Detecting whether a point is above or below a slope

Solution 1:

If the line equation is $y=ax+b$ and the coordinates of a point is $(x_0,y_0)$ then compare $y_0$ and $ax_0+b$, for example if $y_0>ax_0+b$ then the point is above the line, etc.

Solution 2:

Given the following:
point $P(x_1,y_1)$
line $Ax+By+C=0$ (in general form)

We could check the value of $y$ on the line which would correspond to $x_1$.
This will be $y=\frac{-Ax_1-C}{B}$.

The point would then be above the line if $y_1$ is greater than $y$. Therefore,
$y_1-y>0$
$y_1-\frac{-Ax_1-C}{B}>0$
$\frac{Ax_1+By_2+C}{B}>0$

I remember being told by my professor two years ago that the signs of $Ax_1+By_1+C$ and $B$ must be the same if the point $P$ is above the line. I really did not know why. I just tried finding out after digging up my notes.