Let a, b, and c be the coefficients in the balanced reaction:
a Fe + b Cl₂ → c FeCl₃
Count the atoms that show up among the reactants and products. They should be equal on both sides:
Fe : a = c
Cl : 2b = 3c
Let c = 2. Then a = 2 and 2b = 6, so b = 3.
Then the balanced reaction is
2 Fe + 3 Cl₂ → 2 FeCl₃
