Consider the simple linear system: $$ 3 x_1 + 2x_2 = 4 \\ 5x_3 = 9 $$
and the corresponding matrix form Ax=b: $$ \left( \begin{array}{ccc} 3 & 2 & 0 \\ 0 & 0 & 5 \end{array} \right) \left( \begin{array}{ccc} x_1 \\ x_2 \\ x_3 \end{array} \right) = \left( \begin{array}{cc} 4 \\ 9 \end{array} \right) $$
We have that $x_3$ admits exactly one solution (9/5), while $x_1$ and $x_2$ admit more than one solution.
Question Given a system $Ax=b$ can we determine the number of variables that be univocally determined (those with exactly one solution) without solving the system and check "manually" (Gaussian elimination + inspection of the echelon form)?
The matrix $\left(\begin{smallmatrix}3&2&0\\0&0&5\end{smallmatrix}\right)$ is in row echelon form. The columns corresponding to non-pivots are free, and hence may be any value. In this example the pivots are in the first and third columns, so the second variable ($x_2$) is free. However, some of the pivot columns also might vary. If a pivot is the only nonzero element of its row, then it has a unique value. In this example, that's the third column ($x_3$). If instead there are free variables in its row, then that variable can assume any value, as those free variables vary. In this example, that's the first column ($x_1$).
However you can read off all this information only if the matrix is in row echelon form.