# Question #dfc9a

Aug 29, 2016

Point-slope form: $y + \frac{1}{2} = - \frac{3}{2} \left(x - \frac{3}{2}\right)$

Standard form: $y + \frac{3}{2} x = \frac{7}{4}$

#### Explanation:

In order to write the equation of a line in point-slope form, we need the slope of the line, and a point on that line. As we already have a point (in fact, we have two), we need only the slope.

The slope of a line represents the increase (or decrease) in $y$ when we increase $x$ by $1$. That is, $\text{slope" = "change in y"/"change in x}$.

To find the slope $m$ of a line given two points $\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right)$ on that line, then, we find the change in $y$ as the difference ${y}_{2} - {y}_{1}$, the change in $x$ as the difference ${x}_{2} - {x}_{1}$, and then divide:

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

In this case, we have $\left({x}_{1} , {y}_{1}\right) = \left(\frac{3}{2} , - \frac{1}{2}\right)$ and $\left({x}_{2} , {y}_{2}\right) = \left(- \frac{1}{2} , \frac{5}{2}\right)$. Plugging that into the above:

$m = \frac{\frac{5}{2} - \left(- \frac{1}{2}\right)}{- \frac{1}{2} - \frac{3}{2}} = - \frac{3}{2}$

Then, to get the equation of the line in point-slope form, we can choose one of the points, say, $\left(\frac{3}{2} , - \frac{1}{2}\right)$, and plug it and the slope into the format $y - {y}_{1} = m \left(x - {x}_{1}\right)$:

$y - \left(- \frac{1}{2}\right) = - \frac{3}{2} \left(x - \frac{3}{2}\right)$
$\implies y + \frac{1}{2} = - \frac{3}{2} \left(x - \frac{3}{2}\right)$

Next, to change the equation into standard form, we distribute, then gather the constant term on one side and the variable terms on the other:

$y + \frac{1}{2} = - \frac{3}{2} x - \left(- \frac{3}{2}\right) \left(\frac{3}{2}\right)$

$\implies y + \frac{1}{2} = - \frac{3}{2} x + \frac{9}{4}$

$\implies \left(y + \cancel{\frac{1}{2}}\right) - \cancel{\frac{1}{2}} + \frac{3}{2} x = \left(\cancel{- \frac{3}{2} x} + \frac{9}{4}\right) - \frac{1}{2} + \cancel{\frac{3}{2} x}$

$\implies y + \frac{3}{2} x = \frac{9}{4} - \frac{1}{2}$

$\therefore y + \frac{3}{2} x = \frac{7}{4}$