Find the Perpendicular Line (4,-2) that is perpendicular to the line 4x+5y=8

$(4, - 2)$ that is perpendicular to the line $4 x + 5 y = 8$

Step 1

Solve $4 x + 5 y = 8$ .

Subtract $4 x$ from both sides of the equation.

$5 y = 8 - 4 x$

Divide each term in $5 y = 8 - 4 x$ by $5$ and simplify.

Divide each term in $5 y = 8 - 4 x$ by $5$ .

$\frac{5 y}{5} = \frac{8}{5} + \frac{- 4 x}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{8}{5} + \frac{- 4 x}{5}$

Divide $y$ by $1$ .

$y = \frac{8}{5} + \frac{- 4 x}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{8}{5} - \frac{4 x}{5}$

Step 2

Find the slope when $y = \frac{8}{5} - \frac{4 x}{5}$ .

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 $\frac{8}{5}$ and $- \frac{4 x}{5}$ .

$y = - \frac{4 x}{5} + \frac{8}{5}$

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

Reorder terms.

$y = - (\frac{4}{5} x) + \frac{8}{5}$

Remove parentheses.

$y = - \frac{4}{5} x + \frac{8}{5}$

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

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

Step 3

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

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

Step 4

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

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

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

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

Step 5

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

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

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

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

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

Step 6

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

Solve for $y$ .

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

Rewrite.

$y + 2 = 0 + 0 + \frac{5}{4} \cdot (x - 4)$

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

Cancel the common factor of $4$ .

Factor $4$ out of $- 4$ .

$y + 2 = \frac{5 x}{4} + \frac{5}{4} \cdot (4 (- 1))$

Cancel the common factor.

$y + 2 = \frac{5 x}{4} + \frac{5}{4} \cdot (4 \cdot - 1)$

Rewrite the expression.

$y + 2 = \frac{5 x}{4} + 5 \cdot - 1$

Multiply $5$ by $- 1$ .

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

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

Subtract $2$ from both sides of the equation.

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

Subtract $2$ from $- 5$ .

$y = \frac{5 x}{4} - 7$

Reorder terms.

$y = \frac{5}{4} x - 7$

Step 7

Find the Perpendicular Line (4,-2) that is perpendicular to the line 4x+5y=8

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