Find the Perpendicular Line x-4y=7 , through the point (-5,3)

$x - 4 y = 7$ , through the point $(- 5, 3)$

Step 1

Solve $x - 4 y = 7$ .

Subtract $x$ from both sides of the equation.

$- 4 y = 7 - x$

Divide each term in $- 4 y = 7 - x$ by $- 4$ and simplify.

Divide each term in $- 4 y = 7 - x$ by $- 4$ .

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

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Dividing two negative values results in a positive value.

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

Step 2

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

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 $- \frac{7}{4}$ and $\frac{x}{4}$ .

$y = \frac{x}{4} - \frac{7}{4}$

Reorder terms.

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

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

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

Step 3

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

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

Step 4

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 5

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

Use the slope $- 4$ and a given point $(- 5, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = - 4 \cdot (x - (- 5))$

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

$y - 3 = - 4 \cdot (x + 5)$

Step 6

Solve for $y$ .

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

Rewrite.

$y - 3 = 0 + 0 - 4 \cdot (x + 5)$

Simplify by adding zeros.

$y - 3 = - 4 \cdot (x + 5)$

Apply the distributive property.

$y - 3 = - 4 x - 4 \cdot 5$

Multiply $- 4$ by $5$ .

$y - 3 = - 4 x - 20$

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

Add $3$ to both sides of the equation.

$y = - 4 x - 20 + 3$

Add $- 20$ and $3$ .

$y = - 4 x - 17$

Step 7

Find the Perpendicular Line x-4y=7 , through the point (-5,3)

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