re: What are some ways to get out of paying child support?

alimony is for support of an ex-spouse regardless of children. Maintain standard of living she was accustomed to and other factors help determine it.
One simple example: One spouse works to put another through med school. Spouse becomes a neurosurgeon and they divorce, then the ex may get regular alimony payments for some period of time

Ex's can get alimony for a period even without children. With kids, tHey can get alimony AND child support too. Also, different tax rules apply to each