Intuition for the linear equation: $Ax+By+C=0$

Question

Is it possible to gain an intuition for the linear equation in the following situations:

1. When being rearranged into the slope-intercept form. The equation becomes: $y=-\frac{A}{B}x-\frac{C}{B}$. I wonder if there is any intuition for why the slope is $-\frac{A}{B}$ and the y-intercept is $-\frac{C}{B}$?
2. When $B=0$, the equation will represent a vertical line with an x-intercept at $-\frac{C}{A}$. I wonder why does this happen?

Answer 1

For the second question, if $B$ is $0$, then $Ax+C=0$. If you have a value $x_0$ (in this particular case $-C/A$) that verifies the previous equation, then any $(x_0,y)$ pair will be a solution to $Ax+0y+C=0$. That is an equation of a line, where all the points on the line have the same $x$ coordinate, so it's a vertical line. Similarly, if $A=0$, for any $(x,-C/B)$ pair is a solution of the equation, so it's a horizontal line