At the result, if you get $0=-6$, does that mean that there is no solution. If you get $5=5$ or $6=6$, does that mean any value of $x$ (besides those values that are restricted) are solutions?
Yes. When solving linear equations and you obtain a false result, such as $4=5$ or $0=-6$, that means there are no solutions. If the result is always true, i.e. $5=5$ then your linear equations have infinitely many solutions.
$y = x + 5$ and $x = y - 5$ which are the same equations and thus have infinitely many solutions.
Adding these two together:
$y + x = x + y + 5 - 5$
Of course, everything cancels so that $0=0$, but you can also leave the $5$'s and that shows you $5 = 5$