On top of that, you won't find a gaming PC, pre-assembled for $400, actually the budget is pretty tight, even for a home build.
CPU: AMD A6-5400K 3.6GHz Dual-Core Processor ($66.99 @ SuperBiiz)
Motherboard: MSI FM2-A75MA-E35 Micro ATX FM2 Motherboard ($53.99 @ Newegg)
Memory: G.Skill Ripjaws X Series 4GB (2 x 2GB) DDR3-1866 Memory ($39.99 @ Newegg)
Storage: Western Digital Caviar Blue 500GB 3.5" 7200RPM Internal Hard Drive ($53.98 @ Outlet PC)
Case: Rosewill FBM-01 MicroATX Mini Tower Case ($21.99 @ Newegg)
Power Supply: Corsair Builder 430W 80 PLUS Bronze Certified ATX12V Power Supply ($29.99 @ Newegg)
Optical Drive: LG GH24NS95 DVD/CD Writer ($14.99 @ Newegg)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) ($89.94 @ Outlet PC)
Total: $371.86
(Prices include shipping, taxes, and discounts when available.)
The graphics on the AMD A6-5400k paired with super fast DDR3-1866 memory should be able to provide graphics for what you wanting to do. All of this is for under $400. And you are looking at a better quality build then some pre-built system.