Find the Perpendicular Line Through (-4,-3) , perpendicular to 9x-5y=-51

Through $(- 4, - 3)$ , perpendicular to $9 x - 5 y = - 51$

Solve $9 x - 5 y = - 51$ .

Subtract $9 x$ from both sides of the equation.

$- 5 y = - 51 - 9 x$

Divide each term in $- 5 y = - 51 - 9 x$ by $- 5$ and simplify.

Divide each term in $- 5 y = - 51 - 9 x$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 51}{- 5} + \frac{- 9 x}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{- 51}{- 5} + \frac{- 9 x}{- 5}$

Divide $y$ by $1$ .

$y = \frac{- 51}{- 5} + \frac{- 9 x}{- 5}$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$y = \frac{51}{5} + \frac{- 9 x}{- 5}$

Dividing two negative values results in a positive value.

$y = \frac{51}{5} + \frac{9 x}{5}$

Find the slope when $y = \frac{51}{5} + \frac{9 x}{5}$ .

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 $\frac{51}{5}$ and $\frac{9 x}{5}$ .

$y = \frac{9 x}{5} + \frac{51}{5}$

Reorder terms.

$y = \frac{9}{5} x + \frac{51}{5}$

Using the slope-intercept form, the slope is $\frac{9}{5}$ .

$m = \frac{9}{5}$

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

$m_{perpendicular} = - \frac{1}{\frac{9}{5}}$

Simplify $- \frac{1}{\frac{9}{5}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{5}{9}))$

Multiply $\frac{5}{9}$ by $1$ .

$m_{perpendicular} = - \frac{5}{9}$

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

Use the slope $- \frac{5}{9}$ and a given point $(- 4, - 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 3) = - \frac{5}{9} \cdot (x - (- 4))$

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

$y + 3 = - \frac{5}{9} \cdot (x + 4)$

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

Solve for $y$ .

Simplify $- \frac{5}{9} \cdot (x + 4)$ .

Rewrite.

$y + 3 = 0 + 0 - \frac{5}{9} \cdot (x + 4)$

Simplify by adding zeros.

$y + 3 = - \frac{5}{9} \cdot (x + 4)$

Apply the distributive property.

$y + 3 = - \frac{5}{9} x - \frac{5}{9} \cdot 4$

Combine $x$ and $\frac{5}{9}$ .

$y + 3 = - \frac{x \cdot 5}{9} - \frac{5}{9} \cdot 4$

Multiply $- \frac{5}{9} \cdot 4$ .

Multiply $4$ by $- 1$ .

$y + 3 = - \frac{x \cdot 5}{9} - 4 (\frac{5}{9})$

Combine $- 4$ and $\frac{5}{9}$ .

$y + 3 = - \frac{x \cdot 5}{9} + \frac{- 4 \cdot 5}{9}$

Multiply $- 4$ by $5$ .

$y + 3 = - \frac{x \cdot 5}{9} + \frac{- 20}{9}$

Simplify each term.

Move $5$ to the left of $x$ .

$y + 3 = - \frac{5 \cdot x}{9} + \frac{- 20}{9}$

Move the negative in front of the fraction.

$y + 3 = - \frac{5 x}{9} - \frac{20}{9}$

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

Subtract $3$ from both sides of the equation.

$y = - \frac{5 x}{9} - \frac{20}{9} - 3$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = - \frac{5 x}{9} - \frac{20}{9} - 3 \cdot \frac{9}{9}$

Combine $- 3$ and $\frac{9}{9}$ .

$y = - \frac{5 x}{9} - \frac{20}{9} + \frac{- 3 \cdot 9}{9}$

Combine the numerators over the common denominator.

$y = - \frac{5 x}{9} + \frac{- 20 - 3 \cdot 9}{9}$

Simplify the numerator.

Multiply $- 3$ by $9$ .

$y = - \frac{5 x}{9} + \frac{- 20 - 27}{9}$

Subtract $27$ from $- 20$ .

$y = - \frac{5 x}{9} + \frac{- 47}{9}$

Move the negative in front of the fraction.

$y = - \frac{5 x}{9} - \frac{47}{9}$

Reorder terms.

$y = - (\frac{5}{9} x) - \frac{47}{9}$

Remove parentheses.

$y = - \frac{5}{9} x - \frac{47}{9}$

Find the Perpendicular Line Through (-4,-3) , perpendicular to 9x-5y=-51

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