# How do you find the equation through the line (3/4,-8) and (2,-5)?

Apr 16, 2017

See the entire 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:

$m = \frac{\textcolor{red}{- 5} - \textcolor{b l u e}{- 8}}{\textcolor{red}{2} - \textcolor{b l u e}{\frac{3}{4}}} = \frac{\textcolor{red}{- 5} + \textcolor{b l u e}{8}}{\textcolor{red}{\frac{8}{4}} - \textcolor{b l u e}{\frac{3}{4}}} = \frac{3}{\frac{5}{4}} = \frac{12}{5}$

Now, we can use the point-slope formula to write 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 second point gives:

$\left(y - \textcolor{red}{- 5}\right) = \textcolor{b l u e}{\frac{12}{5}} \left(x - \textcolor{red}{2}\right)$

$\left(y + \textcolor{red}{5}\right) = \textcolor{b l u e}{\frac{12}{5}} \left(x - \textcolor{red}{2}\right)$

We can also substitute the slope we calculated and the first point giving:

$\left(y - \textcolor{red}{- 8}\right) = \textcolor{b l u e}{\frac{12}{5}} \left(x - \textcolor{red}{\frac{3}{4}}\right)$

$\left(y + \textcolor{red}{8}\right) = \textcolor{b l u e}{\frac{12}{5}} \left(x - \textcolor{red}{\frac{3}{4}}\right)$

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.

$y + \textcolor{red}{5} = \left(\textcolor{b l u e}{\frac{12}{5}} \times x\right) - \left(\textcolor{b l u e}{\frac{12}{5}} \times \textcolor{red}{2}\right)$

$y + \textcolor{red}{5} = \frac{12}{5} x - \frac{24}{5}$

$y + \textcolor{red}{5} - 5 = \frac{12}{5} x - \frac{24}{5} - 5$

$y + 0 = \frac{12}{5} x - \frac{24}{5} - \frac{25}{5}$

$y = \textcolor{red}{\frac{12}{5}} x - \textcolor{b l u e}{\frac{49}{5}}$