In class, my professor went over a method for finding the net ionic equation without determining the balanced equation (e.g., if finding the balanced equation was too difficult for us students). It involved writing the ions of each of the solutions being mixed and determining which ions would be spectator ions.
For example, finding the net ionic equation for the following mixture of solutions:
$\ce{NaC_2H_3O_3 + BaCl_2 +NH_4NO_3 + K_2CrO_4}$
Unfortunately, I don't quite grasp the process. How does one determine which ions and compounds will remain in the net ionic equation?