# Why do pivot entries correspond to bound variables?

Why do the pivot entries of a matrix in row echelon form correspond to bound variables?

Take this matrix in row echelon form for example with variables $x_1, x_2, x_3, x_4$ (the last column is meant to be augmented).

$$A = \begin{bmatrix} 1 & 2 & 0 & 0 &3\\ 0 & 0 & 1 & 0 &3\\ 0 & 0 & 0 &1&2 \end{bmatrix}$$

It is somewhat an arbitrary rule that mathematicians always choose to say that the pivot entries are bound and the other ones ($x_2$ and $x_5$ in your example above) are "free".
I say that it is is arbitrary because if $x_1 + x_2 = 5$ then we could say $x_1 = 5 - x_2$ or equivalently $x_2 = 5 - x_1$. In any event, the number of pivots corresponds to the number of equations you actually have.