I'm well used to the point-slope form and the $y$-intercept form of straight lines on a Cartesian plane, but I often see lines described as $ax + by + c = 0$. It is not easy for me to picture this the graph of the line this way.
Question. So what's so special about this form? When would a mathematician prefer this form over the others?
I've been trying to build an intuition for how to read the general form. It doesn't seem very intuitive. I think I have to say something like --- walk $b$ units along the $x$-axis and go up (or down) $a$ units and finally move $c$ units vertically. (But I must worry about whether both $a, b$ are negative, which gives me four possibilities. It's not convenient. There must be something interesting about the general form?)
The vector $(a,b)$ is perpendicular (aka, "normal") to the line. This generalizes nicely to "flat" objects in higher dimensions: the vector $(a,b,c)$ is perpendicular to the plane $ax+by+cz+d=0$, etc.
You can think of $ax+by+c$ as a function of $(x,y)$ that evaluates to zero for points on the line, positive for points on one side, and negative for points on the other side. Even better, scaling so that $a^2+b^2=1$, that function gives the exact signed distance from a point to the line. (This also generalizes to higher dimensions.)
The form also treats $x$ and $y$ as "peers", which seems fair. :)
That said, the "preferred" form of a line (or any object) depends upon context. Point-slope form, standard form, parametric form, etc, all have their uses, advantages, and drawbacks. (For instance, point-slope form doesn't like vertical lines; standard and parametric forms don't mind them at all.)
See also some thoughts in this old answer of mine. The topic is ellipse equations, but I start by discussing lines; and the general point about the benefit of flexibility applies.