Find the Perpendicular Line y=4/5x and passes through the point (5,4)

$y = \frac{4}{5} x$ and passes through the point $(5, 4)$

Step 1

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

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.

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

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

Reorder terms.

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

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

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

Step 2

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 3

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

Step 4

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

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

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

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

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

Step 5

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

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

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

Multiply $- 5$ by $- 1$ .

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

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

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

Multiply $5$ by $5$ .

$y - 4 = - \frac{x \cdot 5}{4} + \frac{25}{4}$

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

$y - 4 = - \frac{5 x}{4} + \frac{25}{4}$

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

Add $4$ to both sides of the equation.

$y = - \frac{5 x}{4} + \frac{25}{4} + 4$

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

$y = - \frac{5 x}{4} + \frac{25}{4} + 4 \cdot \frac{4}{4}$

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

$y = - \frac{5 x}{4} + \frac{25}{4} + \frac{4 \cdot 4}{4}$

Combine the numerators over the common denominator.

$y = - \frac{5 x}{4} + \frac{25 + 4 \cdot 4}{4}$

Simplify the numerator.

Multiply $4$ by $4$ .

$y = - \frac{5 x}{4} + \frac{25 + 16}{4}$

Add $25$ and $16$ .

$y = - \frac{5 x}{4} + \frac{41}{4}$

Reorder terms.

$y = - (\frac{5}{4} x) + \frac{41}{4}$

Remove parentheses.

$y = - \frac{5}{4} x + \frac{41}{4}$

Step 6

Find the Perpendicular Line y=4/5x and passes through the point (5,4)

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