# The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions.

Put content here**Theorem: Number of Pivots and Solution Uniqueness**
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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.
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**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**.
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**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.
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**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)
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**Special case:** If A is square (n × n) and invertible, it has n pivots, so `Ax = b` has exactly one solution for every b.

# Parents

* Linear systems and echelon matrices