Find the Perpendicular Line through: (-5,-1) , perp. to y=5/4x-3

through: $(- 5, - 1)$ , perp. to $y = \frac{5}{4} x - 3$

Find the slope when $y = \frac{5}{4} x - 3$ .

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.

$y = m x + b$

Simplify the right side.

Combine $\frac{5}{4}$ and $x$ .

$y = \frac{5 x}{4} - 3$

Reorder terms.

$y = \frac{5}{4} x - 3$

Using the slope-intercept form, the slope is $\frac{5}{4}$ .

$m = \frac{5}{4}$

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

$m_{perpendicular} = - \frac{1}{\frac{5}{4}}$

Simplify $- \frac{1}{\frac{5}{4}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{4}{5}))$

Multiply $\frac{4}{5}$ by $1$ .

$m_{perpendicular} = - \frac{4}{5}$

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

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

$y - (- 1) = - \frac{4}{5} \cdot (x - (- 5))$

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

$y + 1 = - \frac{4}{5} \cdot (x + 5)$

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

Solve for $y$ .

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

Rewrite.

$y + 1 = 0 + 0 - \frac{4}{5} \cdot (x + 5)$

Simplify terms.

Apply the distributive property.

$y + 1 = - \frac{4}{5} x - \frac{4}{5} \cdot 5$

Combine $x$ and $\frac{4}{5}$ .

$y + 1 = - \frac{x \cdot 4}{5} - \frac{4}{5} \cdot 5$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{4}{5}$ into the numerator.

$y + 1 = - \frac{x \cdot 4}{5} + \frac{- 4}{5} \cdot 5$

Cancel the common factor.

$y + 1 = - \frac{x \cdot 4}{5} + \frac{- 4}{5} \cdot 5$

Rewrite the expression.

$y + 1 = - \frac{x \cdot 4}{5} - 4$

Move $4$ to the left of $x$ .

$y + 1 = - \frac{4 x}{5} - 4$

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

Subtract $1$ from both sides of the equation.

$y = - \frac{4 x}{5} - 4 - 1$

Subtract $1$ from $- 4$ .

$y = - \frac{4 x}{5} - 5$

Reorder terms.

$y = - (\frac{4}{5} x) - 5$

Remove parentheses.

$y = - \frac{4}{5} x - 5$

Find the Perpendicular Line through: (-5,-1) , perp. to y=5/4x-3

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