Gauss scheme (need to find the mistake)

[PrfromTexas]

Please, help me find the mistake in the scam answer on forums (they have indicated it in the Gauss scheme):
Multiplying both sides of the last equality by x1, x2, ..., xn. Where (x1, x2, ..., xn) is a solution of the given system, and we will get it together.
Y1 (a11x1 + a12x2 + ... + a1nxn) +
+ y2 (a21x1 + a22x2 + ... + a2nxn) +
.............................. ..
+ yn (an1x1 + an2x2 + ... + annxn) = x1
but the expression in brackets is b1, b2, ..., bn. So
b1y1 + b2y2 + ... + bnyn = x1
the extreme right element of the last equality will be x1. Quite the same, if we took -1 then in the last tape of the second column the extreme element would equal x2 .. The same calculation can be made by taking -1 in any of the first n columns. Consider now a square matrix. Cause I'm not sure if it's a prejudice based result.