We need the augmented matrix to end up with a zero row in order for the system of equations to have a solution (since there's more rows than columns in the left-hand matrix). This means the un-augmented matrix needs to have determinant 0. So if we were given that determinant formula (which is the determinant of the un-augmented matrix), we'd know the candidate values for z (they're the roots of that determinant polynomial), and we can then go on to try and solve for x and y.
