# What is the equation of the line that passes through the point (-2,3) and that is perpendicular to the line represented by 3x-2y= -2?

Jan 13, 2017

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

Or

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

#### Explanation:

First, we need to convert the line into slope-intercept form to find the slope.

The slope-intercept form of a linear equation is:

$y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$

Where $\textcolor{red}{m}$ is the slope and color(blue)(b is the y-intercept value.

We can solve the equation in the problem for $y$:

$3 x - 2 y = - 2$

$3 x - \textcolor{red}{3 x} - 2 y = - 2 - \textcolor{red}{3 x}$

$0 - 2 y = - 3 x - 2$

$- 2 y = - 3 x - 2$

$\frac{- 2 y}{\textcolor{red}{- 2}} = \frac{- 3 x - 2}{\textcolor{red}{- 2}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 2}}} y}{\cancel{\textcolor{red}{- 2}}} = \frac{- 3 x}{\textcolor{red}{- 2}} - \frac{2}{\textcolor{red}{- 2}}$

$y = \frac{3}{2} x + 1$

So for this equation the slope is $\frac{3}{2}$

A line perpendicular to this line will have a slope which is the negative inverse of our line or $- \frac{3}{2}$

We can now use the point-slope formula to write the equation for the perpendicular line:

The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

Substituting the point from problem and the slope we calculated gives:

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- \frac{3}{2}} \left(x - \textcolor{red}{- 2}\right)$

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- \frac{3}{2}} \left(x + \textcolor{red}{2}\right)$

Or, we can put the equation in the more familiar slope-intercept form by solving for $y$:

$y - \textcolor{red}{3} = \textcolor{b l u e}{- \frac{3}{2}} x + \left(\textcolor{b l u e}{- \frac{3}{2}} \times \textcolor{red}{2}\right)$

$y - \textcolor{red}{3} = - \frac{3}{2} x - 3$

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

$y = - \frac{3}{2} x + 0$

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