Balancing by oxidation numbers method multiple atoms
In my Chemistry textbook, the rules for balancing a chemical equation using the Oxidation-Number method are as follows:
1. Assign oxidation numbers to all atoms in the equation.
2. Identify the atoms that are oxidized and the atoms that are reduced.
3. Determine the change in oxidation number for the atoms that are oxidized and for the atoms that are reduced.
4. Make the change in oxidation numbers equal in magnitude by adjusting coefficients in the equation.
5. If necessary, use the conventional method to balance the remainder of the equation.
I understand how to do this for simple equations, such as $$SnCl_4+Fe\rightarrow SnCl_2+FeCl_3$$ Following the steps:
1. $Sn^{4+}+4Cl^-+Fe^0\rightarrow Sn^{2+}+2Cl^-+Fe^{3+}+3Cl^-$
2. 979
oxidation: $Fe^0\rightarrow Fe^{3+}$ (Fe); reduction: $Sn^{4+}\rightarrow Sn^{+2}$ (Sn)
3. oxidation: +3; reduction -2
4. $3SnCl_4+2Fe\rightarrow 3SnCl_2+2FeCl_3$
5. (no change necessary, already balanced)
The Problem: Notice how Sn and Fe (the reduced and oxidized atoms) all have subscripts of 1. However, when their subscripts differ, I'm not sure how to balance it.
For example, I can easily balance the following equations by inspection, but not so easily by this method of oxidation numbers:
$$KClO_3\rightarrow KCl+O_2$$ (O is oxidized, subscripts of 3 and 2)
$$NH_3+NO_2\rightarrow N_2+H_2O$$ (N is oxidized and reduced, oxidation numbers of -3, +2, and 0)
And this gets even more complicated when the subscripts differ and there are more than two elements whose oxidation numbers changed (are reduced or oxidized), e.g.,
$$NH_4ClO_4+Al\rightarrow Al_2O_3+HCl+N_2+H_2O$$ (Al and N are oxidized, Cl is reduced)
For all of these, I get stuck on the fourth step. Determining oxidation numbers and calculating the change is not the problem, but figuring out how to do it with larger or different coefficients as in the examples above is the problem.
The five general steps in my textbook don't help me understand this method much with more complicated methods such as this. Could you give me a more thorough explanation— a better explanation or a general rule— of how to solve all of these? Thanks.
FYI: I am not asking you to do my homework, but this is a homework question this meta answer. I have already done the questions via inspection but want to learn how to use this alternative method. 979