Consider the following unbalanced reaction:
$$\ce{XeF4 + H2O -> Xe + XeO2 + HF + O2}$$
There are six variables to be solved, but only four equations linking them. I solved the set of equations using Gauss elimination and arbitrarily set the stoichiometric coefficient of $\ce{XeF4}$ as $3$:
$$\ce{3XeF4 + 6H2O -> $(4k)$Xe + $(3-4k)$XeO2 + 12HF + $(4k)$O2}$$
where $k$ is the free variable, the parameter, with $0<k<\frac 34$ (since coefficients must be positive).
Clearly, for different values of $k$, it gives radically different yet balanced equations. The very yields are different for fixed amount of $\ce{XeF4}$ and $\ce{H2O}$. This issue has been noted in a previous question.
My textbook, however, gives the balanced equation:
$$\ce{6XeF4 + 12H2O -> 4Xe + 2XeO2 + 24HF + 4O2}$$
clearly setting $k=\frac 12$. The linked question above does not offer any explanation for why a particular value of $k$ should be favoured. So, my exact question is: Why set $k=\frac 12$? Why was $\frac 12$ preferred over any other value for $k$:
- Is it determined by experimentally measuring the stoichiometry?
- Is there a theoretical explanation for it?
