Why does dividing by zero give us no answer whatsoever? I've heard about this and I know that division can be used in one way like this:
For example, if I want to do $30$ divided by $3$, how many times can I subtract $3$ from $30$ to get to $0$?  Well, I can do it this way: $30-3=27-3=24-3=21-3=18-3=15-3=12-3=9-3=6-3=3-3=0$As you just saw, I subtract $3$ from $30$ 10 times to get to $0$.  So, $30$ divided by $3$ is $10$.  Well, what if I divide by a negative number or what if I divide a negative number by a positive number?  Well, I do know how to solve $15$ divided by $-5$ this way.  Just do it like in the mentioned way, but $15$ will get bigger if I subtract $-5$.  So, what should I do.  That's easy; do it backwards by subtracting the opposite of $-5$, or $5$: $15-5=10-5=5-5=0$.  As you can see, I just did it backwards, and because I did it backwards, I need to subtract the number of times I subtract $5$, since it's the opposite of $-5$.  So, I subtracted $-5$ -3 times.  So, $15$ divided by $-5$ is $-3$.
As you have seen, I have used subtraction while doing division.  Let's cut to the chase now.  What if we divided any number by $0$?  Well, this would happen: If I want to solve $10$ divided by $0$, I would just subtract $0$ from $10$ until I make the $10$ a $0$.  So, $10-0=10-0=10-0=10-0=10-0=10-0=10-0=10...$ and it just keeps going on and on forever.  So, this must explain why any number divided by zero has no answer at all.  Also, for right here, right now, I can say it's infinity since it just goes on and on.  So, is this why any number divided by $0$ never has an answer?  Good answers at the bottom down there, if possible!

The reason division by zero is not defined is simply that it's not possible to define it and still preserve the usual rules of algebra. 
For example we know that if $x$ is a nonzero real number, and if $ax = bx$, then $a = b$. This is called the cancellation law and it follows from the fact that each nonzero real x has a multiplicative inverse $x^{-1}$ with the property that $xx^{-1} = 1$. So if $ax = bx$ then $axx^{-1} = bxx^{-1} \Rightarrow a = b$.
The cancellation law is very handy when we're solving equations and doing other algebraic manipulations, so we want to keep it around.
On the other hand, for example $2(0) = 3(0)$ but $2 \neq 3$. What have we learned? We can not cancel zero. And why is this? It's because zero does not have a multiplicative inverse. That is, there is no real number $x$ with $0x = 1$. That's why we can't cancel it.
Ok no problem, we just make a rule that you can't cancel zero. But if you think about it, what is division in the first place? Division is just another name for the cancellation rule. If you tell me that $x$ is nonzero and $ax = bx$ then I am allowed to conclude that $a = b$. But this is just regular old division.
In other words the cancellation law is division. We can cancel any number that has a multiplicative inverse. Therefore we can not cancel 0, because 0 does not have a multiplicative inverse. So we can't cancel zero ... which is just another way of saying that we can't divide by zero.
