Linear systems have 0

Theorem: A linear system has exactly three possible outcomes:

1. 0 solutions — the system is inconsistent
2. 1 solution — the system has a unique solution
3. Infinitely many solutions — the system is consistent with free variables

It is impossible for a linear system to have exactly 2, 3, or any finite number of solutions greater than 1.

Reason: If $x_1$ and $x_2$ are two distinct solutions, then every point on the line $x = (1-t)x_1 + tx_2$ (for $t \in \mathbb{R}$) is also a solution. Since there are infinitely many values of $t$, there are infinitely many solutions.

This is a fundamental property that distinguishes linear systems from nonlinear ones.