**Theorem: Pivots and Free Variables**
The number of pivots in the reduced row echelon form (RREF) of a consistent system determines the number of free variables in the solution set.
**Statement:** For a consistent system `Ax = b` where `A` is `m × n`:
**Number of free variables = n - (number of pivots)**
where `n` is the number of variables (columns of A).
**Explanation:**
- Each **pivot column** corresponds to a **basic variable** (uniquely determined by the system)
- Each **non-pivot column** corresponds to a **free variable** (can be chosen arbitrarily)
- The number of pivot columns = rank(A)
- The number of free variables = n - rank(A) = nullity(A)
**Example:** Consider a system with 4 variables whose RREF has 2 pivots:
```
[ 1 0 2 0 | 3]
[ 0 1 -1 0 | 1]
[ 0 0 0 1 | -2]
[ 0 0 0 0 | 0]
```
Pivot columns: 1, 2, 4 → basic variables: x₁, x₂, x₄
Non-pivot column: 3 → free variable: x₃
Free variables: 4 - 3 = 1
The solution is: x₁ = 3 - 2x₃, x₂ = 1 + x₃, x₄ = -2, where x₃ is free.