TITLE: How prove $a*b=b*a$
QUESTION [6 upvotes]: let $x,y$ be  any real numbers,define $*$,such
$$x=(x*y)*y=y*(y*x)$$
fo any $a,b$,show that
$$a*b=b*a$$
My try:
$$x=(x*y)*y=y*(y*x)$$
then
$$y=(y*x)*x=x*(x*y)$$
so
$$y*x=x*(x*y)*x$$
and
$$x*y=y*(y*x)*y$$
then I can't,Thank you

REPLY [6 votes]: Start from the result and do the only things you can (multiply by things that make something go away):
$\begin{align}
a\ast b&=b\ast a\\
(a\ast b)\ast a &= b\\
(a\ast b)\ast((a\ast b)\ast a) &= (a\ast b) \ast b\\
a&=a
\end{align}$
Now, read upwards to get a proof (you have to left-multiply by $a\ast b$ to get from line 3 to line 2, and right-multiply by $a$ to get from line 2 to line 1).