# How do you write the equation of a line in slope intercept, point slope and standard form given (3,5) and (1,2)?

Jul 2, 2017

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

#### Explanation:

There are generally two techniques for finding the equation of a straight line that passes through the points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$.

The first technique is to use the point-point equation:

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

The second technique is to use the point-slope equation:

$y - {y}_{1} = m \left(x - {x}_{1}\right)$ where the slope $m$ is calculated using:
$m = \frac{\Delta y}{\Delta x} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

So for the given coordinates $\left(3 , 5\right)$ and $\left(1 , 2\right)$.

Using the first method the equation would be:

$\frac{y - 5}{2 - 5} = \frac{x - 3}{1 - 3}$
$\therefore \frac{y - 5}{- 3} = \frac{x - 3}{- 2}$
$\therefore y - 5 = \frac{3}{2} x - \frac{9}{2}$
$\therefore y = \frac{3}{2} x + \frac{1}{2}$

Using the second method the slope would be:

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

So the equation would be :

$y - 5 = \frac{3}{2} \left(x - 3\right)$
$\therefore y - 5 = \frac{3}{2} x - \frac{9}{2}$
$\therefore y = \frac{3}{2} x + \frac{1}{2}$, as before