An equilibrium constant constrains the final composition, while a balanced reaction constrains how concentrations change on the way there. Combining both pieces allows an equilibrium composition to be calculated from initial conditions.

A convenient bookkeeping device is an initial-change-equilibrium table, often called an ICE table.

Consider

$$\mathrm{A\rightleftharpoons B}$$

with

$$K_c=4.0.$$

Suppose initially

$$[\mathrm A]_0=1.00,\mathrm M,\qquad [\mathrm B]_0=0.$$

Because there is initially no B, the system must undergo net forward reaction. Let $x$ be the concentration of A converted to B.

The stoichiometric coefficients determine the relative changes. Substituting the equilibrium concentrations into

$$K_c=\frac{[\mathrm B]}{[\mathrm A]}$$

gives

$$4.0=\frac{x}{1.00-x}.$$

Solving,

$$4.00-4x=x,$$

so

$$x=0.800,\mathrm M.$$

Therefore

$$[\mathrm A]_{eq}=0.200,\mathrm M,$$

$$[\mathrm B]_{eq}=0.800,\mathrm M.$$

A final check gives

$$Q_c=\frac{0.800}{0.200}=4.0=K_c.$$

For a general reaction, each concentration change is the same reaction progress multiplied by its stoichiometric coefficient. A species with coefficient 2 changes by $2x$ when one with coefficient 1 changes by $x$.

More complicated equilibrium equations can produce quadratics or higher-order equations. The conceptual structure remains the same:

1. determine the direction of change if necessary;
2. express all concentration changes using reaction stoichiometry;
3. write equilibrium concentrations in terms of one unknown;
4. impose $Q=K$;
5. solve and reject any mathematical root that gives physically impossible concentrations.

The ICE table is bookkeeping, not an additional chemical law. The chemistry comes from conservation through stoichiometry and the equilibrium condition $Q=K$.