Find the Perpendicular Line perpendicular to 7y=x-4 and passes through the point (-2,2)

perpendicular to $7 y = x - 4$ and passes through the point $(- 2, 2)$

Divide each term in $7 y = x - 4$ by $7$ and simplify.

Divide each term in $7 y = x - 4$ by $7$ .

$\frac{7 y}{7} = \frac{x}{7} + \frac{- 4}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y}{7} = \frac{x}{7} + \frac{- 4}{7}$

Divide $y$ by $1$ .

$y = \frac{x}{7} + \frac{- 4}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{x}{7} - \frac{4}{7}$

Find the slope when $y = \frac{x}{7} - \frac{4}{7}$ .

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.

$y = \frac{1}{7} x - \frac{4}{7}$

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 $(- 2, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - 7 \cdot (x - (- 2))$

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

$y - 2 = - 7 \cdot (x + 2)$

Solve for $y$ .

Simplify $- 7 \cdot (x + 2)$ .

Rewrite.

$y - 2 = 0 + 0 - 7 \cdot (x + 2)$

Simplify by adding zeros.

$y - 2 = - 7 \cdot (x + 2)$

Apply the distributive property.

$y - 2 = - 7 x - 7 \cdot 2$

Multiply $- 7$ by $2$ .

$y - 2 = - 7 x - 14$

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

Add $2$ to both sides of the equation.

$y = - 7 x - 14 + 2$

Add $- 14$ and $2$ .

$y = - 7 x - 12$

Find the Perpendicular Line perpendicular to 7y=x-4 and passes through the point (-2,2)

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