# What is an equation for a line going through the points: (-17, 10) and (9, 0)?

Feb 28, 2017

See the entire solution process below:

#### Explanation:

First, we need to determine the slope. 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:

$m = \frac{\textcolor{red}{0} - \textcolor{b l u e}{10}}{\textcolor{red}{9} - \textcolor{b l u e}{- 17}} = \frac{\textcolor{red}{0} - \textcolor{b l u e}{10}}{\textcolor{red}{9} + \textcolor{b l u e}{17}} = - \frac{10}{26} = - \frac{5}{13}$

Now, we can 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 $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

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

$\left(y - \textcolor{red}{10}\right) = \textcolor{b l u e}{- \frac{5}{13}} \left(x - \textcolor{red}{- 17}\right)$

$\left(y - \textcolor{red}{10}\right) = \textcolor{b l u e}{- \frac{5}{13}} \left(x + \textcolor{red}{17}\right)$

Or, we can substitute the slope we calculated and the second point from the problem giving:

$\left(y - \textcolor{red}{0}\right) = \textcolor{b l u e}{- \frac{5}{13}} \left(x - \textcolor{red}{9}\right)$

$y = \textcolor{b l u e}{- \frac{5}{13}} \left(x - \textcolor{red}{9}\right)$

Or, we can expand this 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.

$y = \left(\textcolor{b l u e}{- \frac{5}{13}} \times x\right) - \left(\textcolor{b l u e}{- \frac{5}{13}} \times \textcolor{red}{9}\right)$

$y = \textcolor{red}{- \frac{5}{13}} x + \textcolor{b l u e}{\frac{45}{13}}$

Three possible solutions are:

$\left(y - \textcolor{red}{10}\right) = \textcolor{b l u e}{- \frac{5}{13}} \left(x + \textcolor{red}{17}\right)$

$y = \textcolor{b l u e}{- \frac{5}{13}} \left(x - \textcolor{red}{9}\right)$

$y = \textcolor{red}{- \frac{5}{13}} x + \textcolor{b l u e}{\frac{45}{13}}$