Find the Perpendicular Line Through (5,5) ; perpendicular to x+4y=-4

Through $(5, 5)$ ; perpendicular to $x + 4 y = - 4$

Solve $x + 4 y = - 4$ .

Subtract $x$ from both sides of the equation.

$4 y = - 4 - x$

Divide each term in $4 y = - 4 - x$ by $4$ and simplify.

Divide each term in $4 y = - 4 - x$ by $4$ .

$\frac{4 y}{4} = \frac{- 4}{4} + \frac{- x}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 4}{4} + \frac{- x}{4}$

Divide $y$ by $1$ .

$y = \frac{- 4}{4} + \frac{- x}{4}$

Simplify the right side.

Simplify each term.

Divide $- 4$ by $4$ .

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

Move the negative in front of the fraction.

$y = - 1 - \frac{x}{4}$

Find the slope when $y = - 1 - \frac{x}{4}$ .

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 $- 1$ and $- \frac{x}{4}$ .

$y = - \frac{x}{4} - 1$

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

Reorder terms.

$y = - (\frac{1}{4} x) - 1$

Remove parentheses.

$y = - \frac{1}{4} x - 1$

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

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

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

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

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

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 4$

Multiply $4$ by $1$ .

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

Multiply $- 1$ by $4$ .

$m_{perpendicular} = 4$

Multiply $- 1$ by $- 4$ .

$m_{perpendicular} = 4$

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

Use the slope $4$ and a given point $(5, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = 4 \cdot (x - (5))$

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

$y - 5 = 4 \cdot (x - 5)$

Solve for $y$ .

Simplify $4 \cdot (x - 5)$ .

Rewrite.

$y - 5 = 0 + 0 + 4 \cdot (x - 5)$

Simplify by adding zeros.

$y - 5 = 4 \cdot (x - 5)$

Apply the distributive property.

$y - 5 = 4 x + 4 \cdot - 5$

Multiply $4$ by $- 5$ .

$y - 5 = 4 x - 20$

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

Add $5$ to both sides of the equation.

$y = 4 x - 20 + 5$

Add $- 20$ and $5$ .

$y = 4 x - 15$

Find the Perpendicular Line Through (5,5) ; perpendicular to x+4y=-4

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