Sorry this is a long answer. But remember, after doing the long way for your question we got (5/9)F-(5/9)*32=C. But early on we got F-32=(9/5)C. Well, we could divide both sides by (9/5), then the (9/5) next to the C would become 1, because anything divided by itself is one. But we still need to divide F and -32 by (9/5). But remember what we said in the (1/2)X=1 example? To divide by a fraction, flip the fraction you're dividing by then multiply the flipped number by whatever you were dividing. So F/(9/5)=F*(5/9)=(5/9)*F, and -32/(9/5)=(5/9)*-32, so the left side becomes (5/9)F-(5/9)*32, just like we got by doing it the long way.
