Nope you don't know basic mathematics.
10x=9.999...
x = 9,999.../10
End of story.

You cannot add x during solving by yourself.

This, to me, is an example of the idea that accepted math rules automatically equate to reality. Does .999 = 1 when you use these rules? Yes. What does that actually mean? That .999 = 1 when you use these rules. Nothing more, nothing less.

How do you actually use numbers that go on into infinity in calculations? You don't. You use representations of those number, then you pretend those representations are the same as the original concept.

The main thing this proves is that the decimal system is not perfect.

I'm not adding x. 9.999 repeating is the same as 9 + 0.999... repeating, so 9.999.... is the same as 9 + x

So that does not disprove the proof (alternatively, you could use the other proof either way, since I provided two).

The problem is that you don't understand you cannot replace 0,999.... with x during the solution.

x is the number you are looking it is unknown.

As said, this shows a bad comprehension of the concept of infinity. A brief resume of Hilbert's Hotel: https://www.youtube.com/watch?v=faQBrAQ87l4

The concept of infinity also implies that no matter which coma place you take, there is still an infinity of coma places to follow. You never actually come closer to the end, however you may transform your current coma number (in normal mathematical operations). You constantly stay at 0% progress towards reaching infinity.

Beyond that, the equality 

So this works with every number then? The math problem that you showed at the beginning, works with any number? Btw, why didn't you add Maff to the poll?

Fine, if you don't believe that proof (and it is correct), then how about the second proof. What's wrong with that?

x isn't unknown, because I've defined x to be 0.999...

"If x=0.999... then"

0.999999 = 1 is wrong.

0.9999999.... = 1 is true. That 0.0000001 part doesn't exist with an infinity of 9.

It is basic mathematics, you can learn it in school (okay, it's more often teached as a "fun fact" in college or engineering schools), it's really well-known and established. 

You don't know how to prove that it is bad maths because it's not. You can find flaws in a demonstration of 0.99999.... = 1 but you will always find other ways to prove it, but not the contrary.

Example (I did'n't read the whole subject, sorry if already mentioned) :

1/3 = 0.333333333333...

3*(1/3) = 3*0.3333333333....

3*(1/3) = 1

3*0.3333333333333.... = 0.999999999999...

So 0.9999999999... = 1

And in the second example you made mistake check it better.