# How do you write the equation of a line given (1,4), (-1,1)?

Mar 19, 2017

See the entire solution process below:

#### Explanation:

First, we need to determine the slope of the line passing through the two points in 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:

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

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 first point from the problem gives:

$\left(y - \textcolor{red}{4}\right) = \textcolor{b l u e}{\frac{3}{2}} \left(x - \textcolor{red}{1}\right)$

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

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

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

Or, we can solve either of these equations for $y$ to put the equation into 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.

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

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

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

$y - 0 = \frac{3}{2} x + \frac{3}{2} + \left(\frac{2}{2} \times 1\right)$

$y = \frac{3}{2} x + \frac{3}{2} + \frac{2}{2}$

$y = \textcolor{red}{\frac{3}{2}} x + \textcolor{b l u e}{\frac{5}{2}}$