Find the Perpendicular Line through: (-1,-4) , perp. to y=2x+5

through: $(- 1, - 4)$ , perp. to $y = 2 x + 5$

Step 1

Use the slope-intercept form to find the slope.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Using the slope-intercept form, the slope is $2$ .

$m = 2$

Step 2

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

$m_{perpendicular} = - \frac{1}{2}$

Step 3

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

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

$y - (- 4) = - \frac{1}{2} \cdot (x - (- 1))$

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

$y + 4 = - \frac{1}{2} \cdot (x + 1)$

Step 4

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

Solve for $y$ .

Simplify $- \frac{1}{2} \cdot (x + 1)$ .

Rewrite.

$y + 4 = 0 + 0 - \frac{1}{2} \cdot (x + 1)$

Simplify by adding zeros.

$y + 4 = - \frac{1}{2} \cdot (x + 1)$

Apply the distributive property.

$y + 4 = - \frac{1}{2} x - \frac{1}{2} \cdot 1$

Combine $x$ and $\frac{1}{2}$ .

$y + 4 = - \frac{x}{2} - \frac{1}{2} \cdot 1$

Multiply $- 1$ by $1$ .

$y + 4 = - \frac{x}{2} - \frac{1}{2}$

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

Subtract $4$ from both sides of the equation.

$y = - \frac{x}{2} - \frac{1}{2} - 4$

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{x}{2} - \frac{1}{2} - 4 \cdot \frac{2}{2}$

Combine $- 4$ and $\frac{2}{2}$ .

$y = - \frac{x}{2} - \frac{1}{2} + \frac{- 4 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = - \frac{x}{2} + \frac{- 1 - 4 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 4$ by $2$ .

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

Subtract $8$ from $- 1$ .

$y = - \frac{x}{2} + \frac{- 9}{2}$

Move the negative in front of the fraction.

$y = - \frac{x}{2} - \frac{9}{2}$

Reorder terms.

$y = - (\frac{1}{2} x) - \frac{9}{2}$

Remove parentheses.

$y = - \frac{1}{2} x - \frac{9}{2}$

Step 5

Find the Perpendicular Line through: (-1,-4) , perp. to y=2x+5

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