The key trick to balance an equation that may involve, for example, a Mn compound undergoing a change in oxidation states from the left to the right side of the equation, is to focus on the fact that the same # of Mn atoms must appear, in total, on both side of the equation. The adjustment factor to the Mn compound, for this to occur, should be somewhat apparent (for example, going from 2 atoms to 3, requires 3 and 2 coefficients to equate each side at 6). Then use this information to assist in computing the required # of oxygen and say hydrogen atoms appearing outside of the Mn compound to arrive at a total balanced equation. This can be may easier, at times, by also noting that each side of the equation's total charges must also be equal (if the sum +/- charges is +1 on one side, it must match the other). Note, that an electron has a charge of -1 and two electrons -2, etc. Also, atom count and charge balance are both required (albeit, the latter often not emphasized as often naturally occurring) for a valid equation, in accord with the preservation of matter and energy.
Do your best to follow the teacher's rule, however, the last rule "The reaction is impossible to balance by manipulating coefficients" requires adjusting in practice.
If the actual resulting mathematics indicates answers for each coefficient that are unexpectedly large (irrational), present the results as a raw unbalanced equation. Clearly, some inputs are missing or not applicable for the reaction you are trying to balance (that is, to correctly explain).
Not surprising, there is a problem with universal rules for balancing chemical reactions, as all reactions are not transparent as to how precisely the observed products were actually formed.
This is evident, for example, when the experimentally estimated coefficients are decimal amounts. Evidently, the suggested reaction equation may a summary of several random occurring more elemental reactions (usually, these can be radical reactions, which are reversible and have temperature sensitive forward and backward reaction rates).
A larger problem arises when the reaction occurs in an open system with say the amount of oxygen from air placed into the system is not known. In case again of say fractional weights and some inputs unknown, how does one present results?
My recommendation is honest reporting, as an unbalanced equation, or otherwise, the ability to speculate on true reaction pathways could be lose under a perception of false certainty.
Per examination of the literature, this problem is also resolved by stating "+ other products". The latter is acceptable, in my opinion, as an honest statement of lack of knowledge as to underlying mechanics and/or product estimate accuracy issues.
Hope this helps!