Solve a system of linear equations to balance the chemical equation Fe + O2 → Fe2O3?

Assign the coefficients x, y and z to the equation, i.e.
x Fe + y O₂ → z Fe₂O₃

Number of Fe atoms : x = 2z ...... [1]
Number of O atoms : 2y = 3z ...... [2]

From [2] : y = 3z/2 ...... [3]

Put [1] and [3] into the ratio of x, y and z, i.e.
x : y : z
= 2z : 3z/2 : z
= 2 : 3/2 : 1
= 2*2 : (3/2)*2 : 1*2
= 4 : 3 : 2 (The simplest integral ratio)

Hence, the balanced equation is :
4 Fe + 3 O₂ → 2 Fe₂O₃