# What is the equation of the line between (5,-6) and (4,2)?

Jan 18, 2017

$\left(y - \textcolor{red}{2}\right) = \textcolor{b l u e}{- 8} \left(x - \textcolor{red}{4}\right)$

Or

$y = - 8 x + 34$

Or

$\left(y + \textcolor{red}{6}\right) = \textcolor{b l u e}{- 8} \left(x - \textcolor{red}{5}\right)$

#### Explanation:

The point-slope formula can be used to find this equation. However, we must first find the slope which can be found using two points on a 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 problem gives:

$m = \frac{\textcolor{red}{2} - \textcolor{b l u e}{- 6}}{\textcolor{red}{4} - \textcolor{b l u e}{5}}$

$m = \frac{\textcolor{red}{2} + \textcolor{b l u e}{6}}{\textcolor{red}{4} - \textcolor{b l u e}{5}}$

$m = \frac{8}{-} 1 = - 8$

The slope and either of the points can now be used with the point-slope formula to find 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 calculate slope and the second point gives:

$\left(y - \textcolor{red}{2}\right) = \textcolor{b l u e}{- 8} \left(x - \textcolor{red}{4}\right)$

Or, we can convert to the more familiar slope-intercept form by solving for $y$:

$y - \textcolor{red}{2} = \left(\textcolor{b l u e}{- 8} \times x\right) - \left(\textcolor{b l u e}{- 8} \times \textcolor{red}{4}\right)$

$y - 2 = - 8 x + 32$

$y - 2 + \textcolor{red}{2} = - 8 x + 32 + \textcolor{red}{2}$

$y - 0 = - 8 x + 34$

$y = - 8 x + 34$

Or, we can use the point-slope formula and the first point to give:

$\left(y - \textcolor{red}{- 6}\right) = \textcolor{b l u e}{- 8} \left(x - \textcolor{red}{5}\right)$

$\left(y + \textcolor{red}{6}\right) = \textcolor{b l u e}{- 8} \left(x - \textcolor{red}{5}\right)$