# The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set.

Put content here**Theorem: Pivots and Free Variables**
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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.
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**Statement:** For a consistent system `Ax = b` where `A` is `m × n`:
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**Number of free variables = n - (number of pivots)**
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where `n` is the number of variables (columns of A).
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**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)
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**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]
```
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Pivot columns: 1, 2, 4 → basic variables: x₁, x₂, x₄
Non-pivot column: 3 → free variable: x₃
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Free variables: 4 - 3 = 1
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The solution is: x₁ = 3 - 2x₃, x₂ = 1 + x₃, x₄ = -2, where x₃ is free.

# Parents

* Linear systems and echelon matrices