# How do you write the equation given (1,-1); (-2,-6)?

Mar 11, 2017

$\left(y + \textcolor{red}{1}\right) = \textcolor{b l u e}{\frac{5}{3}} \left(x - \textcolor{red}{1}\right)$

Or

$\left(y + \textcolor{red}{6}\right) = \textcolor{b l u e}{\frac{5}{3}} \left(x + \textcolor{red}{2}\right)$

Or

$y = \textcolor{red}{\frac{5}{3}} x - \textcolor{b l u e}{\frac{8}{3}}$

#### 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:

$m = \frac{\textcolor{red}{- 6} - \textcolor{b l u e}{- 1}}{\textcolor{red}{- 2} - \textcolor{b l u e}{1}} = \frac{\textcolor{red}{- 6} + \textcolor{b l u e}{1}}{\textcolor{red}{- 2} - \textcolor{b l u e}{1}} = \frac{- 5}{- 3} = \frac{5}{3}$

Now, we can use the point-slope formula to write an equation for the line passing through the two points. 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 gives:

$\left(y - \textcolor{red}{- 1}\right) = \textcolor{b l u e}{\frac{5}{3}} \left(x - \textcolor{red}{1}\right)$

$\left(y + \textcolor{red}{1}\right) = \textcolor{b l u e}{\frac{5}{3}} \left(x - \textcolor{red}{1}\right)$

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

$\left(y - \textcolor{red}{- 6}\right) = \textcolor{b l u e}{\frac{5}{3}} \left(x - \textcolor{red}{- 2}\right)$

$\left(y + \textcolor{red}{6}\right) = \textcolor{b l u e}{\frac{5}{3}} \left(x + \textcolor{red}{2}\right)$

We 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.

$y + \textcolor{red}{6} = \left(\textcolor{b l u e}{\frac{5}{3}} \times x\right) + \left(\textcolor{b l u e}{\frac{5}{3}} \times \textcolor{red}{2}\right)$

$y + \textcolor{red}{6} = \frac{5}{3} x + \frac{10}{3}$

$y + \textcolor{red}{6} - 6 = \frac{5}{3} x + \frac{10}{3} - 6$

$y + 0 = \frac{5}{3} x + \frac{10}{3} - \left(\frac{3}{3} \times 6\right)$

$y = \frac{5}{3} x + \frac{10}{3} - \frac{18}{3}$

$y = \textcolor{red}{\frac{5}{3}} x - \textcolor{b l u e}{\frac{8}{3}}$