What is the best Linux system fro my Vista, Home Premium PC?
It is a HP Model a6317c, Intel Premium Dual cpu e2200, @ 2.20 GHz, memory 3 gb, 32 bit operating sysytem
Answer to the question
I have a similar desktop computer. I did upgrade the memory on mine. According to http://support.hp.com/us-en/product/HP-Pavilion-a6000-Desktop-PC-series/3644673/model/3660281/document/c01312592/, yours supports up to 8GB of memory.
I use Linux Lite 2.8 64 bit on my HP Pavilion a6110n. The performance is so good, I sometimes forget the computer is nearly 9 years old. You can find out about Linux Lite at www.linuxliteos.com.
By the way, you can get 8GB of DDR2 memory on ebay for about $15 including shipping.