I recently finished an antibiotic about a week ago. how long does it stay in your system and how long can it affect my birth control?

Elimination issue: That depends on the antibiotic. Usually we say that it takes 5 half lives of the antibiotic to get out of the body. More importantly, the effect on your normal gut bacteria; can be altered > 1 month depending on how broad spectrum the antibiotic. Flora helps ocp absorption. It would be safest to at least use a secondary mode of protection for a month after you complete your course of antibiotic.