The art of balancing chemical equations is taught very early chemistry degree programs, and understandably so. Correctly balancing a chemical equation is the first step in a great number of chemistry problems including reaction setup, percentage yield determination, and equilibrium constant calculations, among others.
Although the most popular method, "balancing by inspection", works in simple cases, a large number of exceptions and traps makes this method frustrating to learn and difficult to apply to even moderately complex equations. For practical purposes, many equations simply can't be balanced by inspection.
What if there were a systematic method for balancing any chemical equation, regardless of complexity?
A paper published by Lawrence Thorne in 2010 describes such a method. This matrix-based approach balances a large number of equations that can't be balanced by inspection, or even other matrix approaches. An introduction to using this method is given in the video presentation (slidedeck here).
Although operationally simple, Thorne's method does require a lot of arithmetic, which can become tedious. Matrix algebra can be done on many scientific calculators or spreadsheets, but setup requires technical skill and data entry is lengthy at best.