Ok I work on laptops a lot. Many have had bad system boards. I have been able to fix a few without buying parts. Here's what I have done to diagnose them starting with the simplest thing:

1.    Unplug the computer, then take the battery out, push the power button and hold it for about 30 seconds. Plug the system back in (leave battery out) and push the power button. If it starts, Great! If not unplug go to the next step.
2.    You should still have the battery out and the power unplugged. Now turn the laptop over. Take off all the covers and pull out the hard drive and ram and wireless card. Find the screw that has a symbol kind of like this € its the CD rom drive screw. Now pull the drive out. Pick one stick of ram and put it back in the computer. Flip it over open it plug it in and hit the power button. If it does not fire up then unplug it and switch the ram. Plug it back in and hit the power button again. If it doesn't start up, unplug the computer and go to the last step.
3.    Pull the memory out and find the screws that hold the keyboard on. There are symbols in the plastic there should be 2 screws holding it in. Now in the battery compartment, there are several screw at the back edge of the computer take them out and also the ones in the battery in the center strip. Some are silver and some are black. Once you have the screws out of the battery area flip the computer over and open the lid. The strip just below the LCD with the power button and controls will now be able to be lifted up. You will probably have to lightly get an edge started before it will come up. There should be several little cables hooked to it. Flip the control panel over and look for some screws holding the keyboard in. I think there's 2. Lift up the keyboard. Making sure there's nothing touching anything important and then plug the computer in. Carefully hit the power button and see if the computer tries to start. If it does try it will probably beep at you. Unplug it and put a stick of ram in it. Plug it in and try to start it again. If not, unplug the computer and Go to the next step.
4. Carefully put the keyboard and control strip in place. Flip the computer over and loosen all the screws  that hold the computer together. Flip the computer over again and pull the keyboard and control strip up again. Flex the inner aluminum plate that was under the keyboard lightly. Plug in the computer and then hit the power button. If it tries to start, Great! put it together again in the reverse process.

If it does not, there are 2 possible things I would suspect the Control strip or the mainboard. If you don't get the computer to start back up please put it back together so you remember where all the screws go. Even if you plan to get another mainboard put it back together.