Given the law of composition x*y= xy+4mx+2ny, determine m and n if the law is commutative.
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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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
So, for the law to be commutative, we find m = n/2, for any real value of m and n.