# What is the equation of the line between (-1,12) and (31,16)?

May 29, 2018

See a solution process below:

#### Explanation:

Fist, 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:

$m = \frac{\textcolor{red}{16} - \textcolor{b l u e}{12}}{\textcolor{red}{31} - \textcolor{b l u e}{- 1}} = \frac{\textcolor{red}{16} - \textcolor{b l u e}{12}}{\textcolor{red}{31} + \textcolor{b l u e}{1}} = \frac{4}{32} = \frac{1}{8}$

Now, we can use this 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 and the values from the first point in the problem gives:

$\left(y - \textcolor{b l u e}{12}\right) = \textcolor{red}{\frac{1}{8}} \left(x - \textcolor{b l u e}{- 1}\right)$

$\left(y - \textcolor{b l u e}{12}\right) = \textcolor{red}{\frac{1}{8}} \left(x + \textcolor{b l u e}{1}\right)$

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

$\left(y - \textcolor{b l u e}{16}\right) = \textcolor{red}{\frac{1}{8}} \left(x - \textcolor{b l u e}{31}\right)$