In certain problems an extent of reaction balance method is simply not practical (or even possible). This can happen if the number of total reactions (and therefore necessary extent variables) is large, or if we don't even know exactly what reactions take place (for example, when burning coal or some other complex material).
In this case, there is another technique which proves to be quite simple for solving reactive balances.
An atomic balance is one in which we balance each individual atomic species rather than each independent molecular species (i.e., instead of balancing water we note that it contains two H's and one O, and balance those).
How does our differential material balance simplify in this case?! Note that performing an atomic species balance is much like balancing a stoichiometric expression.
Both reactive balance methods should always give you the same answer! We simply learn two because sometimes one is easier than the other!
Obviously our degrees of freedom analysis will change relative to the extent of reaction balances:
Perform atomic species balances
Re-do the yield example and make certain an atomic balance gives the same answer