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

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.