The symmetry of the equations suggests to look at the particular case first. What happens in this case?
Then, suppose . You can try to solve the system just as you would have with known numbers: for instance, using the second and the last equation, you can get an equation without . Repeating this with another subset of two equations, you find a system of two equations in and (where the coefficients depend on ). Do these equations have one solution? none? infinitely many?
I hope this helps. If you know matrices and determinants, there's a quicker solution by looking at the system only for the values of which make the determinant of the system equal to 0 (you know is one of them), since you know that if it is non-zero then there's a unique solution.