3 Simultaneous equations help please.

Find the value of k which the simultaneous equations have solutions:

2y - z = k

2x + 3y - z = 2

2x - y + z = 0

The answer is k = 1 and its done by eliminating x from equations 2 and 3 and getting 4y - 2z = 2....Then comparing that with the first equation.

BUT... I don't get how you can do that? Because if you eliminate x from one equation, you can't then use that equation to eliminate x in another equation...because it won't have an x?

Thanks for any help you can offer :).