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German state moving 30,000 PCs to Linux and LibreOffice - The Document Foundation Blog
(blog.documentfoundation.org)
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Linux is a family of open source Unix-like operating systems based on the Linux kernel, an operating system kernel first released on September 17, 1991 by Linus Torvalds. Linux is typically packaged in a Linux distribution (or distro for short).
Distributions include the Linux kernel and supporting system software and libraries, many of which are provided by the GNU Project. Many Linux distributions use the word "Linux" in their name, but the Free Software Foundation uses the name GNU/Linux to emphasize the importance of GNU software, causing some controversy.
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That, again, is not how governments work.
What you depict is how companies work: You save amount X on something, so there are X moneys left to invest in something.
Governments work with separated and highly regulated budgets. That is sometimes bullshit, but sometimes necessary to make sure government aids are spent fairly, for example. So: You save amount X on something, you aren't allowed to just give this amount to someone. There has to be either a program, a law, or (most often) an entirely different budget somewhere else that this someone is allowed to receive.
So the “trade-off” logic cannot be fulfilled by governments, and it shouldn't be. Think about the myriad of bullshit, money would just be dumped into by the government if this wasn't the case. On top of the myriad of bullshit that already made it through the nets, that is.