Point 9 — Conditions of solvability

No solutions, one solution, infinite solutions.

Slope–intercept form: for m₁x + c₁ = m₂x + c₂:

• If m₁ ≠ m₂: intersecting lines — one solution.
• If m₁ = m₂ but c₁ ≠ c₂: parallel lines — no solution.
• If m₁ = m₂ and c₁ = c₂: coincident lines — infinite solutions.

General form: for a₁x + b₁y = c₁ and a₂x + b₂y = c₂:

• If $\frac{a_1}{a_2}$ ≠ $\frac{b_1}{b_2}$: one solution.
• If $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ ≠ $\frac{c_1}{c_2}$: no solution.
• If $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$: infinitely many solutions.