Why does $x$ divided by zero not equal $x$? Why does $x$ divided by zero not equal $x$? After all, $x$ is not being divided by anything.
Think of division as an "inverse" operation to multiplication.

For instance, let us look at $\frac{a}{b}$ means.

Let $x = \frac{a}{b}$. What it means is when $x$ is multiplied by $b$ it gives $a$.

So, if you want to make sense of $\frac{a}{0}$. 

Say $\frac{a}{0}$ is $y$.

So, we want $y$ such that $y \times 0$ gives us $a$.

But we know that $y \times 0 = 0$, $\forall y$.

So there is a logical inconsistency here.
Who says X divided by zero does not equal X?  Let's challenge the conventional thinking (wording) and agree that, yes, if you divide something by nothing, there is no division.  Likewise, if you multiply something by nothing, there is no multiplication.  Don't treat zero as though it were a number, as though it were a part of the continuity of infinitely shrinking numbers less than one.  It's not.  It's a limit those numbers will never reach.  Zero is not another point on a line.  It's a gap.  It comes from the Sanskrit word meaning "empty."

Divide that apple on the table by ... nothing at all, and you've performed no operation.  You're left with the apple, whole and undivided.  Seems perfectly logical. You've done nothing to the apple.  You've done something to the operation, which was to abandon it.  So let's go with that and invite Yuan or another contributor to explain more convincingly why zero isn't the same as nothing.

That's going to be a hard sell, don't you think?  Of course, I like to argue negative numbers don't exist because you can't have less than nothing; you can't take two apples away from one.  But that's another topic.