Given the law of composition xyxy2ax2y determine a and b if

Given the law of composition x*y=xy+2ax+2y determine a and b if the law is communicative and associative.

x*y = xy + 2ax + 2y

Since the law is communicative :

==> x*y = y*x

==> x*y = xy + 2ax + 2y.........(1)

==> y*x = yx + 2ay + 2x ........(2)

we know that (1) = (2) :

==> xy + 2ax + 2y = yx + 2ay + 2x

Combin terms with a on the left side:

==> 2ax - 2ay = 2x - 2y

Now factor 2a on left and 2 on right.

==> 2a(x-y) = 2(x-y)

Divide by (x-y) :

==> 2a = 2

==> a = 2/2 = 1

==> a= 1