Will sapphire radeon hd 7750 work in my 537s q8200?

You need a better power supply at least 300watts, your processor might bottleneck the card. You can try it on your current power supply but most probably it won't work.
Also do your PSU has a 6pin connector? It is required by the card.

xttony . i opened it up and found one power line called p1 and it has 22 wires going to motherboard, and one p3 that has 10 wires going to mother board and one that is loose looks like 4 wires but not connected. im 39 yrs old just want to boost gaming on this slim series. works ok now but i know it could be better. i have 2 pci slota and 1 pci express slot.

What exactly is your processor is it a core 2 quad (quad core)? The 4 pin connector is probably a molex pin.

You can first get a HD 7750 and try to run it in your system if your requirements are low you will be good with it.

You do not need a 6 pin power connector for HD 7750 just plug it into the pcie slot and it will start running.
You PSU will be able to handle your card as long as both the card and the processor are not being utilized. You can play games.

Ignore what I said previously.

what do u mean as long as card and processor are not being utilized. i dont know much ,but try to reasearch things.how do i create that environment. card and processor not being utilized
and thank u for your time.
and if i double my ram, is that easy process,go from 4 gb to 8gb
this card should be easy if i get drivers right and settings. fan and such. ive seen posts of others with very similar set up to mine and that card works well. even the 530s series it worked well on.yes i have quad core , only thing inside is hard drive and cd drive pci express card and two fans , i think 4 fans on in psu one on mother board and one on card and one at base i think.

I said that both the card and and processor are being utilized at that same time. This case is rare. You should not face any problems. If you are getting the 7750 only for playing games then it is fine. You cannot multitask while playing games (eg video editting and gaming at the same time).
Ram will be only beneficial if your requirement is more than 4gB or you are using the second card for a dual channel set up.

I accidentally read 7750 as 7770 so I told you to search for the 6pin connector and said the setup won't work.

thanks

The 7750 uses a max of 56w while benchmarking, so your power supply should suffice.