Counting atoms
If you code each species in a vector showing the number of atoms of a given type, you can turn the chemical equation into a meaningful mathematical equation. In these vectors, the first value would be the number of hydrogen atoms, the second those of helium, and so forth. For example, the formula of water ($\ce{H2O}$ in case you forgot) would be:
$$ \newcommand\mycolv[1]{\begin{bmatrix}#1\end{bmatrix}}
\mycolv{2\\0\\0\\0\\0\\0\\0\\1\\0\\\vdots} $$
Now, you can multiply these vectors with the stoichiometric coefficients, and check that the sum for the reactants is equal to the sum of the products.
$$1 \mycolv{4\\0\\0\\0\\0\\1\\0\\0\\0\\\vdots} + 2 \mycolv{0\\0\\0\\0\\0\\0\\0\\2\\0\\\vdots} = 1 \mycolv{0\\0\\0\\0\\0\\1\\0\\2\\0\\\vdots} + 2 \mycolv{2\\0\\0\\0\\0\\0\\0\\1\\0\\\vdots}$$
Adding reactions
When you have two reactions, you can add them, using the rules of addition on both sides of the equation. For example, consider reactions (1) and (2):
$$\ce{CO2(g) + H2O(l) <=> H2CO3(aq)}\tag{1}$$
$$\ce{H2CO3 <=> HCO3-(aq) + H+(aq)}\tag{2}$$
If you add these, you get:
$$\ce{CO2(g) + H2O(l) + H2CO3(aq) <=> H2CO3(aq) + HCO3-(aq) + H+(aq)}\tag{1+2}$$
If you cancel the carbonic acid (i.e. substract on both side), you get the net equation:
$$\ce{CO2(g) + H2O(l) <=> HCO3-(aq) + H+(aq)}\tag{1+2}$$
In a similar manner, you get from
$$\ce{H2 + 1/2 O2 -> H2O}\tag{3}$$
to
$$\ce{2 H2 + O2 -> 2 H2O}\tag{3+3}$$
Stoichiometry
While the interpretation above is great to check whether an equation is balanced, I would argue that the most common use of the stoichiometric coefficients is to figure out the stoichiometry of how the amounts of different species change in the course of a reaction. If we translate the chemical equation into a mathematical formula for this purpose, it has a very different structure and the species are not added up. Again, we can use vectors but with the different species as the rows of the vector. For example, we could defined the composition of the reaction mix as:
$$\mathrm{mix} = \mycolv{n_\ce{CH4}\\n_\ce{O2}\\n_\ce{CO2}\\n_\ce{H2O}}$$
And we could encode the stoichiometric coefficients into another vector (notice the negative sign for the reactants):
$$\mathrm{reaction} = \mycolv{-1\\-2\\1\\2}$$
Now we can do math with these objects. If we want to show how the mix changes when 0.3 mol of carbon dioxide are formed, we would write:
$$ \mathrm{mix_{after}} = \mathrm{mix_{before}} + \pu{0.3 mol} \cdot\mathrm{reaction}$$
This is essentially the calculation we would do in an ICE table. From this equation, you can also get the following relationship:
$$ \frac{\Delta n_\ce{CH4}}{-1} = \frac{\Delta n_\ce{O2}}{-2} = \frac{\Delta n_\ce{CO2}}{1}= \frac{\Delta n_\ce{H2O}}{2} = \xi$$
So for stoichiometry, thinking of the plus sign as an addition operation is not so useful (and thinking of the stoichiometric coefficient as a multiplier is not so useful either). Instead, seasoned teachers sometimes translate the equation into "for each methane molecule, two oxygen molecules react, and one carbon dioxide and two water molecules are formed". In effect, the chemical equation defines fixed ratios of species being used up and being made.