???

Not sure what you’re aiming for. It proves that the setup works, I suppose.

x = 0.555…

10x = 5.555…

10x = 5 + 0.555…

10x = 5+x

9x = 5

x = 5/9

5/9 = 0.555…

So it shows that this approach will indeed provide a result for x that matches what x is supposed to be.

Hopefully it helped?