$y\ln y - y = x\ln x - x$ ; How do I write $y$ explicitly as a function of $x$?

I used Desmos and found that $y\ln y - y = x\ln x - x$ can be splited into two functions.

It looks clear that one of them is $y=x$.However, I am having trouble finding out the second one.How do I derive it, and how can I prove rigorously that $y\ln y - y = x\ln x - x$ can be split only by these two?

p.s. I am currently a student studying calculus 2, so I would appreciate it if you could make this as easy to understand as possible. Thank you.