**Theorem: Number of Pivots and Solution Uniqueness**

For a consistent linear system, the number of pivots in the reduced row echelon form (RREF) determines whether the solution is unique or there are infinitely many solutions.

**Statement:** Let `A` be an `m × n` matrix. If the system `Ax = b` is consistent, then:
- If the RREF of `[A | b]` has **n pivots** (one in every variable column), the solution is **unique**.
- If the RREF has **fewer than n pivots**, there are **infinitely many solutions**.

**Explanation:** Each pivot corresponds to a basic (leading) variable that is uniquely determined. Each non-pivot column corresponds to a free variable that can take any value. The presence of even one free variable generates infinitely many solutions.

**Examples:**
- 3 variables, 3 pivots → unique solution
- 3 variables, 2 pivots → 1 free variable → infinitely many solutions (a line of solutions)
- 3 variables, 1 pivot → 2 free variables → infinitely many solutions (a plane of solutions)

**Special case:** If A is square (n × n) and invertible, it has n pivots, so `Ax = b` has exactly one solution for every b.