When you balance chemical equations you are showing mathematically that the number of each type of atom is the same on both sides of the equation. This is necessary because chemical reactions obey the Law of Conservation of Mass. No mass is gained or lost, the atoms are just rearranged.
If the number of each type of atom isn't the same on each side, we add coefficients which are multipliers that change the quantitites represented. The subscripts in the chemical formulas can't be changed because they represent the ratio of atoms in each molecule.
As an example, here's the unbalanced equation for the combustion of methane:
`CH_4 + O_2 -> H_2O + CO_2`
Atoms on the reactant side are:
1 carbon, 4 hydrogen, 2 oxygen
On the product side we have:
1 carbon, 2 hydrogen, 3 oxygen
1. Adding a coefficient of 2 to `H_2O ` on the product side gives 4 H and 4 O. Hydrogen is now balanced but oxygen isn't.
`CH_4 + O_2 -> 2 H_2O + CO_2`
2. Adding a coefficient of 2 to `O_2` on the reactant side give 4 O, so oxygen is now balanced.
`CH_4 + 2O_2 -> 2 H_2O + CO_2`
3. Double checking shows the same number of atoms on both sides:
1 carbon, 4 hydrogen and 4 oxygen.
Balance uncombined elements last because changing their coefficients doesn't change the amount of any other atoms.
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