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Former IT support guy here. There's a few things you can do to narrow down the problem.
If it's not turning on at all, this would point to the PSU. If you have another one handy, the easiest thing to do would be to swap it out and try turning on the PC. If it works, you have your answer - it's the PSU. However, if you don't have an extra one handy, then you can try something as simple as just disconnecting everything from the PSU, and only plugging in a cooling fan. Any cooling fan. We just want to see if it's drawing power from the PSU. DOes it turn on? No? Then that's probably what it is.
But suppose your PSU did turn on a fan. Now we know that it is providing power. So the next step is to make sure all the power connections to the motherboard are connected and properly seated. When you try to turn on the power, does it make any kind of noise? Does anything light up? If it didn't, then it's probably your motherboard. If it did, let's move on to the next step.
We know that the motherboard is receiving power because something lit up or made noise. Ok good. Now, does anything come up on the monitor? Anything at all? Doesn't have to be Windows, just text. We just want to see the BIOS info come up. No? Ok, are you getting any kind of beeping noise?
Click here and on the right side you'll see info on beep codes. If a code came up on the screen, that same link can give you more info on the problem. Just look up your motherboard make.
But wait, nothing came up on the screen? Ok, let's try this. You mentioned you had a graphics card installed. But let's see what happens when you plugin your monitor into the onboard graphics. Plug it in. Does anything come up? If so, then it's your graphics card causing the problem. If nothing comes up, then it could be your motherboard, but it could also be your monitor. Try a different monitor if you have another. Also try a different vga/dvi cable to each of the monitors.
Try taking out one of the memory chips, leave just one in there. Do the same for the other one. Does anything change? Faulty memory has been known to lockup systems before then can even boot up.
Assuming you got this far, you probably should have found your problem by now. But from your description, it's more than likely going to be either your PSU, or your Motherboard. My money's on the PSU.
Keep up posted!