# How do you find equation of line passing through the point S(-1,-4) and T (3,4)?

Feb 6, 2017

See the entire solution process below:

#### Explanation:

We can use the point-slope formula to find an equation. However, we must first find the slope. 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 and solving gives:

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

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

$m = \frac{8}{4} = 2$

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.

We can substitute the slope we calculated and the values from the first point to give:

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

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

We can also substitute the slope we calculated and the values from the second point to give:

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

Or we can solve for $y$ to put this equation in the more familiar slope-intercept form:

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

$y - \textcolor{red}{4} = 2 x - 6$

$y - \textcolor{red}{4} + 4 = 2 x - 6 + 4$

$y - 0 = 2 x - 2$

$y = 2 x - 2$

These equations are solutions to this problem:

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

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

$y = 2 x - 2$