Balancing chemical reactions, issues of Gauss's elimination method I am trying to balance some chemical reactions by using the Gauss' elimination method and I noticed that this method does not work if the creation of a matrix with exactly one degree of freedom is not possible.

In this example

$$\ce{H2O2 + NO3- + H+ → O2 + NO + H2O}$$ 

I succeeded in creating just 4 equations

$$\ce{\alpha H2O2 + \beta NO3- + \gamma H+ → \delta O_2 + \epsilon NO + \zeta H2O}$$

$$
\left\{
\begin{aligned}
2\alpha+\gamma&amp;=2\zeta &amp;\qquad &amp;(\ce{H})\\
2\alpha+3\beta&amp;=2\delta+\epsilon+\zeta &amp;\qquad &amp;(\ce{O})\\
\beta&amp;=\epsilon &amp;\qquad &amp;(\ce{N})\\
-\beta+\gamma&amp;=0 &amp;\qquad &amp;(\text{electrons})
\end{aligned}
\right.
$$

obtaining

$$(\alpha, \beta, \gamma, \delta, \epsilon, \zeta)=\left(\zeta-\frac{1}{2}\epsilon, \epsilon,\epsilon,\frac{\zeta+\epsilon}{2}, \epsilon, \zeta\right)$$

I know that the solution should be $(3,2,2,3,2,4)$, but, in having two degrees of freedom, I am not able to choose the two free variables in order to obtain what I expect, i.e. $\epsilon=2$ and $\zeta=4$.

Is there any (just one more) equation that I missed?
Is there anything else that I have to add further?
Or do I have to consider that Gauss' elimination fails with problems with a number of free variables $\neq 1$?
Your "should-be" solution,

$$\ce{3H2O2 + 2NO3- + 2H+ → 3O2 + 2NO + 4H2O}$$

is a linear combination of

$$\ce{H2O2 + 2NO3- + 2H+ → 2O2 + 2NO + 2H2O}$$

and 

$$\ce{2H2O2  → O2 +  2H2O}$$

Any other linear combination is valid as well.