# What is the equation of the line passing through (21,15) and (11,-3)?

May 13, 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:

$m = \frac{\textcolor{red}{- 3} - \textcolor{b l u e}{15}}{\textcolor{red}{11} - \textcolor{b l u e}{21}} = \frac{- 18}{-} 10 = \frac{9}{5}$

We can now use the point-slope formula to write and 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 values from the first point in the problem gives:

Solution 1: $\left(y - \textcolor{red}{15}\right) = \textcolor{b l u e}{\frac{9}{5}} \left(x - \textcolor{red}{21}\right)$

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

$\left(y - \textcolor{red}{- 3}\right) = \textcolor{b l u e}{\frac{9}{5}} \left(x - \textcolor{red}{11}\right)$

Solution 2: $\left(y + \textcolor{red}{3}\right) = \textcolor{b l u e}{\frac{9}{5}} \left(x - \textcolor{red}{11}\right)$

We can also solve the first 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.

$y - \textcolor{red}{15} = \left(\textcolor{b l u e}{\frac{9}{5}} \cdot x\right) - \left(\textcolor{b l u e}{\frac{9}{5}} \cdot \textcolor{red}{21}\right)$

$y - \textcolor{red}{15} = \frac{9}{5} x - \frac{189}{5}$

$y - \textcolor{red}{15} + 15 = \frac{9}{5} x - \frac{189}{5} + 15$

$y - 0 = \frac{9}{5} x - \frac{189}{5} + \left(\frac{5}{5} \times 15\right)$

$y = \frac{9}{5} x - \frac{189}{5} + \frac{75}{5}$

Solution 3: $y = \textcolor{red}{\frac{9}{5}} x - \textcolor{b l u e}{\frac{114}{5}}$