What does it mean if the bottom row of a matrix is all zeros followed by a constant?
Example Solve the given system of equations using either Gaussian or Gauss-Jordan elimination.
x - y + z = 0
-x + 3y + z = 5
3x + y + 7z = 2
Reduces to $\begin{bmatrix}1 & -1 & 1 & 0\\0 & 1 & 1 & 5/2\\0 & 0 &0&-8\end{bmatrix}$
What does that mean? No solution?
In general, when you get a row consisting of all zeros, except for the last column, this means the system is inconsistent and has no solution.
This situation means that, according to this system of equations, $3*0+0+7*0=-8$, which is the reason for inconsistency mentioned above.
If the reduction had produced a zero on the fourth column, that means that $z$ is a parameter and values of $x$ and $y$ would depend on the parameter $z$. This would lead to infinite number of solutions.
One further example of this is shown here (among other many sources).
Note: I did not check that you have made the correct reduction!