How do you know when an algebraic equation has infinitely many solutions?

A system of equations has an infinite set of solutions when the equations define the same line, such that for ax + by = c, the values for two equations is a1/a2 + b1/b2 = c1/c2.

Equations where a variable drops out completely, e.g.

3x - y = 6x -2y

there are either an infinite number of solutions, or no solution at all.