If I back up my iphone on a computer, can it restore backup from another computer that it was authorized on?

While trying to install a siri port (not knowing it was dangerous), I put my iphone in reboot mode and it is currently stuck on the apple logo. If there is no other way, I am prepared to restore it. (Let me know if you can answer that question). However, the computer that I backed it up on will not let me connect to it. So I was wondering, that since this phone is authorized on 4 other computers, will the back up be saved on those ones too? Im sorry that this question is a bit “geeky” but I REALLY need help. Thanks to ANYONE who answers this question.