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

justaguide | Certified Educator

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If an operation % is commutative it implies that x%y = y%x.

As * is commutative for x*y = xy + 4mx + 2ny, we have:

xy + 4mx +...

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giorgiana1976 | Student

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.

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