# What is the equation of the line passing through the points (3,3) and (-2, 17)?

May 2, 2017

$y = - 2.8 x + 11.4$

#### Explanation:

For any two points on a straight line (as given by a linear equation)
the ratio of the difference between the $y$ coordinate values divided by the difference between the $x$ coordinate values (called the slope) is always the same.

For the general point $\left(x , y\right)$ and specific points $\left(3 , 3\right)$ and $\left(- 2 , 17\right)$
this means that:
the slope $= \frac{\Delta y}{\Delta x} = \frac{y - 3}{x - 3} = \frac{y - 17}{x - \left(- 2\right)} = \frac{3 - 17}{3 - \left(- 2\right)}$

Evaluating the last expression we have that
the slope $= \frac{3 - 17}{3 - \left(- 2\right)} = \frac{- 14}{5} = - 2.8$

and therefore both
$\left.\left(\frac{y - 3}{x - 3} = - 2.8 , \textcolor{w h i t e}{\text{XX")andcolor(white)("XX}} \frac{y - 17}{x - \left(- 2\right)} = - 2.8\right)\right.$

We could use either of these to develop our equation; the first one seems easier to me (but feel free to test this with the second version to see that you get the same result).

If $\frac{y - 3}{x - 3} = - 2.8$
then (assuming $x \ne 3$, otherwise the expression is meaningless)
after multiplying both sides by $\left(x - 3\right)$
$\textcolor{w h i t e}{\text{XX}} y - 3 = - 2.8 x + 8.4$
and therefore (after adding $3$ to both sides)
$\textcolor{w h i t e}{\text{XX}} y = - 2.8 x + 11.4$