# How do you write an equation of a line going through (-1,3), (2,9)?

Jul 22, 2017

See a solution process below:

#### Explanation:

First, we need to 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:

$m = \frac{\textcolor{red}{9} - \textcolor{b l u e}{3}}{\textcolor{red}{2} - \textcolor{b l u e}{- 1}} = \frac{\textcolor{red}{9} - \textcolor{b l u e}{3}}{\textcolor{red}{2} + \textcolor{b l u e}{1}} = \frac{6}{3} = 2$

Now, we can use the point-slope formula to write 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:

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

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

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

$\left(y - \textcolor{red}{9}\right) = \textcolor{b l u e}{2} \left(x - \textcolor{red}{2}\right)$

If necessary, we can convert this equation to the slope-intercept form by solving for $y$. 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.

$y - \textcolor{red}{9} = \left(\textcolor{b l u e}{2} \times x\right) - \left(\textcolor{b l u e}{2} \times \textcolor{red}{2}\right)$

$y - \textcolor{red}{9} = 2 x - 4$

$y - \textcolor{red}{9} + 9 = 2 x - 4 + 9$

$y - 0 = 2 x + 5$

$y = \textcolor{red}{2} x + \textcolor{b l u e}{5}$