I have a desktop PC that when I turn it on it will power up and then after 6 seconds turn back off.

I saw a computer with a similar problem. Probably the problem is with your sink fan. Probably you wanted to clean inside computer for dust, or for other reason you replaced or moved the sink fan and after cleaning maybe, you did not fix it properly on the CPU. This way, when the CPU is hot enough it is not cooled and to protect the CPU from overheating, the computer shuts down.

Unless you recently made a hardware component change, you have a problem with a bad Power Supply, or bad Electrolytic Capacitors on the motherboard.

I also suspect the inside of the computer, as well as the inside of the Power Supply, is dirty inside.

Not pointing fingers. The leading two causes of desktop computer failure, is the computer is dirty inside, and a bad Power Supply.
However more, and more, bad capacitors are found on older desktop computers, also.

With as fast as the computer is turning off, it could go either way.
Bad Power Supply, or bad capacitors.

Power Supply:
Just enough power to light LED lights, and maybe spin fans, but not enough power to keep them going, AND not near enough power in this particular case, to turn the Processor on.

1) IF, all of the LED lights were on at once, they would use less than 1 Watt of power.

(Typical eMachines desktop computer, uses a Power Supply with either 250 Watts, or 300 Watts, stated rated maximum available power )

2) EACH fan uses 2 to 3 Watts of power.

3) A typical Processor as used in your eMachines, can use from 51 to 125 Watts of power.
Just depends on what Processor it is. (AMD or Intel, and model number)

State in a Comment as to the model number of your eMachines.
Let's see if it uses a common ATX (Form Factor) type power supply, or it is a type of slimline computer, and only one style of power supply will fit.

If it is the common ATX style, perhaps there is an unused computer with a KNOWN to be good, compatible power supply, that you can borrow for a test.

Or you could use a multimeter, and check the voltages coming out of the Power Supply.
An economical multimeter can be purchased for around $5 to $12.
A multitude of stores carry them. An auto parts store is but one example.

The voltage going INTO the Power Supply is dangerous. Depending on your country it can be from 100 Volts AC, to 220/240 Volts AC.
It is contained inside the case of the Power Supply. The case is Not to be opened.

The voltages coming OUT of the Power Supply is not dangerous.
(Unless you are standing in a body of water, barefoot, and the computer is on......maybe )

The incoming AC voltage is changed into much lower DC voltages.
There are 3 main voltages produced;

A) The 3.3 Volt power rail (DC)
B) The 5 Volt power rail (DC)
C) The 12 Volt power rail (DC)

In comparison two D cell flashlight batteries produce 3 Volts DC

Bad Capacitors:

Capacitors used on your eMachines motherboard are Electrolytic Capacitors.
More specifically, Radial Aluminum Electrolytic Capacitors.

They are used as either Filters, or Voltage Regulators.
The ones used as voltage regulators are in the Motherboard Voltage Regulator Circuit.

Part of what the motherboard voltage circuit does, is to regulate voltage for the Processor.
A Processor MUST have a steady, 'clean', supply of voltage, AND it has to be within a very TIGHT tolerance range.

If a Processor has too little, or too much voltage, it turns off.
(Actually BIOS turns it off)

A) http://www.capacitorlab.com/index.htm

B) http://www.hardwaresecrets.com/article/How-to-Identify-Japanese-Electrolytic-Capacitors/595

C) http://www.mouser.com/Passive-Components/Capacitors/Aluminum-Electrolytic-Capacitors/_/N-75hqt/

D) http://www.youtube.com/watch?v=YCSNWi3UHf4

E) http://www.youtube.com/watch?v=KN6_-8fYHo0

F) A DC circuit has a Positive, and a Negative. Negative being - Ground.

Orange insulated wires are 3.3 Volts (DC)
Red wires are 5 Volts
Yellow wires are 12 Volts.
ALL Black wires are Ground wires.

Function knob is set to DC Voltage. If a symbol it is a dotted line over a solid line. If more than one DC Voltage scale, set it to the 0 - 50 Volt setting.

Positive (Red) probe lead of the multimeter goes to the power wire.
(3.3 Volts, or 5 Volts, or 12 Volts)
Negative (Black) probe lead goes to a Ground wire. (ANY Black wire)

http://en.wikipedia.org/wiki/Power_supply_unit_%28computer%29

(The power supply in your computer is also an SMPS.
Switched-Mode Power Supply)

For additional questions post in a Comment.

Regards,
joecoolvette

The first solution is right on the money, It's the video card, or many of us today forget to plug the damm thing in and the computer doesn't read and we think that there is a serious problem. If none of this helps then try troubleshooting through this PC repair guide. Hope this helps.