Based on some Linux distributions can do commercial OS? Can I distribute such a system without source code?
No, according to the current laws you can't close the source AFAIK. I'm not a lawyer, but what I understand is you can use the source code to build your own version of linux as long as you make the source available.
See "what is copyleft" here.
There are no rules, however about closing the source on programs that you write for linux and include with the distribution.
"Can I distribute such a system without source code?"
No.
You would lose the right to run, distribute, or modify such software as you lose rights granted under the GPL if/when you breach it.
The FSF and SFLC have plenty of resources to assist you in compliance. Their goal is compliance, not lawsuits.
Honestly, be compliant with the GPL--it's easier. Distributing source code isn't the most onerous licensing provision. You could be paying per-copy royalties.
This does not preclude you from doing things like TomTom does. Their GPS mapping & navigation software runs on top of Linux. Due to Linus Torvalds' kernel exception they need only distribute the source code to Linux and their changes, not to the mapping software that runs on top of it.
Open Source is by definition compatible with commercial software, that is, software that you sell for money. However, you must comply with the licenses of the software you redistribute, in original or modified form. For GPL-licensed components such as the Linux kernel or Bash, this includes the requirement to provide source code.
As mentioned, note the emphasis on redistributing others' code - code you write yourself you can license however you want.
Yes You can BUT if user ask for source code you MUST provide it! (You can charge only for costs of delivering that code to them)
PS You can sell your commercial Linux although!!