# What's the equation of a line that passes through (1,2) (3,5)?

Jun 26, 2015

In slope-intercept form, the equation of the line is:

$y = \frac{3}{2} x + \frac{1}{2}$

as derived below...

#### Explanation:

First let's determine the slope $m$ of the line.

If a line passes through two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ then its slope $m$ is given by the formula:

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

In our example, $\left({x}_{1} , {y}_{1}\right) = \left(1 , 2\right)$ and $\left({x}_{2} , {y}_{2}\right) = \left(3 , 5\right)$, so

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{5 - 2}{3 - 1} = \frac{3}{2}$

In slope-intercept form, the line has the equation:

$y = m x + c$ where $m$ is the slope and $c$ the intercept.

We know $m = \frac{3}{2}$, but what about $c$?

If we substitute the values for $\left(x , y\right) = \left(1 , 2\right)$ and $m = \frac{3}{2}$ into the equation, we get:

$2 = \left(\frac{3}{2}\right) \cdot 1 + c = \frac{3}{2} + c$

Subtract $\frac{3}{2}$ from both sides to get:

$c = 2 - \frac{3}{2} = \frac{1}{2}$

So the equation of the line can be written:

$y = \frac{3}{2} x + \frac{1}{2}$