Entropy question for 2 input 1 output system

[perplexabot]

Hey all. I have a question regarding the solution to a question, both shown below (only part of the solution is shown). Specifically the line that states: H(Y|X) = H(Z|X). Why does this equality hold? Expanding and using the definition of entropy, I can see that for the above equality to hold, p(Y,X) must equal p(Z,X), right? I am not able to show this. Please some help will be greatly appreciated.

Here is the question with part of the solution:

 

[perplexabot]

Hey again! I found the answer to my question, here it is:

PS: Maybe a better name for this thread would have been "Entropy of sum question." I would have renamed it if I could.
Thanks for reading.