Algebra Examples

$5 x - 2 y = - 10$

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Solve $5 x - 2 y = - 10$ .

Tap for more steps...

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

$- 2 y = - 10 - 5 x$

Divide each term in $- 2 y = - 10 - 5 x$ by $- 2$ and simplify.

Tap for more steps...

Divide each term in $- 2 y = - 10 - 5 x$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- 10}{- 2} + \frac{- 5 x}{- 2}$

Simplify the left side.

Tap for more steps...

Cancel the common factor of $- 2$ .

Tap for more steps...

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{- 10}{- 2} + \frac{- 5 x}{- 2}$

Divide $y$ by $1$ .

$y = \frac{- 10}{- 2} + \frac{- 5 x}{- 2}$

Simplify the right side.

Tap for more steps...

Simplify each term.

Tap for more steps...

Divide $- 10$ by $- 2$ .

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

Dividing two negative values results in a positive value.

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

Find the slope when $y = 5 + \frac{5 x}{2}$ .

Tap for more steps...

Rewrite in slope-intercept form.

Tap for more steps...

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 $5$ and $\frac{5 x}{2}$ .

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

Reorder terms.

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

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

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

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

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

Simplify $- \frac{1}{\frac{5}{2}}$ to find the slope of the perpendicular line.

Tap for more steps...

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Tap for more steps...

Use the slope $- \frac{2}{5}$ and a given point $(0, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

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

Tap for more steps...

Solve for $y$ .

Tap for more steps...

Add $y$ and $0$ .

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

Simplify $- \frac{2}{5} \cdot (x + 0)$ .

Tap for more steps...

Add $x$ and $0$ .

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

Combine $x$ and $\frac{2}{5}$ .

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

Move $2$ to the left of $x$ .

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

Reorder terms.

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

Remove parentheses.

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