Find the Perpendicular Line through: (3,4) , perp. to y=-2x-4

through: $(3, 4)$ , perp. to $y = - 2 x - 4$

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

Simplify $- \frac{1}{- 2}$ to find the slope of the perpendicular line.

Move the negative in front of the fraction.

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

Multiply $- - \frac{1}{2}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{2})$

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

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

Step 4

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

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

$y - (4) = \frac{1}{2} \cdot (x - (3))$

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

$y - 4 = \frac{1}{2} \cdot (x - 3)$

Step 5

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

Solve for $y$ .

Simplify $\frac{1}{2} \cdot (x - 3)$ .

Rewrite.

$y - 4 = 0 + 0 + \frac{1}{2} \cdot (x - 3)$

Simplify by adding zeros.

$y - 4 = \frac{1}{2} \cdot (x - 3)$

Apply the distributive property.

$y - 4 = \frac{1}{2} x + \frac{1}{2} \cdot - 3$

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

$y - 4 = \frac{x}{2} + \frac{1}{2} \cdot - 3$

Combine $\frac{1}{2}$ and $- 3$ .

$y - 4 = \frac{x}{2} + \frac{- 3}{2}$

Move the negative in front of the fraction.

$y - 4 = \frac{x}{2} - \frac{3}{2}$

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

Add $4$ to both sides of the equation.

$y = \frac{x}{2} - \frac{3}{2} + 4$

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

$y = \frac{x}{2} - \frac{3}{2} + 4 \cdot \frac{2}{2}$

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

$y = \frac{x}{2} - \frac{3}{2} + \frac{4 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{x}{2} + \frac{- 3 + 4 \cdot 2}{2}$

Simplify the numerator.

Multiply $4$ by $2$ .

$y = \frac{x}{2} + \frac{- 3 + 8}{2}$

Add $- 3$ and $8$ .

$y = \frac{x}{2} + \frac{5}{2}$

Reorder terms.

$y = \frac{1}{2} x + \frac{5}{2}$

Step 6

Find the Perpendicular Line through: (3,4) , perp. to y=-2x-4

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