Find the Slope of a Perpendicular Line 5x-2y=3

$5 x - 2 y = 3$

Step 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.

$y = m x + b$

Subtract $5 x$ from both sides of the equation.

$- 2 y = 3 - 5 x$

Divide each term in $- 2 y = 3 - 5 x$ by $- 2$ and simplify.

Divide each term in $- 2 y = 3 - 5 x$ by $- 2$ .

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Dividing two negative values results in a positive value.

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

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

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

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

Reorder terms.

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

Step 2

Using the slope-intercept form, the slope is $\frac{5}{2}$ .

$m = \frac{5}{2}$

Step 3

The equation of a perpendicular line to $y = \frac{5}{2} x - \frac{3}{2}$ must have a slope that is the negative reciprocal of the original slope.

$m_{perpendicular} = - \frac{1}{\frac{5}{2}}$

Step 4

Simplify the result.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{2}{5}))$

Multiply $\frac{2}{5}$ by $1$ .

$m_{perpendicular} = - \frac{2}{5}$

Step 5