# What is the equation of the line perpendicular to y=-3/2x  that passes through  (2,-4) ?

Dec 20, 2015

$y = \frac{2}{3} x - \frac{16}{3}$

#### Explanation:

The slope-intercept form of a line is written in the form:

$y = m x + b$

where:
$y =$y-coordinate
$m =$slope
$x =$x-coordinate
$b =$y-intercept

Start by finding the slope that is perpendicular to $- \frac{3}{2} x$. Recall that when a line is perpendicular to another line, it is ${90}^{\circ}$ to it.

We can find the slope of the line perpendicular to $- \frac{3}{2} x$ by finding the negative reciprocal. Recall that the reciprocal of any number is $\frac{1}{\text{number}}$. In this case, it is $\frac{1}{\text{slope}}$. To find the negative reciprocal we can do:

$- \left(\frac{1}{\text{slope}}\right)$
$= - \left(\frac{1}{- \frac{3}{2} x}\right)$
$= - \left(1 \div - \frac{3}{2} x\right)$
$= - \left(1 \cdot - \frac{2}{3} x\right)$
$= - \left(- \frac{2}{3} x\right)$
$= \frac{2}{3} x \Rightarrow$ negative reciprocal, perpendicular to $- \frac{3}{2} x$

So far, our equation is: $y = \frac{2}{3} x + b$

Since we do not know the value of $b$ yet, this is going to be what we are trying to solve for. We can do this by substituting the point, $\left(2 , - 4\right)$, into the equation:

$y = m x + b$
$- 4 = \frac{2}{3} \left(2\right) + b$
$- 4 = \frac{4}{3} + b$
$- \frac{16}{3} = b$

Now that you know all your values, rewrite the equation in slope-intercept form:

$y = \frac{2}{3} x - \frac{16}{3}$