I would work with determinants. Check this thread out. That might give you a few ideas. Make sure to read all the way to the end, if you're interested in a rigorous proof, as you can see that my first idea was invalid.
I solved this using Maxima and its answer agrees with Oiler's reduced echelon form.
Just use that form to get expressions for your variables, Oiler. You can read them right off. Looking at the last row, what's z? Then find x and y similarly (these depend on k). Is there any k that won't give an answer, or will give more than one answer?