# The number of solutions to a linear system

Put content here.**Theorem:** A linear system has either:
- **0 solutions** (inconsistent)
- **1 solution** (unique)
- **Infinitely many solutions**
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No other number of solutions is possible.
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**Why:** If a system has two distinct solutions $x_1$ and $x_2$, then the line connecting them contains infinitely many solutions. For homogeneous systems, any point on the line through $x_1$ and $x_2$ is also a solution.
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**How to determine which case:**
1. Row reduce the augmented matrix $[A | b]$ to echelon form
2. If there is a row $[0 \ \cdots \ 0 \ | \ b]$ with $b \neq 0$ → **0 solutions**
3. If consistent and no free variables → **1 solution**
4. If consistent and at least one free variable → **infinitely many solutions**

# Parents

* Linear systems of equations