A bit to tired to build a PC right now but here's some advice.
Give your monitor resolution because that hugely affects performance.
If your playing - to be honest- there is no 550$ dollar build that will play BF3 on medium at 1080p even at - unless your ok with 30FPS on PC at 1080p.
. So if your playing at 1920x1080p resolution expect to be playing on low at maybe 40 FPS max. By your price ceiling i can see that your not looking to spend a lot of money, but BF3 is a very hard game to run, especially with low standard gear. So if I were to give advice I'd say this:
1) Save up, or just BEG mommy&daddy -- no offense intended-- for more money to buy better parts, 700 would be a very secure place, looking at your situation.
no offense intended.
2) Use a 1280x720 resoltion monitor if your going to use 550$. A lower resolution decreases the amount of computer resources a game will have to utilize because it's not powering as many pixels, making it easier to play with lower-tier equipment, which is what you will most likely end up buying at 550$
3) If you really can't go any higher - either just choose a less computer intensive game to play(there are a lot of fun games out there that don't tak expensive parts to play) or really just buy a console and sacrifice quality for affordability. You can buy a 200$ xbox and a decent sized 1080p Tv and still play BF3 - although the graphics and performance will be much lower than PC standards, but it will still be very playable.
Conclusion - Try and get more money if you can, the PC experience is worth every penny !
Good Luck!