I have read a lot of articles that state that SHA-512 hashing cannot be unhashed.
Yes. That is the definition of "hash". This has nothing to do with SHA-512. The definition of a hash function is that it cannot be reversed. Period. If it can be reversed, it's not a hash.
I would like to know if it is possible to reverse this coding, in a way, to decrypt SHA-512 hashed text.
No, you can't decrypt it, because it isn't encrypted, it's hashed.
Linux encrypts their passwords with SHA-512 hashing.
No, it doesn't. It hashes them, it doesn't encrypt them.
As a systems administrator, I would prefer to simply decrypt or unhash this information as needed, rather than guessing whether a password is correct or incorrect and see if the hash matches. Creating new passwords can cause a lot of extra time and money. If you do not feel comfortable publishing this information and would like to discuss it privately, feel free to request my contact information.
As a systems administrator, if you don't understand the difference between encryption and hashing, please tell me where you work, so that I never ever accidentally become of customer of yours! The Pigeonhole Principle is so simple and obvious that it can be understood by a child.