through (5,-9) and perpendicular to the line whose equation is x+7y-12=0

passing through $(5, - 9)$ and perpendicular to the line whose equation is $x + 7 y - 12 = 0$

Step 1

Solve $x + 7 y - 12 = 0$ .

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

Subtract $x$ from both sides of the equation.

$7 y - 12 = - x$

Add $12$ to both sides of the equation.

$7 y = - x + 12$

Divide each term in $7 y = - x + 12$ by $7$ and simplify.

Divide each term in $7 y = - x + 12$ by $7$ .

$\frac{7 y}{7} = \frac{- x}{7} + \frac{12}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y}{7} = \frac{- x}{7} + \frac{12}{7}$

Divide $y$ by $1$ .

$y = \frac{- x}{7} + \frac{12}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{x}{7} + \frac{12}{7}$

Step 2

Find the slope when $y = - \frac{x}{7} + \frac{12}{7}$ .

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.

$y = m x + b$

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

Reorder terms.

$y = - (\frac{1}{7} x) + \frac{12}{7}$

Remove parentheses.

$y = - \frac{1}{7} x + \frac{12}{7}$

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.

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

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

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

Move the negative in front of the fraction.

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

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 $7$ .

$m_{perpendicular} = 7$

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 $(5, - 9)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 9) = 7 \cdot (x - (5))$

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

$y + 9 = 7 \cdot (x - 5)$

Step 6

Solve for $y$ .

Simplify $7 \cdot (x - 5)$ .

Rewrite.

$y + 9 = 0 + 0 + 7 \cdot (x - 5)$

Simplify by adding zeros.

$y + 9 = 7 \cdot (x - 5)$

Apply the distributive property.

$y + 9 = 7 x + 7 \cdot - 5$

Multiply $7$ by $- 5$ .

$y + 9 = 7 x - 35$

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

Subtract $9$ from both sides of the equation.

$y = 7 x - 35 - 9$

Subtract $9$ from $- 35$ .

$y = 7 x - 44$

Step 7

Find the Perpendicular Line passing through (5,-9) and perpendicular to the line whose equation is x+7y-12=0

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