Find the Perpendicular Line (-1,4) and x-6y=12

$(- 1, 4)$ and $x - 6 y = 12$

Solve $x - 6 y = 12$ .

Subtract $x$ from both sides of the equation.

$- 6 y = 12 - x$

Divide each term in $- 6 y = 12 - x$ by $- 6$ and simplify.

Divide each term in $- 6 y = 12 - x$ by $- 6$ .

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

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Divide $12$ by $- 6$ .

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

Dividing two negative values results in a positive value.

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

Find the slope when $y = - 2 + \frac{x}{6}$ .

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.

$y = m x + b$

Reorder $- 2$ and $\frac{x}{6}$ .

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

Reorder terms.

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

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

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

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $6$ by $1$ .

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

Multiply $- 1$ by $6$ .

$m_{perpendicular} = - 6$

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

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

$y - (4) = - 6 \cdot (x - (- 1))$

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

$y - 4 = - 6 \cdot (x + 1)$

Solve for $y$ .

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

Rewrite.

$y - 4 = 0 + 0 - 6 \cdot (x + 1)$

Simplify by adding zeros.

$y - 4 = - 6 \cdot (x + 1)$

Apply the distributive property.

$y - 4 = - 6 x - 6 \cdot 1$

Multiply $- 6$ by $1$ .

$y - 4 = - 6 x - 6$

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

Add $4$ to both sides of the equation.

$y = - 6 x - 6 + 4$

Add $- 6$ and $4$ .

$y = - 6 x - 2$

Find the Perpendicular Line (-1,4) and x-6y=12

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