The geometric image of the one-variable trichotomy: two lines cross once (slopes differ, a_1/a_2 ≠ b_1/b_2), are parallel and distinct (no solution, a_1/a_2 = b_1/b_2 ≠ c_1/c_2), or are the same line twice (infinitely many, all three ratios equal). There are no other possibilities.