Find the Perpendicular Line y-2=-1/6(x+1) ; (-1,2)

$y - 2 = - \frac{1}{6} (x + 1)$ ; $(- 1, 2)$

Step 1

Solve $y - 2 = - \frac{1}{6} \cdot (x + 1)$ .

Simplify $- \frac{1}{6} (x + 1)$ .

Rewrite.

$y - 2 = 0 + 0 - \frac{1}{6} (x + 1)$

Simplify by adding zeros.

$y - 2 = - \frac{1}{6} (x + 1)$

Apply the distributive property.

$y - 2 = - \frac{1}{6} x - \frac{1}{6} \cdot 1$

Combine $x$ and $\frac{1}{6}$ .

$y - 2 = - \frac{x}{6} - \frac{1}{6} \cdot 1$

Multiply $- 1$ by $1$ .

$y - 2 = - \frac{x}{6} - \frac{1}{6}$

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

Add $2$ to both sides of the equation.

$y = - \frac{x}{6} - \frac{1}{6} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = - \frac{x}{6} - \frac{1}{6} + 2 \cdot \frac{6}{6}$

Combine $2$ and $\frac{6}{6}$ .

$y = - \frac{x}{6} - \frac{1}{6} + \frac{2 \cdot 6}{6}$

Combine the numerators over the common denominator.

$y = - \frac{x}{6} + \frac{- 1 + 2 \cdot 6}{6}$

Simplify the numerator.

Multiply $2$ by $6$ .

$y = - \frac{x}{6} + \frac{- 1 + 12}{6}$

Add $- 1$ and $12$ .

$y = - \frac{x}{6} + \frac{11}{6}$

Step 2

Find the slope when $y = - \frac{x}{6} + \frac{11}{6}$ .

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.

$y = m x + b$

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

Reorder terms.

$y = - (\frac{1}{6} x) + \frac{11}{6}$

Remove parentheses.

$y = - \frac{1}{6} x + \frac{11}{6}$

Using the slope-intercept form, the slope is $- \frac{1}{6}$ .

$m = - \frac{1}{6}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{1}{6}}$

Step 4

Simplify $- \frac{1}{- \frac{1}{6}}$ to find the slope of the perpendicular line.

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

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{1}{6}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{1}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 6$

Multiply $- - (1 \cdot 6)$ .

Multiply $6$ by $1$ .

$m_{perpendicular} = - (- 1 \cdot 6)$

Multiply $- 1$ by $6$ .

$m_{perpendicular} = 6$

Multiply $- 1$ by $- 6$ .

$m_{perpendicular} = 6$

Step 5

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

Use the slope $6$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = 6 \cdot (x - (- 1))$

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

$y - 2 = 6 \cdot (x + 1)$

Step 6

Solve for $y$ .

Simplify $6 \cdot (x + 1)$ .

Rewrite.

$y - 2 = 0 + 0 + 6 \cdot (x + 1)$

Simplify by adding zeros.

$y - 2 = 6 \cdot (x + 1)$

Apply the distributive property.

$y - 2 = 6 x + 6 \cdot 1$

Multiply $6$ by $1$ .

$y - 2 = 6 x + 6$

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

Add $2$ to both sides of the equation.

$y = 6 x + 6 + 2$

Add $6$ and $2$ .

$y = 6 x + 8$

Step 7

Find the Perpendicular Line y-2=-1/6(x+1) ; (-1,2)

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