For the real numbers, the equation a x = b has infinitely many solutions if a = b = 0, no solutions if a = 0 but b ≠ 0, and exactly one solution whenever a ≠ 0. Because there's nearly always exactly one solution, it's convenient to have a symbol for "the one solution to the equation a x = b" and that symbol is b / a; b but you can't write that if a = 0 because then there isn't exactly one solution.

This is true of any field, almost by definition.