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