When more than one equilibrium are established in a vessel at the same time and any one of the reactant or product is common in more than one equilibrium, then the equilibrium concentration of the common species in all the equilibrium would be same.
For example, if we take CaCO3(s) and C(s) together in a vessel of capacity V liter and heat it at temperature T kelvin, then CaCO3 decomposes to CaO(s) and CO2(g). Further, evolved CO2 combines with C(s) to give carbon monoxide. Let the moles of CaCO3 and carbon taken initially be a and b, respectively.
Thus, as CO2 is common in both equilibria, its concentration is same in both the equilibrium constant expressions.
Equilibrium constant for first equilibrium,
Equilibrium constant for second equilibrium,