Find the Perpendicular Line y=3/5x , (5,3)

$y = \frac{3}{5} x$ , $(5, 3)$

Step 1

Find the slope when $y = \frac{3}{5} x$ .

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 $\frac{3}{5}$ and $x$ .

$y = \frac{3 x}{5}$

Reorder terms.

$y = \frac{3}{5} x$

Using the slope-intercept form, the slope is $\frac{3}{5}$ .

$m = \frac{3}{5}$

Step 2

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

$m_{perpendicular} = - \frac{1}{\frac{3}{5}}$

Step 3

Simplify $- \frac{1}{\frac{3}{5}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{5}{3}))$

Multiply $\frac{5}{3}$ by $1$ .

$m_{perpendicular} = - \frac{5}{3}$

Step 4

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

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

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

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

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

Step 5

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

Solve for $y$ .

Simplify $- \frac{5}{3} \cdot (x - 5)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $x$ and $\frac{5}{3}$ .

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

Multiply $- \frac{5}{3} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

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

Combine $5$ and $\frac{5}{3}$ .

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

Multiply $5$ by $5$ .

$y - 3 = - \frac{x \cdot 5}{3} + \frac{25}{3}$

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

$y - 3 = - \frac{5 x}{3} + \frac{25}{3}$

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

Add $3$ to both sides of the equation.

$y = - \frac{5 x}{3} + \frac{25}{3} + 3$

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

$y = - \frac{5 x}{3} + \frac{25}{3} + 3 \cdot \frac{3}{3}$

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

$y = - \frac{5 x}{3} + \frac{25}{3} + \frac{3 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = - \frac{5 x}{3} + \frac{25 + 3 \cdot 3}{3}$

Simplify the numerator.

Multiply $3$ by $3$ .

$y = - \frac{5 x}{3} + \frac{25 + 9}{3}$

Add $25$ and $9$ .

$y = - \frac{5 x}{3} + \frac{34}{3}$

Reorder terms.

$y = - (\frac{5}{3} x) + \frac{34}{3}$

Remove parentheses.

$y = - \frac{5}{3} x + \frac{34}{3}$

Step 6

Find the Perpendicular Line y=3/5x , (5,3)

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