Gigabyte Model GA-7DXR Motherboard

This unfortunately is a very common problem with eMachines computers built between 2001 thru 2005 especially those with 478 sockets (processor-P4/Celeron) & 462/A sockets (processor-AMD K7) & Intel OEM (Original Equipment Manufactured) Motherboards. I assume your motherboard is the original OEM? In cases like this it's usually time to replace the motherboard. A new one will have this problem addressed and corrected. However, if you like to 'tinker' I will share with you how I usually solve this problem when my customer doesn't have the finances for a new board as many of my customers are retirees and not in too much of a hurry. This may sound ridiculous to some but I assure you it does work and My own machine #3 in my personal network can attest to it for the last two years.
First and most important you must identify the exact transistor that has the problem. Usually there are 3-5 depending on the motherboard that control the memory circuit. Assuming you've done this you will then need to find another used Intel motherboard of comparable design that's not working and is going to be or has been discarded which you can usually get for free. Don't worry it doesn't need be an exact match as your not going to be robbing your replacement from it's memory control. Once you've located the appropriate 'scrapped' board look around it's area in between and toward the it's rear across from the "Northbridge" for where the front audio is or where it's cables would plug in for a comparable transistor to yours in size, contacts, solder points, etc both on front and BACK of the motherboard. Once you have located an appropriate replacement remove it carefully heating each solder point just enough to remove it so as to avoid overheating. Replace your shorted transistor with this one and you should get your video back.
Don't worry you have nothing to lose! As I stated above in cases such as this it's time to replace the board. I have though replaced these transistors with comparable ones from the audio circuit's of many different boards and have had about a 75% success rate in doing so.
Hope it helps and good luck!