Find the Perpendicular Line 7-x=-8(1+2y) , (-7,-5)

$7 - x = - 8 (1 + 2 y)$ , $(- 7, - 5)$

Step 1

Solve $7 - x = - 8 (1 + 2 y)$ .

Rewrite the equation as $- 8 (1 + 2 y) = 7 - x$ .

$- 8 (1 + 2 y) = 7 - x$

Divide each term in $- 8 (1 + 2 y) = 7 - x$ by $- 8$ and simplify.

Divide each term in $- 8 (1 + 2 y) = 7 - x$ by $- 8$ .

$\frac{- 8 (1 + 2 y)}{- 8} = \frac{7}{- 8} + \frac{- x}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 (1 + 2 y)}{- 8} = \frac{7}{- 8} + \frac{- x}{- 8}$

Divide $1 + 2 y$ by $1$ .

$1 + 2 y = \frac{7}{- 8} + \frac{- x}{- 8}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$1 + 2 y = - \frac{7}{8} + \frac{- x}{- 8}$

Dividing two negative values results in a positive value.

$1 + 2 y = - \frac{7}{8} + \frac{x}{8}$

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

Subtract $1$ from both sides of the equation.

$2 y = - \frac{7}{8} + \frac{x}{8} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$2 y = \frac{x}{8} - \frac{7}{8} - 1 \cdot \frac{8}{8}$

Combine $- 1$ and $\frac{8}{8}$ .

$2 y = \frac{x}{8} - \frac{7}{8} + \frac{- 1 \cdot 8}{8}$

Combine the numerators over the common denominator.

$2 y = \frac{x}{8} + \frac{- 7 - 1 \cdot 8}{8}$

Simplify the numerator.

Multiply $- 1$ by $8$ .

$2 y = \frac{x}{8} + \frac{- 7 - 8}{8}$

Subtract $8$ from $- 7$ .

$2 y = \frac{x}{8} + \frac{- 15}{8}$

Move the negative in front of the fraction.

$2 y = \frac{x}{8} - \frac{15}{8}$

Divide each term in $2 y = \frac{x}{8} - \frac{15}{8}$ by $2$ and simplify.

Divide each term in $2 y = \frac{x}{8} - \frac{15}{8}$ by $2$ .

$\frac{2 y}{2} = \frac{\frac{x}{8}}{2} + \frac{- \frac{15}{8}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{\frac{x}{8}}{2} + \frac{- \frac{15}{8}}{2}$

Divide $y$ by $1$ .

$y = \frac{\frac{x}{8}}{2} + \frac{- \frac{15}{8}}{2}$

Simplify the right side.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x}{8} \cdot \frac{1}{2} + \frac{- \frac{15}{8}}{2}$

Multiply $\frac{x}{8} \cdot \frac{1}{2}$ .

Multiply $\frac{x}{8}$ by $\frac{1}{2}$ .

$y = \frac{x}{8 \cdot 2} + \frac{- \frac{15}{8}}{2}$

Multiply $8$ by $2$ .

$y = \frac{x}{16} + \frac{- \frac{15}{8}}{2}$

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x}{16} - \frac{15}{8} \cdot \frac{1}{2}$

Multiply $- \frac{15}{8} \cdot \frac{1}{2}$ .

Multiply $\frac{1}{2}$ by $\frac{15}{8}$ .

$y = \frac{x}{16} - \frac{15}{2 \cdot 8}$

Multiply $2$ by $8$ .

$y = \frac{x}{16} - \frac{15}{16}$

Step 2

Find the slope when $y = \frac{x}{16} - \frac{15}{16}$ .

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.

$y = \frac{1}{16} x - \frac{15}{16}$

Using the slope-intercept form, the slope is $\frac{1}{16}$ .

$m = \frac{1}{16}$

Step 3

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

$m_{perpendicular} = - \frac{1}{\frac{1}{16}}$

Step 4

Simplify $- \frac{1}{\frac{1}{16}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 16)$

Multiply $- (1 \cdot 16)$ .

Multiply $16$ by $1$ .

$m_{perpendicular} = - 1 \cdot 16$

Multiply $- 1$ by $16$ .

$m_{perpendicular} = - 16$

Step 5

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

Use the slope $- 16$ and a given point $(- 7, - 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 5) = - 16 \cdot (x - (- 7))$

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

$y + 5 = - 16 \cdot (x + 7)$

Step 6

Solve for $y$ .

Simplify $- 16 \cdot (x + 7)$ .

Rewrite.

$y + 5 = 0 + 0 - 16 \cdot (x + 7)$

Simplify by adding zeros.

$y + 5 = - 16 \cdot (x + 7)$

Apply the distributive property.

$y + 5 = - 16 x - 16 \cdot 7$

Multiply $- 16$ by $7$ .

$y + 5 = - 16 x - 112$

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

Subtract $5$ from both sides of the equation.

$y = - 16 x - 112 - 5$

Subtract $5$ from $- 112$ .

$y = - 16 x - 117$

Step 7

Find the Perpendicular Line 7-x=-8(1+2y) , (-7,-5)

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