# How do you write an equation for the line through (6,8) and (2,-10)?

Dec 21, 2016

$y - 8 = \frac{9}{2} \left(x - 6\right)$ or $y = \frac{9}{2} x - 19$

#### Explanation:

To write an equation for a line through two points we need to use a two step process.

Step 1) Find the slope.

Step 2) Using the slope, one of the points and the point-slope formula determine the equation.

The slope can be found by using the formula: color(red)(m = (y_2 - y_1)/(x_2 - x_1)
Where $m$ is the slope and $\left(\textcolor{red}{\left({x}_{1} , {y}_{1}\right)}\right)$ and $\left(\textcolor{red}{\left({x}_{2} , {y}_{2}\right)}\right)$ are the two points.

Substituting the points we are given in this problem the slope is:

$m = \frac{- 10 - 8}{2 - 6}$

$m = \frac{- 18}{- 4}$

$m = \frac{- 2 \times 9}{- 2 \times 2}$

$m = \frac{- 2}{- 2} \times \frac{9}{2}$

$m = 1 \times \frac{9}{2}$

$m = \frac{9}{2}$

Now we can use one of the points we were given, the slope we determined and the point-slope formula to find the equation for the line.

The point-slope formula states: $\textcolor{red}{\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)}$
Where $\textcolor{red}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

$y - 8 = \frac{9}{2} \left(x - 6\right)$

If we want to put this into slope-intercept form we can solve for $y$:

$y - 8 = \frac{9}{2} x - \left(\frac{9}{2} \times 6\right)$

$y - 8 = \frac{9}{2} x - \left(9 \times 3\right)$

$y - 8 = \frac{9}{2} x - 27$

$y - 8 + 8 = \frac{9}{2} x - 27 + 8$

$y - 0 = \frac{9}{2} x - 19$

$y = \frac{9}{2} x - 19$