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

For a consistent system with coefficient matrix $A\in\mathbb F^{m\times n}$ of rank $r$, there are $n-r$ free variables. Each pivot column corresponds to a basic variable, and every remaining variable is free.

Example: $\begin{bmatrix}1&0&2&0&3\\0&1&-1&0&1\\0&0&0&1&-2\\0&0&0&0&0\end{bmatrix}$ is an augmented RREF with **three** pivots and four variable columns. Thus $x_3=t$ is the single free variable, and $(x_1,x_2,x_3,x_4)=(3-2t,1+t,t,-2)$.Copy