Finding moles of different compounds using balancing
Never having heard of the Principle of Atom Conservation (POAC) before I'm curious. 

Looking up the concept, I understand it simply to mean that atoms can't be created or destroyed in a chemical reaction. (So we're ignoring radioactive decay for example.) I start blindly with a variable for each reactant and product, and will then look for linear relationships in the variables. 

$$\ce{$a$\,K2Cr2O7 + $b$\,H2SO4 + $c$\,SO2 -&gt; $d$\,Cr2(SO4)3 + $e$\,K2SO4 + $f$\,H2O}$$ 

So with $a$ $\ce{K2Cr2O7}$ molecules, I must get $a$ $\ce{Cr2(SO4)3}$ molecules to balance $\ce{Cr}$, and $a$ $\ce{K2SO4}$ molecules to balance $\ce{K}$. So: 

$$\ce{$a$\,K2Cr2O7 + $b$\,H2SO4 + $c$\,SO2 -&gt; $a$\,Cr2(SO4)3 + $a$\,K2SO4 + $f$\,H2O}$$ 

Now $\ce{H}$ is only in $\ce{H2SO4}$ on the left and $\ce{H2O}$ on the right so $f=b$.

$$\ce{$a$\,K2Cr2O7 + $b$\,H2SO4 + $c$\,SO2 -&gt; $a$\,Cr2(SO4)3 + $a$\,K2SO4 + $b$\,H2O}$$ 

To balance $\ce{S}$:


  $$b + c = 4a$$


To balance $\ce{O}$: 


  $$7a + 4b + 2c = 16a + b$$ which reduces to $$3b +2c = 9a$$


At this point just balancing atoms leaves you with 2 linear equations and 3 unknowns. So there are not enough equations to explicitly solve the problem using linear equations. However relationships can be established. 

\begin{array}{cccc}\\
3b&amp; +2c &amp; = &amp; 9a \\
-2(b&amp; +c &amp; = &amp; 4a)\\ \hline
 &amp; b &amp; = &amp;a \\
\end{array}

$\therefore c = 3a = 3b$

The final step in this problem is to realize that an integer solution (a Diophantine problem in mathematics) is desired. So $a=b=1$ and thus $c=3$. So the final chemical equation is:

$$\ce{K2Cr2O7 + H2SO4 + 3SO2 -&gt;Cr2(SO4)3 + K2SO4 + H2O}$$



PS - Many thanks to user Mathew Mahindaratne for suggesting a cleaner solution in the comments.