# Question #1024c

Feb 18, 2017

$y = \textcolor{red}{\frac{2}{3}} x - \textcolor{b l u e}{1}$

#### 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:

$m = \frac{\textcolor{red}{5} - \textcolor{b l u e}{1}}{\textcolor{red}{9} - \textcolor{b l u e}{3}} = \frac{4}{6} = \frac{2}{3}$

Next, we can 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 first point from the problem gives:

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

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. Solving for $y$ gives:

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

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

$y - \textcolor{red}{1} = \frac{2}{3} x - 2$

$y - \textcolor{red}{1} + 1 = \frac{2}{3} x - 2 + 1$

$y - 0 = \frac{2}{3} x - 1$

$y = \textcolor{red}{\frac{2}{3}} x - \textcolor{b l u e}{1}$