# What is the equation of the line between (3,-2) and (5,1)?

May 30, 2018

See a solution process below:

#### Explanation:

First, we need to determine the slope of the line. The formula for find the slope of a line is:

$m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Where $\left(\textcolor{b l u e}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right)$ and $\left(\textcolor{red}{{x}_{2}} , \textcolor{red}{{y}_{2}}\right)$ are two points on the line.

Substituting the values from the points in the problem gives:

$m = \frac{\textcolor{red}{1} - \textcolor{b l u e}{- 2}}{\textcolor{red}{5} - \textcolor{b l u e}{3}} = \frac{\textcolor{red}{1} + \textcolor{b l u e}{2}}{\textcolor{red}{5} - \textcolor{b l u e}{3}} = \frac{3}{2}$

Now, we can use the point-slope formula to write an equation for the line. The point-slope form of a linear equation is:

$\left(y - \textcolor{b l u e}{{y}_{1}}\right) = \textcolor{red}{m} \left(x - \textcolor{b l u e}{{x}_{1}}\right)$

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

Substituting the slope we calculated above and the values from the first point in the problem gives:

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

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

We can also substitute the slope we calculated above and the values from the second point in the problem giving:

$\left(y - \textcolor{b l u e}{1}\right) = \textcolor{red}{\frac{3}{2}} \left(x - \textcolor{b l u e}{5}\right)$

May 30, 2018

$y = \frac{3}{2} x - \frac{13}{2}$

#### Explanation:

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{1 + 2}{5 - 3} = \frac{3}{2}$
So
$y = \frac{3}{2} x + n$
we have
$1 = \frac{15}{2} + n$

so

$n = - \frac{13}{2}$