# Linear systems have 0

Put content here**Theorem:** A linear system has exactly three possible outcomes:
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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
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It is impossible for a linear system to have exactly 2, 3, or any finite number of solutions greater than 1.
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**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.
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This is a fundamental property that distinguishes linear systems from nonlinear ones.

# Parents

* The number of solutions to a linear system