# Question f83a7

Mar 18, 2017

See the entire solution process below:

#### Explanation:

First, we need to determine the slope of the line passing through the two points from the problem. 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:

$m = \frac{\textcolor{red}{11} - \textcolor{b l u e}{- 3}}{\textcolor{red}{5} - \textcolor{b l u e}{2}} = \frac{\textcolor{red}{11} + \textcolor{b l u e}{3}}{\textcolor{red}{5} - \textcolor{b l u e}{2}} = \frac{14}{3}$

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:

$\left(y - \textcolor{red}{- 3}\right) = \textcolor{b l u e}{\frac{14}{3}} \left(x - \textcolor{red}{2}\right)$

color(green)(Solution 1)) $\left(y + \textcolor{red}{3}\right) = \textcolor{b l u e}{\frac{14}{3}} \left(x - \textcolor{red}{2}\right)$

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

color(green)(Solution 2)) $\left(y - \textcolor{red}{11}\right) = \textcolor{b l u e}{\frac{14}{3}} \left(x - \textcolor{red}{5}\right)$

We can also solve one of these equations for $y$ to put it into the 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.

$y + \textcolor{red}{3} = \left(\textcolor{b l u e}{\frac{14}{3}} \times x\right) - \left(\textcolor{b l u e}{\frac{14}{3}} \times \textcolor{red}{2}\right)$

$y + \textcolor{red}{3} = \frac{14}{3} x - \frac{28}{3}$

$y + \textcolor{red}{3} - 3 = \frac{14}{3} x - \frac{28}{3} - 3$

$y + 0 = \frac{14}{3} x - \frac{28}{3} - \left(3 \times \frac{3}{3}\right)$

$y = \frac{14}{3} x - \frac{28}{3} - \frac{9}{3}$

color(green)(Solution 3))# $y = \textcolor{red}{\frac{14}{3}} x - \textcolor{b l u e}{\frac{37}{3}}$