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.
How to determine which case:
Row reduce the augmented matrix $[A | b]$ to echelon form
If there is a row $[0 \ \cdots \ 0 \ | \ b]$ with $b \neq 0$ → 0 solutions
If consistent and no free variables → 1 solution
If consistent and at least one free variable → infinitely many solutions
**Theorem:** A linear system has either:
- **0 solutions** (inconsistent)
- **1 solution** (unique)
- **Infinitely many solutions**
No other number of solutions is possible.
**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.
**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**