If the equilibrium favours much one compound, you know in advance how much there is of this one, rather accurately. You can deduce the others and check the assumption.
If the assumption were slightly optimistic, you can subtract a bit from the favoured compound and loop the computation once or twice for improved accuracy.
If no compound is much favoured, you have to write the sums (reaction equation) and ratios (equilibrium constants) of all partial pressures and solve for good - by hand for CO2 vs CO, with a software if burning aluminium and polybutadiene in ammonium perchlorate.
For that, you don't want to compute the equilibrium between every pair of possible products, but rather refer the constants of the products to a limited set of compounds. The elements would have seemed a natural set, but they lead to impractical equilibrium constants that may overflow or underflow computer capabilities.