# Which equation represents the line that passes through the points (6,-3) and (-4,-9)?

Jan 7, 2017

$\left(y + \textcolor{red}{9}\right) = \textcolor{b l u e}{\frac{3}{5}} \left(x + \textcolor{red}{4}\right)$

or

$y = \textcolor{b l u e}{\frac{3}{5}} x - \frac{33}{5}$

#### Explanation:

Because we are given two points we can calculate the slope and then we can use the slope and either point and use the point-slope formula to find the equation for 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 two points from the problem gives:

$m = \frac{\textcolor{red}{- 9} - \textcolor{b l u e}{- 3}}{\textcolor{red}{- 4} - \textcolor{b l u e}{6}}$

$m = \frac{\textcolor{red}{- 9} + \textcolor{b l u e}{3}}{\textcolor{red}{- 4} - \textcolor{b l u e}{6}}$

$m = \frac{- 6}{- 10}$

$m = \frac{6}{10} = \frac{3}{5}$

Now we have the slope and can use either point and the point-slope formula to find the 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 $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

Substituting the slope we calculated and one of the points gives:

$\left(y - \textcolor{red}{- 9}\right) = \textcolor{b l u e}{\frac{3}{5}} \left(x - \textcolor{red}{- 4}\right)$

$\left(y + \textcolor{red}{9}\right) = \textcolor{b l u e}{\frac{3}{5}} \left(x + \textcolor{red}{4}\right)$

We can convert to the more familiar slope-intercept form by solving for $y$:

$y + \textcolor{red}{9} = \textcolor{b l u e}{\frac{3}{5}} x + \left(\textcolor{b l u e}{\frac{3}{5}} \times \textcolor{red}{4}\right)$

$y + \textcolor{red}{9} = \textcolor{b l u e}{\frac{3}{5}} x + \frac{12}{5}$

$y + \textcolor{red}{9} - 9 = \textcolor{b l u e}{\frac{3}{5}} x + \frac{12}{5} - 9$

$y + 0 = \textcolor{b l u e}{\frac{3}{5}} x + \frac{12}{5} - \left(9 \times \frac{5}{5}\right)$

$y = \textcolor{b l u e}{\frac{3}{5}} x + \frac{12}{5} - \frac{45}{5}$

$y = \textcolor{b l u e}{\frac{3}{5}} x - \frac{33}{5}$