Find the Perpendicular Line x+2y=6 through (8,3)

$x + 2 y = 6$ through $(8, 3)$

Step 1

Solve $x + 2 y = 6$ .

Subtract $x$ from both sides of the equation.

$2 y = 6 - x$

Divide each term in $2 y = 6 - x$ by $2$ and simplify.

Divide each term in $2 y = 6 - x$ by $2$ .

$\frac{2 y}{2} = \frac{6}{2} + \frac{- x}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{6}{2} + \frac{- x}{2}$

Divide $y$ by $1$ .

$y = \frac{6}{2} + \frac{- x}{2}$

Simplify the right side.

Simplify each term.

Divide $6$ by $2$ .

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

Move the negative in front of the fraction.

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

Step 2

Find the slope when $y = 3 - \frac{x}{2}$ .

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 $3$ and $- \frac{x}{2}$ .

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

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

Reorder terms.

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

Remove parentheses.

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

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

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

Step 3

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

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

Step 4

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

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 2$

Multiply $- - (1 \cdot 2)$ .

Multiply $2$ by $1$ .

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

Multiply $- 1$ by $2$ .

$m_{perpendicular} = 2$

Multiply $- 1$ by $- 2$ .

$m_{perpendicular} = 2$

Step 5

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

Use the slope $2$ and a given point $(8, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = 2 \cdot (x - (8))$

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

$y - 3 = 2 \cdot (x - 8)$

Step 6

Solve for $y$ .

Simplify $2 \cdot (x - 8)$ .

Rewrite.

$y - 3 = 0 + 0 + 2 \cdot (x - 8)$

Simplify by adding zeros.

$y - 3 = 2 \cdot (x - 8)$

Apply the distributive property.

$y - 3 = 2 x + 2 \cdot - 8$

Multiply $2$ by $- 8$ .

$y - 3 = 2 x - 16$

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

Add $3$ to both sides of the equation.

$y = 2 x - 16 + 3$

Add $- 16$ and $3$ .

$y = 2 x - 13$

Step 7

Find the Perpendicular Line x+2y=6 through (8,3)

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