Perhaps the computer is Not turning on at all, and only seems like it is. Monitor's will go to 'Power Saving Mode' due to No Signal.
No video signal.
The computer isn't sending a video signal to the monitor, therefore the monitor goes into a power saving mode.
I suggest first checking the monitor cable, if it is a detachable unit. Use a Known to be Good monitor cable in it's place, or use the monitor, AND the monitor cable, on a computer that is known to be good.
Cable checks out?
Then I suggest cleaning out the inside of the computer, if the computer is dirty inside, and also the Power Supply.
Use a can of compressed air for computers.
[Attach the plastic straw provided with the can of air, to the can of air's nozzle.
When you go to clean out the inside of the Power Supply, you will have to use the plastic straw, through the ventilation holes in the Power Supply's case.
Hold on to the plastic straw with one hand, use the other hand to hold the can of air.
The plastic straw can shoot off of the can of air's nozzle, and wind up inside the case of the Power Supply, if you don't.
Real riot trying to fish out the straw ]
The above is a prerequisite before working on the computer.
Also heavily suggest you observe Anti-Static Precautions BEFORE you reach inside the computer.
Your body carries Static electricity.
Static WILL fry out (Short Circuit) the delicate hardware components inside a computer.
Relieve your body of Static.
Computer off, UNPLUGGED from power, computer on a table, computer case open.
TOUCH the metal frame of the open computer case, to relieve your body of Static.
Should you get up, and walk away in the middle of working on your computer, upon your return be SURE to touch the metal frame again ]
Problem could be with the 'graphics engine' of your computer, but experience, and observation through the years makes me lean more towards a bad Power Supply.
['Graphics engine'. The GPU. Graphics Processing Unit.
Can be Integrated Graphics, or a dedicated graphics adapter card.
Integrated graphics is also known as OnBoard graphics.
On the motherBoard.
The GPU is soldered to the motherboard.
A dedicated graphics card, (Graphics adapter card), has it's own GPU.http://en.wikipedia.org/wiki/GPU
Weak Voltage power rail.
[Voltage Power Rail:
The Power Supply in your eMachines computer, converts AC electricity from your home, or business, to DC electricity.
(An example of stored DC electricity, is a D cell battery used in a flashlight )
There are three main DC voltages produced,
A) 3.3 Volts (3 point 3)
B) 5 Volts
C) 12 Volts
(Two D cell batteries produce 3 Volts DC)
Each of the Voltages indicated above emit from one power rail, inside the Power Supply.
The 3.3 Volt has it's own power rail, same with the 5 Volt, and the 12 Volt.
(Your eMachines power supply, does not have multiple 12 Volt power rails. Unless it isn't the stock power supply that came with the computer ]http://en.wikipedia.org/wiki/Computer_power_supply
Weak Voltage power rail.
1) ALL the LED lights combined, use less than 1 Watt of power.
2) EACH fan uses 2 to 3 Watts
3) A typical Processor can use 51 to 125 Watts. (Less for older Processors. Such as Pentium III's for example)
With a power supply that has a weak voltage power rail, the lights will light, the fans may spin, (Or spin a few times then stop), but the Processor won't have enough power to turn on.
Lights, and fans can spin all day, the Harddrive may sound like the Platters are spinning inside, but if there is no Processor operating, there is No computer.
[Information on Harddrives, and the aforementioned Platters,http://computer.howstuffworks.com/hard-disk.htm
Suggest you check the voltages produced by the Power Supply, or use a Known to be Good, Compatible power supply for a test unit.
Right size, and shape of power supply case.
Correct amount of power cables, and the correct power cables,http://www.playtool.com/pages/psuconnectors/connectors.html
For any questions regarding any of my statements above, please post in a Comment.