|The most likely thing is the power supply is failing.|
emachines systems have a reputation of using el-cheapo power supplies in their systems that tend to malfunction and fail a lot more often than average, and when they DO fail, they are a lot more likely than average to damage something else while failing, often the mboard. This is especially likely if the make of the PS is BESTEC.
If you DO have a BESTEC PS I recommend you DO NOT continue using it until it quits completely, because if it dies there's a strong possibility it will damage your mboard.
There was a successful class action suit against emachines in the US not all that long ago regarding problems mostly caused by those el-cheapo power supplies!
Failing power supplies are common and can cause your symptoms.
Check your PS.
They often partially work, fans and hard drives may spin, leds may come on, yet you get no video and the mboard will not boot all the way.
See response 4 in this:
You could try this but it probably won't help.
Unplug the case/power supply.
Power off your monitor.
Open up the case by removing the left panel as seen when you're looking at the front of the case.
Check all the connections of the wiring to make sure they are all the way onto their pins and into their sockets, especially the main connector from the power supply. The wires close to the mboard going into the main connector/socket should be more or less perpendicular to the mboard surface rather than at an angle. Make sure all cards in slots are all the way down in their slots.
See response 2 in this - try cleaning the contacts on the ram modules, and making sure the modules are properly seated:
While you're in there, if the cpu fan/heatsink has mung (dust, lint, etc.) on it, clean it off, but DO NOT use a vaccuum cleaner to do that - use canned air, or an air nozzle if you have access to an air compressor, or an artist's brush that can be used in small spaces, etc. If may be difficult to clean the top of the heatsink under the cpu fan - the most likely place to have mung on it - and the bottom side of the cpu fan blades unless you remove the fan. If you have a case fan, clean that too if it needs it.