# How do you find the equation given A(-2, 1) and B(3, 7)?

Jun 29, 2017

See a solution process below:

#### Explanation:

First, we must determine the slope of the line. The slope can be found by using the formula: $m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

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

Substituting the values from the points in the problem gives:

$m = \frac{\textcolor{red}{7} - \textcolor{b l u e}{1}}{\textcolor{red}{3} - \textcolor{b l u e}{- 2}} = \frac{\textcolor{red}{7} - \textcolor{b l u e}{1}}{\textcolor{red}{3} + \textcolor{b l u e}{2}} = \frac{6}{5}$

We can now use the point-slope formula to find an equation for the 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 $\left(\textcolor{red}{{x}_{1} , {y}_{1}}\right)$ is a point the line passes through.

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

$\left(y - \textcolor{red}{1}\right) = \textcolor{b l u e}{\frac{6}{5}} \left(x - \textcolor{red}{- 2}\right)$

$\left(y - \textcolor{red}{1}\right) = \textcolor{b l u e}{\frac{6}{5}} \left(x + \textcolor{red}{2}\right)$

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

$\left(y - \textcolor{red}{7}\right) = \textcolor{b l u e}{\frac{6}{5}} \left(x - \textcolor{red}{3}\right)$

We can now solve this equation for $y$ to put the equation in slope-intercept form. 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 $\textcolor{b l u e}{b}$ is the y-intercept value.

$y - \textcolor{red}{7} = \left(\textcolor{b l u e}{\frac{6}{5}} \times x\right) - \left(\textcolor{b l u e}{\frac{6}{5}} \times \textcolor{red}{3}\right)$

$y - \textcolor{red}{7} = \frac{6}{5} x - \frac{18}{5}$

$y - \textcolor{red}{7} + 7 = \frac{6}{5} x - \frac{18}{5} + 7$

$y - 0 = \frac{6}{5} x - \frac{18}{5} + \left(7 \times \frac{5}{5}\right)$

$y = \frac{6}{5} x - \frac{18}{5} + \frac{35}{5}$

$y = \textcolor{red}{\frac{6}{5}} x + \textcolor{b l u e}{\frac{17}{5}}$