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Installing the Ubuntu Linux Steam gaming client basics
If you want to try Steam on other Linux distributions, start at Valve's Steam under Linux page. Good luck!
Moving on, to run Steam games you should be running Ubuntu 12.04 or newer on a dual-core system with 2GBs or more of RAM running at least 2.8GHz. You'll also need an nVidia GeForce 8500/9600GT or an ATI/AMD Radeon HD 2500/3600 or better graphics and their latest drivers. Finally, you'll need the newest version of Flash, that's 11.2 for Linux, and OpenGL 2.1.
In addition, if you're running a 64-bit Linux distribution, you may need to download some 32-bit libraries during the installation process. At this time, there are no fully native 64-bit version of the Linux Steam client.
That said, if you've met all the preconditions, open the Ubuntu Software Center and select Steam.
If the Steam client is not on the front-page of the Software Center simply search for in the search window and then select it.
This will bring you to the program window. From here, hit the buy button. Yes, the program is free but you still need to hit the buy button. You won't be asked for a credit-card number, so just click and go.