Is there a theorem that disproves this or is this just some made up meaningless thing?

Question

I find this slightly funny. I saw this on a meme:$$\begin{align}a=x\\ a+a=a+x\\ 2a=a+x\\ 2a-2x=a+x-2x\\ 2(a-x)=a+x-2x\\ 2(a-x)=a-x\\ 2=1\end{align}$$ How can these strange algebraic manipulations not be true? I feel that this is a stupid question but it's just weird how people come up with these things. Is there some kind of theorem or something that would make this untrue?

Answer 1

Yes there is:

Theorem: If you divide by $0$ you can prove anything ;)

Answer 2

you can't devide two side by $(a-x)$, beacuse $a-x=0$.

Answer 3

You first assumed $a=x$ so the line before the last one is $2(a-a)=a-a$.

Indeed $2\cdot 0 = 0$ but you can not deduce $2=1$ from it since you divided by $a-a=0$.

Answer 4

As soon as you get to the fourth step, your equations are all equivalent to $0 = 0$, and you need to be careful about dividing.