Dividing one equation by another equation

It's just like any other operation: if you have an equation that is true, and you do the same thing to both sides, you get another equation that is true.

For example, if we know $a=b$ and $c=d$ are both true, then we know $a+c = b+d$ is true: since $a$ and $b$ are the same thing, we've done the same thing to both sides of the equation $c+d$.

The only thing that differentiates division in this respect is that you have to pay a little more attention that the thing you do is not nonsense. i.e. dividing by $x$ is nonsense if you don't know $x$ is nonzero.

The left-hand side of an equation represents some number (although it might be unknown or undetermined which one). The equals sign $=$ symbolises that the right hand side represents the same number (whichever number it is).

If you form a fraction with the left-hand side of one equation being the numerator, and the left-hand side of another equation as the denominator, the resulting number (still unknown/undetermined) must necessarily be the same as the number formed by the fraction of corresponding right-hand sides. The two fractions must be equal since their numerators are the same, as are their denominators.

We express this "sameness" as we always do in mathematics, by putting an equals sign between them. This forms a new equation, and we say that this new equation is formed by dividing one equation by the other.