I have a problem with a roommate. I pay all of the rent. If I sue her and she can't pay will the court evict her?

No. Suing her (you don't even say what the problem is - she is supposed to pay rent?) is separate from eviction proceedings. Eviction follows state law, and in some cases, local law, so ask at your courthouse. Give her written notice to vacate the premises for non payment as a start.

Need more info. Any question on law needs your general locale as laws vary by area. Are you both on the lease? Or is she leasing from you. Is the only problem non payment?

But first you give her a pay or quit notice. Then you file with the courts for an eviction order if she doesn't leave. THEN you sue her for what she owes.

What is the paying agreement? In writing? Is she on the lease agreement?

If the roommate is a co-tenant with OP, only the landlord can sue to evict her, not the OP. In other words, while OP and the co-tenant are jointly and severally liable for the rent, OP's only remedy would be to sue the co-tenant for rent (in excess of OP's agreed share, presumably half) paid by the OP.