Find the Perpendicular Line What is the equation of the line that is perpendicular to the line defined by the equation 2x=4y-8 and goes through the point (1,2) ?

What is the equation of the line that is perpendicular to the line defined by the equation $2 x = 4 y - 8$ and goes through the point $(1, 2)$ ?

Solve $2 x = 4 y - 8$ .

Rewrite the equation as $4 y - 8 = 2 x$ .

$4 y - 8 = 2 x$

Add $8$ to both sides of the equation.

$4 y = 2 x + 8$

Divide each term in $4 y = 2 x + 8$ by $4$ and simplify.

Divide each term in $4 y = 2 x + 8$ by $4$ .

$\frac{4 y}{4} = \frac{2 x}{4} + \frac{8}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{2 x}{4} + \frac{8}{4}$

Divide $y$ by $1$ .

$y = \frac{2 x}{4} + \frac{8}{4}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 x$ .

$y = \frac{2 (x)}{4} + \frac{8}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 x}{2 \cdot 2} + \frac{8}{4}$

Cancel the common factor.

$y = \frac{2 x}{2 \cdot 2} + \frac{8}{4}$

Rewrite the expression.

$y = \frac{x}{2} + \frac{8}{4}$

Divide $8$ by $4$ .

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

Find the slope when $y = \frac{x}{2} + 2$ .

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 terms.

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

Using the slope-intercept form, the slope is $\frac{1}{2}$ .

$m = \frac{1}{2}$

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

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

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 2)$

Multiply $2$ by $1$ .

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

Multiply $- 1$ by $2$ .

$m_{perpendicular} = - 2$

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

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

$y - (2) = - 2 \cdot (x - (1))$

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

$y - 2 = - 2 \cdot (x - 1)$

Solve for $y$ .

Simplify $- 2 \cdot (x - 1)$ .

Rewrite.

$y - 2 = 0 + 0 - 2 \cdot (x - 1)$

Simplify by adding zeros.

$y - 2 = - 2 \cdot (x - 1)$

Apply the distributive property.

$y - 2 = - 2 x - 2 \cdot - 1$

Multiply $- 2$ by $- 1$ .

$y - 2 = - 2 x + 2$

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

Add $2$ to both sides of the equation.

$y = - 2 x + 2 + 2$

Add $2$ and $2$ .

$y = - 2 x + 4$

Find the Perpendicular Line What is the equation of the line that is perpendicular to the line defined by the equation 2x=4y-8 and goes through the point (1,2) ?

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