# How do you find an equation of the line containing the given pair of points (-7, -4) and ( -2, -6)?

May 20, 2017

See a solution process below:

#### Explanation:

First, we need to determine the slope of the line running through the two points. 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:

$m = \frac{\textcolor{red}{- 6} - \textcolor{b l u e}{- 4}}{\textcolor{red}{- 2} - \textcolor{b l u e}{- 7}} = \frac{\textcolor{red}{- 6} + \textcolor{b l u e}{4}}{\textcolor{red}{- 2} + \textcolor{b l u e}{7}} = \frac{- 2}{5} = - \frac{2}{5}$

We can 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 values from the first point in the problem gives:

$\left(y - \textcolor{red}{- 4}\right) = \textcolor{b l u e}{- \frac{2}{5}} \left(x - \textcolor{red}{- 7}\right)$

$\left(y + \textcolor{red}{4}\right) = \textcolor{b l u e}{- \frac{2}{5}} \left(x + \textcolor{red}{7}\right)$

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

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

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

May 20, 2017

See the explanation.

#### Explanation:

First find the slope , $m$ of the line, using the two points.

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

Either point can be 1 or 2. I'm going by the order listed in the question. Point 1$=$$\left(- 7 , - 4\right)$, Point 2$=$$\left(- 2 , - 6\right)$

Insert the points into the equation.

$m = \frac{- 6 - \left(- 4\right)}{- 2 - \left(- 7\right)} = - \frac{2}{5}$

Now use the point slope form for a straight line.

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

Insert either point as $\left({x}_{1} , {y}_{1}\right)$. I'm going to use Point 1 from the first part of this answer: $\left(- 4 , - 7\right)$.

$y - \left(- 4\right) = - \frac{2}{5} \left(x - \left(- 7\right)\right)$

Simplify.

$y + 4 = - \frac{2}{5} \left(x + 7\right)$

$y + 4 = - \frac{2}{5} x - 14$

Solve for $y$ to get the slope intercept form for a straight line: $y = m x + b$, where $m$ is the slope and $b$ is the y-intercept.

Subtract $4$ from both sides and simplify.

$y + 4 = - \frac{2}{5} x - 14$

$y = - \frac{2}{5} x - 14 - 4$

$y = - \frac{2}{5} x - 18$ graph{y=-2/5x-18 [-16.42, 15.6, -25.96, -9.94]}