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

Apr 29, 2017

See the entire solution process below:

#### Explanation:

First, 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:

$m = \frac{\textcolor{red}{4} - \textcolor{b l u e}{- 4}}{\textcolor{red}{1} - \textcolor{b l u e}{- 1}} = \frac{\textcolor{red}{4} + \textcolor{b l u e}{4}}{\textcolor{red}{1} + \textcolor{b l u e}{1}} = \frac{8}{2} = 4$

Now, use the point-slope formula to write an equation. 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 calculated slope and the values from the first point in the problem gives:

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

Solution 1) $\left(y + \textcolor{red}{4}\right) = \textcolor{b l u e}{4} \left(x + \textcolor{red}{1}\right)$

You can also substitute the calculated slope and the values from the second point in the problem giving:

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

You can also solve this 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.

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

$y - \textcolor{red}{4} = 4 x - 4$

$y - \textcolor{red}{4} + 4 = 4 x - 4 + 4$

$y - 0 = 4 x - 0$

Solution 3) $y = \textcolor{red}{4} x + \textcolor{b l u e}{0}$

Or

Solution 4) $y = \textcolor{red}{4} x$