I hate to hear that. I have a much older sis so my nephew is fairly close to me in age. Even as a young child his art (drawing freeform, painting, etc.) was amazing. He was a quiet kid who came alive when he drew. Even through high school he made a small amount of money doing adds for local events, etc. When it came time for college, she would only pay for a trade school to "study something practical you can support yourself with." He did that and did make decent money but was always miserable. He gave up his dream to go to an art college, even graphic arts which wasn't much more than a trade school. There was no real reason they couldn't afford it, she just thought it was "a waste of money since jobs are hard to find in the field." It broke my heart as I don't think anyone w/ a God given talent should pass on it. All of us are not blessed with a talent- any talent.
So, you need to talk to your mom. The economy IS really bad and she or your parents may truly not be able to afford the cost of dance classes. That is why it is important for parents to not keep their kids in the dark totally about the household finances. You don't say how old you are, are you old enough to contribute to it in any way? If so, get a part time job, even an evening or two should cover it.
At the very least, since you absolutely hate science olympiad, that is money she is truly wasting. If you gave up band and that, would that cover the dance costs? Talk to her. I can't imagine after 10 years of dance your mom would take that away unless finances truly are dire. Be adult about it, meaning don't get mad. Tell her what you did us, "Dance is my everything." and that the science olympiad makes you miserable, whereas dance helps you with your emotions. That means you truly love it and it is great for you physically. List that as a pro. The only con seems to be the cost, so offer to get a job or even to do more chores, depending on your age. Dropping the science thing, adding a bit of money of your own should make dance more affordable. Do it when it is just the two of you and she isn't tired or rushed.
Also, how much do you love it? Enough to call it your birthday and Christmas gifts, etc. and stick to meaning/ saying that? Becoming an adult means you sometimes have to compromise and pick your priorities. If dance is a priority to you, which it sounds like it is, you may have to give up some other things so your mom can afford it.
Good luck to you.
