Validity of rationale for balancing chemical equations
When you write the equation

$$\ce{H2 + O2 -&gt; H2O}$$

you are implicitly saying that the stoichiometric coefficient of every substance is $1$. It is like writing the equation $a + b = c$. By not having coefficients in front of $a$, $b$, or $c$, you are implying that $1a + 1b = 1c$ and not $2a + 3b = 4c$ or anything like that.

Obviously that cannot be true for the reaction above because if you react one $\ce{H2}$ molecule and one $\ce{O2}$ molecule, you get one $\ce{H2O}$ molecule and one leftover oxygen atom that doesn't know what to do with itself. The only balanced equation that does not lead to stray atoms is

$$\ce{2H2 + O2 -&gt; 2H2O}$$

or any integer multiple of that, i.e.

$$\ce{$(2n)$H2 + $(n)$O2 -&gt; $(2n)$H2O}$$

Humans prefer to work with small integers or half-integers, so $n$ is usually either chosen to be $1$ or $1/2$.