Have you considered a Determinant?
2k + 18 - 8 - 12 + 24 - k = 0
I am really concerned with your opinion that this systems "really does" have a solution. You cannot know that until you determine the value of k. When you did those row operations, there was only one way to get rid of the k. That one way is to make an error. It's the only 'k' in the problem. Parity would suggest that you can't get rid of it. Whatever you do, there will always be a 'k'.
Do you REALLY believe a system would be entirely independent of one of it's coefficients? No kidding, think about the implications for a minute. That '6y' is bothering me. Since the solution to the system doesn't care about that 6, let's just make it a 4. Awesome! That's much better. It is a mass of confusion.