Divide by zero

We can’t. Division has a uniqueness property that would be violated.

$$\frac{a}{c}\ne \frac{b}{c};\text{if}a\ne b$$

Think about this intuitively: if $\frac{a}{0}=x$, where $a\ne 0$ and $x\in \mathbb{R}$, then $a=x\cdot 0$. However, $\forall x\in \mathbb{R}$, $x\cdot 0=0$, so we have a contradiction.