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

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We have to determine a and b given that for x*y = xy + 2ax + by, the * operator is commutative.

x*y = y*x

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

=> 2ax + by = 2ay + bx

equating the coefficients of x and y we get

2a = b

**If the numbers a and b are related as 2a = b, then * 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.**