If your computer is not turning on or working properly it may be the ATX power supply is at fault. This video teaches you how to remove it from the computer and test it using only an inexpensive multimeter.
There should be +12V, +5V and +3.3V DC output lines on every working ATX power supply. If any of these values stray too far beyond its proper value (half a volt or more), you probably have a power supply issue and it should be replaced. For example, if your +5V line is measured at only +4.1V, your PSU isn’t delivering power as it should and that indicates a problem.
We also have a video that teaches how to use an inexpensive power supply tester and where to get one here: http://www.youtube.com/watch?v=Ukcmp2EkYA0
If you do not have a multimeter to measure voltages you can get one for as little as if not free, as Harbor Freight Tools often publishes coupons for a complimentary unit just for visiting their store.
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How to Easily Test a Computers Power Supply with Simple Tools.