There is no Illinois "contract law" per se, it depends on what your contract states. Does it give you the right to rescind? If not, you may be out of luck.
Why are you backing out? Is the car defective? If so, you may be protected by Lemon Laws. Contact an attorney who handles such issues if that is the case.
There is a common misconception that you have a 3 day right to cancel all contracts, but that simply isn't true. There are several Illinois statutes that allow for a 3 day right to cancel period, such as when you purchase merchandise from a door-to-door salesman at your home, or when you purchase a gym membership. There is no such statute for vehicle purchases.
Generally there is no absolute right to cancel a motor vehicle sale unless it is written into your sales contract, but often there are violations of the law in the paperwork, or something wrong done in the sales process by the dealer, and that might well give you the legal right to cancel the deal. But, contrary to what a lot of folks think, there is no 3 day right to cancel (or any other time length) if it isn't written into your sales contract. There is a 3 day right to cancel law (a federal one and a lot of states have a similar one) but it only covers sales that occur at your home such as by door to door sales people, and car dealers aren't that desperate yet. To find out for sure about the law in Illinois, though, you should talk to a local Consumer Law attorney in your state. Call your attorney Bar Association or you can find one on this National List of Consumer Protection Law Lawyers here: http://ohiolemonlaw.com/locate-a-local-attorney.html