# What is the equation of the line between (30,2) and (-23,11)?

Dec 5, 2017

See a solution process below:

#### Explanation:

First, we must 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:

$m = \frac{\textcolor{red}{11} - \textcolor{b l u e}{2}}{\textcolor{red}{- 23} - \textcolor{b l u e}{30}} = \frac{9}{-} 53 = - \frac{9}{53}$

We can now use the point-slope formula to find an equation for the line between the two points. The point-slope form of a linear equation is: $\left(y - \textcolor{b l u e}{{y}_{1}}\right) = \textcolor{red}{m} \left(x - \textcolor{b l u e}{{x}_{1}}\right)$

Where $\left(\textcolor{b l u e}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right)$ is a point on the line and $\textcolor{red}{m}$ is the slope.

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

$\left(y - \textcolor{b l u e}{2}\right) = \textcolor{red}{- \frac{9}{53}} \left(x - \textcolor{b l u e}{30}\right)$

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

$\left(y - \textcolor{b l u e}{11}\right) = \textcolor{red}{- \frac{9}{53}} \left(x - \textcolor{b l u e}{- 23}\right)$

$\left(y - \textcolor{b l u e}{11}\right) = \textcolor{red}{- \frac{9}{53}} \left(x + \textcolor{b l u e}{23}\right)$

We can also solve the first equation for $y$ to transform the equation to 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.

$y - \textcolor{b l u e}{2} = \left(\textcolor{red}{- \frac{9}{53}} \times x\right) - \left(\textcolor{red}{- \frac{9}{53}} \times \textcolor{b l u e}{30}\right)$

$y - \textcolor{b l u e}{2} = - \frac{9}{53} x - \left(- \frac{270}{53}\right)$

$y - \textcolor{b l u e}{2} = - \frac{9}{53} x + \frac{270}{53}$

$y - \textcolor{b l u e}{2} + 2 = - \frac{9}{53} x + \frac{270}{53} + 2$

$y - 0 = - \frac{9}{53} x + \frac{270}{53} + \left(\frac{53}{53} \times 2\right)$

$y - 0 = - \frac{9}{53} x + \frac{270}{53} + \frac{106}{53}$

$y = \textcolor{red}{- \frac{9}{53}} x + \textcolor{b l u e}{\frac{376}{53}}$