Agreed. Westwood prices are extremely high. ([Apartment prices in LA in general are very expensive](Every Single Part of Los Angeles is Unaffordable on $15/Hour - Curbed LA), for that matter.) They’re aware they can charge students higher prices thanks to location, and they do. Most undergrads don’t know any better or simply prefer living in Westwood; either way, they pay as much for a shared room in Westwood as they could for a private bedroom a little further from campus.
Rent is cheaper west of the 405 in Sawtelle/Little Osaka or south of Wilshire in Palms. Splitting a 2 bedroom and getting your own room would run about $10-11K per year for a 12 month lease. For a shared bedroom in west LA, you could be looking at ~$8000 or a little under per year. Subletting your apartment in the summer would save 2-3 months of rent. The important thing is to start looking early – competition for affordable housing on the west side is completely insane. (It’s not unusual for 30 or 40 people to show up and apply for a nice apartment, and apartments on Craigslist can be taken in a couple of hours.)
Grocery bills aren’t bad if you shop around for deals, and good cheap produce is especially easy to find. Lots of farmers’ markets around, including a pretty decent one in Westwood.
Becoming an RA would get you room and board. It’s competitive but definitely an option to consider.