Kerson Huang, Statistical Mechanics problem 7.9

A dilute mixture of $H_2$ and $O_2$ gases is kept at constant temperature T. Initially the
density of $H_2$ was $n_0$, the density of $O_2$ was $n_o/2$, and there was no $H_2O$ present. After a
certain time, the mixture becomes an equilibrium mixture of $H_2, O_2,$ and $H_2O$. Find the
equilibrium densities of the three components $n_1, n_2, n_3$, as a function of $T$ and $n_0$.

My take:

$frac{dN_i}{nu_i}$ is independent of $i$ lead to the two conversation laws:

$n_1-2n_2=0$ –(i)

$n_1+n_3=n_0$ –(ii)

Using chemical potential , $mu =kTlog(lambda^3n)$ and equilibrium condition, $sum_imu_inu_i=0$

$2log(lambda^3_1n_1)+log(lambda^3_2n_2)-2log(lambda^3_3n_3)$

Upon simplification I get

$2c^3n_1^2n_2=n_3^2$ –(iii)

where $c$ is constant with temperature $T$.

Now i am trying to solve eq (i),(ii) and (iii) to obtain relations $n_1/n_o,n_2/n_o$ and $n_3/n_0$ but the equations (from wolfromalpha) are long and complex which I am not sure are correct.

Any help would be appreciated.