UPDATE--

I think DH and I have come to a consensus. We have decided NOT to punish DD16 for failing algebra 2, even though we did tell her in 9th grade that dance would have to go if she failed classes. It doesn't matter whether she was the only one who failed or if everyone failed--she did her best and it wasn't good enough. It doesn't make good sense to punish her for not understanding algebra. I don't understand algebra either, but nobody has punished me for it.
I talked to her guidance counselor today and she concurred that the online class would probably work for DD given that she has good self-management skills. She was also concerned about the idea of giving up dance because DD plans to major in dance and theater. So I felt better after talking with her.
Here's the plan: Dance will continue. It is her life. She needs to stick with it least for the next 1-1/2 years until she goes to college. For her, dance education is probably as important as algebra, Latin, and lit.
DD is now signed up for the online class. The program requires that she log in at least 4 times a week for at least 2 hours a day. She can work on it 7 days a week on her own schedule. She has to meet a few "live chat" classes for test credit. DD16 understands and agrees with this.
DD as offered to help pay for the class. She has $2000 in her savings. I think I will let her do that, but if she does well I'll replace the money at the end.
DDis not allowed to get a part-time job other than pet-sitting within the neighborhood.
And the piece d'resistance--DD will be allowed to attend a sleepover once a month only. She has 4 BFFs and they spend a lot of time together; it isn't unusual for them to sleep over at one of their homes on Friday or Saturday nights. I think this will be stimulus enough to keep her on track and a much better choice than removing her from dance. She also has a new boyfriend. We'll probably limit them to one date/week this semester. I think thats plenty, since they see each other every day in school.
DD is on-board with this program. She was relieved that we didn't take away her dance, show choir, and spring musical performances

(we had not even mentioned it, but she had already imagined the worst. )