Balancing the reaction of potassium permanganate and sulfuric acid with hydrogen peroxide A problem on redox reactions invites the reader to balance the actual reaction from which the redox simplification arose. After some hunting I found the balanced equation and attempted to re-derive the coefficients using:

$$\ce{a H_2SO_4 + b KMnO_4 + cH_2O_2 -&gt; dK_2SO_4 + eMnSO_4 + fH_2O + gO}$$

We can quickly derive some valid relations by looking at each element:

$$
\begin{array}{cll}\\
\ce{H}&amp;\quad a+c &amp;=&amp; f\\
\ce{O}&amp;\quad 4a+4b+2c&amp;=&amp; 4d+4e+f+g\\
\ce{S}&amp;\quad a&amp;=&amp; d+e\\
\ce{K}&amp;\quad b&amp;=&amp; 2d\\
\ce{Mn}&amp;\quad b&amp;=&amp; e
\end{array}
$$

This can be massaged a little but because there are 5 equations and 7 unknowns it seems that the system is underdetermined. 

If we let $a=3,~b=2,~c=5,~d=1,~e=2,~f=8,~g=10$ the equations all work. 

My question is: 


  Could we arrive at these values without guessing or actually doing an experiment to measure (say) the oxygen evolved? 


FWIW, I notice an old paper - J. Am. Chem. Soc. 1889, 11 (6), 94-98 - of which I can only see the first page seems to take a similar approach and gets two possible equations, maybe for the reason above.
It seems to me that this is simply a redox reaction in the presence of acid.  

If you write the 2 half reactions, and balance in the presence of acid (sulfuric acid), you end up with the same answer (with 5 O2 instead of 10 O on the product side).  

This seems to be more of a mathematical manipulation rather than a chemistry problem, IMHO.