Why must a chemical equation be balanced to solve stoichiometry problems?
Because the chemical equation is the shorthand, or map if you like, of complex molecular rearrangement, which will always conserve mass.
Take the oxidation of hexanes,
I can represent its combustion reaction simply:
This reaction scheme tells me that for each 86 g of hexanes I burn, I am going to get approx. 264 g carbon dioxide as a by-product. This reaction scheme tells me a little bit more than this, in that this combustion will release a certain and measurable amount of energy, which I could also put in the reaction scheme (as an amount in Joules!). If the equations are not balanced correctly (or stoichiometrically) I can make no such predictions.
Please note that you practise such stoichiometry all the time; for instance, when you buy stuff at the supermarket; the value of the goods must equal the value of money that you give to the shop. These amounts must be stoichiometrically balanced, otherwise someone has been ripped off; make sure it's not you!