Find the Perpendicular Line Through (7,-6) , perpendicular to 5x+6y=-1

Through $(7, - 6)$ , perpendicular to $5 x + 6 y = - 1$

Solve $5 x + 6 y = - 1$ .

Subtract $5 x$ from both sides of the equation.

$6 y = - 1 - 5 x$

Divide each term in $6 y = - 1 - 5 x$ by $6$ and simplify.

Divide each term in $6 y = - 1 - 5 x$ by $6$ .

$\frac{6 y}{6} = \frac{- 1}{6} + \frac{- 5 x}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{- 1}{6} + \frac{- 5 x}{6}$

Divide $y$ by $1$ .

$y = \frac{- 1}{6} + \frac{- 5 x}{6}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{1}{6} + \frac{- 5 x}{6}$

Move the negative in front of the fraction.

$y = - \frac{1}{6} - \frac{5 x}{6}$

Find the slope when $y = - \frac{1}{6} - \frac{5 x}{6}$ .

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 $- \frac{1}{6}$ and $- \frac{5 x}{6}$ .

$y = - \frac{5 x}{6} - \frac{1}{6}$

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

Reorder terms.

$y = - (\frac{5}{6} x) - \frac{1}{6}$

Remove parentheses.

$y = - \frac{5}{6} x - \frac{1}{6}$

Using the slope-intercept form, the slope is $- \frac{5}{6}$ .

$m = - \frac{5}{6}$

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

$m_{perpendicular} = - \frac{1}{- \frac{5}{6}}$

Simplify $- \frac{1}{- \frac{5}{6}}$ to find the slope of the perpendicular line.

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

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{5}{6}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{5}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{6}{5})$

Multiply $\frac{6}{5}$ by $1$ .

$m_{perpendicular} = \frac{6}{5}$

Multiply $- - \frac{6}{5}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{6}{5})$

Multiply $\frac{6}{5}$ by $1$ .

$m_{perpendicular} = \frac{6}{5}$

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

Use the slope $\frac{6}{5}$ and a given point $(7, - 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 6) = \frac{6}{5} \cdot (x - (7))$

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

$y + 6 = \frac{6}{5} \cdot (x - 7)$

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

Solve for $y$ .

Simplify $\frac{6}{5} \cdot (x - 7)$ .

Rewrite.

$y + 6 = 0 + 0 + \frac{6}{5} \cdot (x - 7)$

Simplify by adding zeros.

$y + 6 = \frac{6}{5} \cdot (x - 7)$

Apply the distributive property.

$y + 6 = \frac{6}{5} x + \frac{6}{5} \cdot - 7$

Combine $\frac{6}{5}$ and $x$ .

$y + 6 = \frac{6 x}{5} + \frac{6}{5} \cdot - 7$

Multiply $\frac{6}{5} \cdot - 7$ .

Combine $\frac{6}{5}$ and $- 7$ .

$y + 6 = \frac{6 x}{5} + \frac{6 \cdot - 7}{5}$

Multiply $6$ by $- 7$ .

$y + 6 = \frac{6 x}{5} + \frac{- 42}{5}$

Move the negative in front of the fraction.

$y + 6 = \frac{6 x}{5} - \frac{42}{5}$

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

Subtract $6$ from both sides of the equation.

$y = \frac{6 x}{5} - \frac{42}{5} - 6$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{6 x}{5} - \frac{42}{5} - 6 \cdot \frac{5}{5}$

Combine $- 6$ and $\frac{5}{5}$ .

$y = \frac{6 x}{5} - \frac{42}{5} + \frac{- 6 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = \frac{6 x}{5} + \frac{- 42 - 6 \cdot 5}{5}$

Simplify the numerator.

Multiply $- 6$ by $5$ .

$y = \frac{6 x}{5} + \frac{- 42 - 30}{5}$

Subtract $30$ from $- 42$ .

$y = \frac{6 x}{5} + \frac{- 72}{5}$

Move the negative in front of the fraction.

$y = \frac{6 x}{5} - \frac{72}{5}$

Reorder terms.

$y = \frac{6}{5} x - \frac{72}{5}$

Find the Perpendicular Line Through (7,-6) , perpendicular to 5x+6y=-1

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