How is a coefficient used to balance an equation

Let's start to understand this question by a simple combustion reaction involving oxidation of Ethane in the presence of Oxygen. When Ethane is burned in the presence of Oxygen it produces Carbon Dioxide and Water respectively. Therefore, the equation is as,

                                C₂H₆  +  O₂    →    CO₂  +  H₂O

Above reaction shows the reaction and the equation is unbalanced. Balancing chemical equation is important because according to law of conservation of mass, mass can neither be created nor destroyed. Hence, we should balance the number of elements on both side.

                                       LHS                      RHS

Carbon Atoms                  2                            1

Hydrogen Atoms              6                           2

Oxygen Atoms                  2                           3

It means this equation is not obeying the law. Now, how to balance? One way is as follow,

                                C₂H₆  +  O₃    →    C₂O₂  +  H₆O

                                       LHS                      RHS

Carbon Atoms                  2                            2

Hydrogen Atoms              6                           6

Oxygen Atoms                  3                           3

We have balanced the equation by changing the subscripts. But, we have messed up the chemical composition of compounds and molecules like Oxygen is converted into Ozone.

Therefore, we will change the coefficients (moles) to balance the equation as,

                                C₂H₆  +  7/2 O₂    →    2 CO₂  +  3 H₂O

                                       LHS                      RHS

Carbon Atoms                  2                            2

Hydrogen Atoms              6                           6

Oxygen Atoms                  7                           7

Now, by changing the coefficients we have balanced the equation without disturbing the chemical composition of compounds and molecules.