The answer to 'supply line won't flush' is that the most common causes are a clogged supply line, a faulty flush valve, or a problem with the toilet tank. Symptoms include the toilet not flushing properly, slow or incomplete flushing, or the tank not refilling after a flush.
Some solutions you can try at home include checking for clogs in the supply line, cleaning the flush valve, and checking the tank fill mechanism. However, if the issue persists or you're not comfortable making repairs, it's best to call a professional plumber to diagnose and fix the problem.