# How do you write an equation in point slope form given (-0.5, 0.9), (-3.3, -0.5)?

Jul 25, 2017

$y - 0.9 = \frac{1}{2} \left(x + 0.5\right)$

#### Explanation:

For the line segment from $\left(- 0.5 , 0.9\right)$ to $\left(- 3.3 , - 0.5\right)$
the change in $y$ is $\delta y = \left(\left(- 0.5\right) - \left(0.9\right)\right) = - 1.4$
and
the change in $x$ is $\delta x = \left(\left(- 3.3\right) - \left(- 0.5\right)\right) = - 2.8$

Slope is defined as $\textcolor{g r e e n}{m} = \frac{\delta y}{\delta x}$
So, in this case,
$\textcolor{w h i t e}{\text{XXX}}$the slope is $\textcolor{g r e e n}{m} = \frac{- 1.4}{- 2.8} = \textcolor{g r e e n}{\frac{1}{2}}$

For a line with a slope of $\textcolor{g r e e n}{m}$ through a point $\left(\textcolor{red}{{x}_{0}} , \textcolor{b l u e}{{y}_{0}}\right)$
the point-slope form of the equation is
$\textcolor{w h i t e}{\text{XXX}} y - \textcolor{b l u e}{{y}_{0}} = \textcolor{g r e e n}{m} \left(x - \textcolor{red}{{x}_{0}}\right)$

We could use either of the given points as $\left(\textcolor{red}{{x}_{0}} , \textcolor{b l u e}{{y}_{0}}\right)$

As an example if we choose $\left(\textcolor{red}{{x}_{0}} , \textcolor{b l u e}{{y}_{0}}\right) = \left(\textcolor{red}{- 0.5} , \textcolor{b l u e}{0.9}\right)$
then the equation of our line would be
$\textcolor{w h i t e}{\text{XXX}} y - \textcolor{b l u e}{0.9} = \textcolor{g r e e n}{\frac{1}{2}} \left(x - \textcolor{red}{\left(- 0.5\right)}\right)$
or
$\textcolor{w h i t e}{\text{XXX}} y - 0.9 = \frac{1}{2} \left(x + 0.5\right)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Alternately, if we had chosen $\left(\textcolor{red}{{x}_{0}} , \textcolor{b l u e}{{y}_{0}}\right) = \left(\textcolor{red}{- 3.3} , \textcolor{b l u e}{- 0.5}\right)$
the resulting equation would be
$\textcolor{w h i t e}{\text{XXX}} y + 0.5 = \frac{1}{2} \left(x + 3.3\right)$

With a bit of manipulation these equations can be shown to be identical.