Find the Perpendicular Line through (6,-4) , perpendicular to 7y=x-14

through $(6, - 4)$ , perpendicular to $7 y = x - 14$

Divide each term in $7 y = x - 14$ by $7$ and simplify.

Divide each term in $7 y = x - 14$ by $7$ .

$\frac{7 y}{7} = \frac{x}{7} + \frac{- 14}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y}{7} = \frac{x}{7} + \frac{- 14}{7}$

Divide $y$ by $1$ .

$y = \frac{x}{7} + \frac{- 14}{7}$

Simplify the right side.

Divide $- 14$ by $7$ .

$y = \frac{x}{7} - 2$

Find the slope when $y = \frac{x}{7} - 2$ .

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.

$y = \frac{1}{7} x - 2$

Using the slope-intercept form, the slope is $\frac{1}{7}$ .

$m = \frac{1}{7}$

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

$m_{perpendicular} = - \frac{1}{\frac{1}{7}}$

Simplify $- \frac{1}{\frac{1}{7}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 7)$

Multiply $7$ by $1$ .

$m_{perpendicular} = - 1 \cdot 7$

Multiply $- 1$ by $7$ .

$m_{perpendicular} = - 7$

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

Use the slope $- 7$ and a given point $(6, - 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 4) = - 7 \cdot (x - (6))$

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

$y + 4 = - 7 \cdot (x - 6)$

Solve for $y$ .

Simplify $- 7 \cdot (x - 6)$ .

Rewrite.

$y + 4 = 0 + 0 - 7 \cdot (x - 6)$

Simplify by adding zeros.

$y + 4 = - 7 \cdot (x - 6)$

Apply the distributive property.

$y + 4 = - 7 x - 7 \cdot - 6$

Multiply $- 7$ by $- 6$ .

$y + 4 = - 7 x + 42$

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

Subtract $4$ from both sides of the equation.

$y = - 7 x + 42 - 4$

Subtract $4$ from $42$ .

$y = - 7 x + 38$

Find the Perpendicular Line through (6,-4) , perpendicular to 7y=x-14

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