Determine the numbers a,b if the law of composition x*y=xy+2ax+by is commutative.

If a law of composition is commutative, that means that
x*y = y*x, for any value of x and y.

We'll substitute x*y
and y*x by the given expression:

x*y = xy + 2ax + by
(1)

y*x = yx + 2ay + bx
(2)

We'll put (1) = (2) and we'll
get:

xy + 2ax + by = yx + 2ay +
bx

We'll remove like
terms:

2ax + by = 2ay +
bx

We'll move the terms in a to the left side and the terms
in b to the right side:

2ax - 2ay = bx  -
by

We'll factorize and we'll
get:

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

We'll
divide by x - y:

2a = b

a =
b/2

So, for the law to be commutative, we
find a = b/2, for any value of a and b.