Find the Perpendicular Line Passes through (4,4) and is perpendicular to the line defined by -5x+4y=-6

Passes through $(4, 4)$ and is perpendicular to the line defined by $- 5 x + 4 y = - 6$

Step 1

Solve $- 5 x + 4 y = - 6$ .

Add $5 x$ to both sides of the equation.

$4 y = - 6 + 5 x$

Divide each term in $4 y = - 6 + 5 x$ by $4$ and simplify.

Divide each term in $4 y = - 6 + 5 x$ by $4$ .

$\frac{4 y}{4} = \frac{- 6}{4} + \frac{5 x}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 6}{4} + \frac{5 x}{4}$

Divide $y$ by $1$ .

$y = \frac{- 6}{4} + \frac{5 x}{4}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 6$ and $4$ .

Factor $2$ out of $- 6$ .

$y = \frac{2 (- 3)}{4} + \frac{5 x}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 \cdot - 3}{2 \cdot 2} + \frac{5 x}{4}$

Cancel the common factor.

$y = \frac{2 \cdot - 3}{2 \cdot 2} + \frac{5 x}{4}$

Rewrite the expression.

$y = \frac{- 3}{2} + \frac{5 x}{4}$

Move the negative in front of the fraction.

$y = - \frac{3}{2} + \frac{5 x}{4}$

Step 2

Find the slope when $y = - \frac{3}{2} + \frac{5 x}{4}$ .

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 $- \frac{3}{2}$ and $\frac{5 x}{4}$ .

$y = \frac{5 x}{4} - \frac{3}{2}$

Reorder terms.

$y = \frac{5}{4} x - \frac{3}{2}$

Using the slope-intercept form, the slope is $\frac{5}{4}$ .

$m = \frac{5}{4}$

Step 3

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

$m_{perpendicular} = - \frac{1}{\frac{5}{4}}$

Step 4

Simplify $- \frac{1}{\frac{5}{4}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{4}{5}))$

Multiply $\frac{4}{5}$ by $1$ .

$m_{perpendicular} = - \frac{4}{5}$

Step 5

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

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

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

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

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

Step 6

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

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $x$ and $\frac{4}{5}$ .

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

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

Multiply $- 4$ by $- 1$ .

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

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

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

Multiply $4$ by $4$ .

$y - 4 = - \frac{x \cdot 4}{5} + \frac{16}{5}$

Move $4$ to the left of $x$ .

$y - 4 = - \frac{4 x}{5} + \frac{16}{5}$

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

Add $4$ to both sides of the equation.

$y = - \frac{4 x}{5} + \frac{16}{5} + 4$

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{4 x}{5} + \frac{16}{5} + 4 \cdot \frac{5}{5}$

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

$y = - \frac{4 x}{5} + \frac{16}{5} + \frac{4 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = - \frac{4 x}{5} + \frac{16 + 4 \cdot 5}{5}$

Simplify the numerator.

Multiply $4$ by $5$ .

$y = - \frac{4 x}{5} + \frac{16 + 20}{5}$

Add $16$ and $20$ .

$y = - \frac{4 x}{5} + \frac{36}{5}$

Reorder terms.

$y = - (\frac{4}{5} x) + \frac{36}{5}$

Remove parentheses.

$y = - \frac{4}{5} x + \frac{36}{5}$

Step 7

Find the Perpendicular Line Passes through (4,4) and is perpendicular to the line defined by -5x+4y=-6

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