Given the law of composition x*y= xy+4mx+2ny, determine m and n if the law is commutative.

We'll write the commutative property of a law of
composition:

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 + 4mx + 2ny
(1)

y*x = yx + 4my + 2nx
(2)

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

xy + 4mx + 2ny = yx + 4my +
2nx

We'll remove like
terms:

4mx + 2ny = 4my +
2nx

We'll move the terms in "m" to the left side and the
terms in "n" to the right side:

4mx - 4my = 2nx  -
2ny

We'll factorize and we'll
get:

4m(x-y) = 2n(x-y)

We'll
divide by x - y:

4m = 2n

m =
2n/4

m=n/2

So,
for the law to be commutative, we find m = n/2, for any real value of m and
n.