Find the Perpendicular Line through (2,6) and perpendicular to y=-5/4x+1

through $(2, 6)$ and perpendicular to $y = - \frac{5}{4} x + 1$

Find the slope when $y = - \frac{5}{4} x + 1$ .

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.

$y = m x + b$

Simplify the right side.

Simplify each term.

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

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

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

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

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

Reorder terms.

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

Remove parentheses.

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

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

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

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

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

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

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

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

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

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

Use the slope $\frac{4}{5}$ and a given point $(2, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = \frac{4}{5} \cdot (x - (2))$

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

$y - 6 = \frac{4}{5} \cdot (x - 2)$

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

Solve for $y$ .

Simplify $\frac{4}{5} \cdot (x - 2)$ .

Rewrite.

$y - 6 = 0 + 0 + \frac{4}{5} \cdot (x - 2)$

Simplify by adding zeros.

$y - 6 = \frac{4}{5} \cdot (x - 2)$

Apply the distributive property.

$y - 6 = \frac{4}{5} x + \frac{4}{5} \cdot - 2$

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

$y - 6 = \frac{4 x}{5} + \frac{4}{5} \cdot - 2$

Multiply $\frac{4}{5} \cdot - 2$ .

Combine $\frac{4}{5}$ and $- 2$ .

$y - 6 = \frac{4 x}{5} + \frac{4 \cdot - 2}{5}$

Multiply $4$ by $- 2$ .

$y - 6 = \frac{4 x}{5} + \frac{- 8}{5}$

Move the negative in front of the fraction.

$y - 6 = \frac{4 x}{5} - \frac{8}{5}$

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

Add $6$ to both sides of the equation.

$y = \frac{4 x}{5} - \frac{8}{5} + 6$

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

$y = \frac{4 x}{5} - \frac{8}{5} + 6 \cdot \frac{5}{5}$

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

$y = \frac{4 x}{5} - \frac{8}{5} + \frac{6 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = \frac{4 x}{5} + \frac{- 8 + 6 \cdot 5}{5}$

Simplify the numerator.

Multiply $6$ by $5$ .

$y = \frac{4 x}{5} + \frac{- 8 + 30}{5}$

Add $- 8$ and $30$ .

$y = \frac{4 x}{5} + \frac{22}{5}$

Reorder terms.

$y = \frac{4}{5} x + \frac{22}{5}$

Find the Perpendicular Line through (2,6) and perpendicular to y=-5/4x+1

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