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English spelling doesn't match sound, it's about sound

European is (depending on exact dialect) /ˌjoː.ɹəˈpɪ.jan/, so it begins with a consonant. So you don't need "an"

Using "Next"

So true! I can also confirm it happens cross-linguistically

Hey, we have the same profile picture!

Surely the US government won't like that if they're US citizens, right?

that can’t stop giving her money

well… I mean… you don't have to

For any a, b, c, if a = b and b = c, then a = c, right? The transitive property of equality.
For any a, b, x, if a = b, then x + a = x + b. The substitution property.
By combining both of these properties, for any a, b, x, y, if a = b and y = b + x, it follows that b + x = a + x and y = a + x.

In our example, a is x' (notice the ') and b is 0.999… (by definition). y is 10x' and x is 9. Let's fill in the values.

If x' = 0.9999… (true by definition) and 10x = 0.999… + 9 (true by algebraic manipulation), then 0.999… + 9 = x' + 9 and 10x' = x' + 9.

if you are rearranging algebra you have to do the exact same thing on both sides

If you actually change any of the sides. Since, after substitution, the numeric value doesn't change (literally the definition of equality), I don't have to do anything – as I'm not rearranging. I'm merely presenting the same value in an equivalent manner. By contrast, when multiplying both sides by 10, since multiplication by 10 changes the concrete numeric value, I have to do it on both sides to maintain the equality relation (ditto for subtracting x'). But substitution never changes a numeric value – only rearranges what we already know.

(Edit)

Take the following simple system of equations.

5y = 3
x + y = 6

How would you solve it? Here's how I would:

\begin{gather*} %% Ignore the LaTeX boilerplate, just so I could render it
\begin{cases}
y = \frac{3}{5} \\ % Isolate y by dividing both sides by 5
x = 6 - y % Subtract y from both sides
\end{cases} \\
x = 6 - \frac{3}{5} \\ % SUBSTITUTE 3/5 for y
x = 5.4 \\
(x, y) = (5.4, 0.6)
\end{gather*}

Here's how Microsoft Math Solver would do it.

The substitution property of equality is a part of its definition; you can substitute anywhere.

The choice is obvious to you. Not to the average American.

[…] until we wake up and Kemi Badenoch doesn’t exist anymore […]

Just an unfortunate choice of words. It would have been easy to foresee how this could be interpreted. That doesn't mean she's right, though.

Also, does anyone have a proper definition of optics (in this context)? I've heard this word a lot, but I still don't quite understand it

Similarly, 1/3 = 0.3333…
So 3 times 1/3 = 0.9999… but also 3/3 = 1

Another nice one:

Let x = 0.9999… (multiply both sides by 10)
10x = 9.99999… (substitute 0.9999… = x)
10x = 9 + x (subtract x from both sides)
9x = 9 (divide both sides by 9)
x = 1

North Korea: 316 downloads

Interesting that an Israeli newspaper provides a more balanced report than US outlets… how did that happen?