$ax+by+c=0$ in computer graphics

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I'm doing a course in computer graphics at the university.

We are talked about this Lineare Equation: $ax+by+c=0$ for $2D$ and of course the one for $3D$.

I don't really get it for what this equation is good for. I already used some tools do visualize it and i think it was $45^\circ$ if $y, x$ and $c =0$, and if $c$ was something else the height of the line was shifted.

I also read something about intersection but I really don't get the concept of this equation in the sense of computer graphics.