Fair enough. What's the method using simultaneous equations? I like to apply algebra to these things.
I haven't tried but my idea was something like:
a *(C + 0.5 O2 -CO ) +b *( CO + 0.5 O2 -CO2) + .= 4 CO + 8H2 -3 CH4 -CO2 -2 H2O
now on left side collect terms containing CO, O2, H2 etc.
Eventually the expression for CO equates to 4, for H2 8 and so on.
For O2 etc. equate to zero since they are absent in the target. You should get 7 equations in 7 unknowns (if my calculations are right!)
That ought to give you all potential solutions.
Again, I haven't tried it but since you seem so interested try it out and let me know. I hope your enthusiasm can motivate you to do the number crunching?