Problem could be a $5 ATX power on switch, or the Power Supply.
I'll show you an easy test to determine which one it is.
Dell Support > Dimension 5150 Desktop PC > Service Manual,http://support.dell.com/support/edocs/systems/dim5150/en/sm/index.htm
1) Work on a table.
A couch, a bed, or directly on a carpet floor, are HIGH areas of Static electricity.
2) FOLLOW Anti-Static Precautions.
Your body carries Static electricity. Static WILL fry out (Short Circuit), the delicate hardware components inside a computer.
Relieve your body of Static BEFORE working on your computer.
Computer on a table, computer Unplugged from power, computer case open.
TOUCH an unpainted surface of the metal frame, of the open computer case.
This action will relieve your body of Static.
Should you leave your computer in the middle of working on it, be SURE to touch the metal frame again upon your return.
3) In the list of the Service Manual, left-click on - Removing the computer cover.
Once the computer cover is removed, I want you to click on -
Technical Overview - in the Service Manual list.
On this page click on - System Board Components
Here you see an illustration of your motherboard. There are no hardware parts installed on it. Makes it better to see without all the cables in the way.
(There is no Processor, Ram Memory, add-on cards, or power cables plugged in, for a few examples)
Looking at the illustration of the motherboard, look down at the bottom right.
Look at the connector on the motherboard that has -
power connector (ATXPWR1) pointing to it.
Long connector that has 24 square socket holes in it. (Two columns of 12 socket holes)
Here is a photo of the motherboard,http://www.ascendtech.us/dell-dimension-5150-5100-e510-board_i_mb4delrd203.aspx
Left-click on the photo to enlarge.
In this photo you see the connector again. It is to the bottom right.
White in color, and two columns of 12 socket holes.
The 24-pin ATX main power cable, from the Power Supply, plugs in here.
This is a more in-depth look at the connector on the motherboard, and the 24-pin ATX main power cable, that plugs into it,http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain24
NOTE* Color of connector on motherboard, and connector on power cable, DOES NOT MATTER. Can be ANY color.
What I wish you to do is bypass the Power On switch.
This has NOTHING to do with the Power On switch.
(The Power On switch, is located inside the plastic Power On button assembly)
Look at the photo all the way to the right. This shows the 24-pin ATX main power cable, plugged into the motherboard. That is the way you want it.
Looking at the photo notice the Green wire. This is the Soft Power On wire.
Abbreviated as PS_ON.
Look at the Black wires next to it. ANY Black wire is a Ground wire.
You will use a jumper wire to connect the Green wire, to ANY Black wire.
The preferred jumper wire is a paper clip.
Straighten the paper clip out. __________________
Now bend the paper clip into a U-shape.
Wrap black plastic electrical tape around the middle of the U.
This is for your fingers, and thumb to hold onto.
The 'legs' of the U are what you use.
(It is turned upside down)
Again look at the green wire, and the socket hole it goes into.
The wire, and it's insulation (Green) go down into the socket hole, pretty far.
At the end of the wire is a metal terminal. Looks like this,http://www.molex.com/molex/products/datasheet.jsp?part=active/0002081202_CRIMP_TERMINALS.xml&channel=Products&Lang=en-US
Power cable plugged into the motherboard you won't see it. Just wanted you to know what it looks like.
One leg of the U-shaped jumper wire, (Paper clip), goes down into the socket hole, with the Green wire.
Goes down RIGHT NEXT TO the Green wire, and MUST touch that metal terminal.
The paper clip leg slides down in the socket hole, right next to the Green wire, and touches the metal terminal.
The other leg of the paper clip goes down into a socket hole, RIGHT NEXT TO
ANY Black wire. Has to touch the metal terminal.
Computer plugged into power, hold the black plastic electrical tape, and slide the legs of the paper clip, into those two socket holes.
The contact time made is no more than 2 seconds.
(Worried about being shocked? Don't be. It is 5 Volts DC.
In comparison two D cell flashlight batteries produce 3 Volts DC.
HOWEVER, there may be a small spark. Just want to let you know in advance, so you don't freak out )
Power Supply comes on? You have a bad ATX Power On switch.
Power Supply does NOT come on? You have a bad Power Supply.
Nothing to do with the motherboard.
Need recommendations for a Power Supply, and how to install it?
Post back in a Comment.
For additional questions please post in a Comment.