What are the meanings of the beeps that the computer makes when it starts up?

Dell systems use a series of "Beep Codes" to determine the system’s condition at POST (Power On Self-Test). Here are the most common beep codes and how they are used to troubleshoot the system:

WARNING: Before you perform any of the following steps, read the Precautionary Measures warning at the end of this document for your personal safety and to prevent damage to the computer system from electrostatic discharge.

1 Short beep - System is operating normally.
1 Long beep - Memory refresh circuitry has failed which indicates a problem with the motherboard. Try reseating the memory DIMM or SIMM modules. Make sure if you have added memory that it is rated at the correct speed for the system.
1 Long 2 Short beeps - The video adapter has failed, or the video memory is having a read/write failure. Try reseating the video card. If the video card is a PCI video card, try moving it to another slot.
1 Long 3 Short beeps - The system does not detect a monitor. Verify that the monitor cable is properly attached to the video card or the onboard video and to the monitor (if the monitor has a detachable cable). If possible, try another monitor on the system to verify that the video adapter is operating properly.
2 or 3 beeps - There is a problem with the system’s memory. Reseat the memory DIMM or SIMM modules.
4 beeps - This indicates a non-functional timer. This indicates a problem with the motherboard.
5 beeps - This indicates a CPU (Central Processing Unit or processor) failure. Try reseating the processor.
6 beeps - This indicates a keyboard controller failure. Try reseating the keyboard connector.
7 beeps - Another indication of a processor problem. Reseat the processor.
8 beeps - This indicates a video memory read/write failure. Reseat the video card (if any) and memory DIMM or SIMM modules.
9 or 10 beeps - This indicates a ROM (Read-Only Memory) checksum error or shutdown register failure. This indicates a defective BIOS chip(s). Reseat the BIOS chips.
11 beeps - Indicates an error in the cache memory. If the system has a Pentium class processor, reseat the processor. If the system has a cache memory module, try reseating it. These beep codes will vary depending on the system.

WARNING FOR YOUR PERSONAL SAFETY AND PROTECTION OF THE EQUIPMENT

Before you start to work inside the computer, perform the following steps in the sequence listed:

Turn off the computer and any attached peripherals.
Disconnect the computer and any attached peripherals from their AC power sources to reduce the potential for personal injury.
Disconnect any telephone or telecommunication lines from the computer.
To help avoid possible damage to the system board, wait 5 seconds after turning off the system before removing a component from the system board or disconnecting a peripheral device from the computer.
Wear a wrist-grounding strap, and clip it to any unpainted metal surface on the computer chassis. If a wrist-grounding strap is not available, touch any unpainted metal surface on the back of the computer chassis to discharge any static charge from your body.

OK if I read this right...
Three beeps then a pause and then one beep would seem to indicate something going on with the memory.
I start by checking that and go from there.

I would definitely check for a bad memory chip.

I would start looking for a new computer,the 5100 is not worth the cost of repairing it. If it is the mother board it will be very expensive to replace it. With Vista coming out in a few months most Xp computers will be discounted.
Did you try blowing out the vents on the computer with compressed air? The 5100 was known to over heat because of a dirty heat sink.

Now that I've got where I can try some things...
I've about come to the conclusion that something is definately fishy. For the purposes of testing I labeled one memory "A" and the other "B". Sometimes I could put "A" into memory slot A and it would work other times not. The same with putting "B" into slot B or any other combination. Meaning sometimes the computer would start the boot sequence. Other times it would come on and immediately back off, or I would get the series of beeps. Other times it would power up but not boot and one of the indicator lights would blink with one steady or other different combinations. There seems to be no consistency.
Now I'm wondering if its not a motherboard or something more serious.
Where should I go from here?
Thanks

http://repair4laptop.org/disassembly_dell.html

Given all I've tried so far I'm wondering if it would be worth the small investment ($50) to get a new stick of memory and see what happens. I saw someone sells new motherboards for ~$169 so I can do that that if needed. There is also a service manual on the Dell website for complete disassembly instructions.
Any thoughts?
Thanks!

Its your call, I still think it would be more cost effective to get a new laptop.