# mathCalculate a and b if x*y = y*x and (x*y)*z = x*(y*z) x*y = xy + 2ax + 2by .

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If the law has the property x*y = y*x, then the law is commutative.

We'll write the law of composition for x*y:

x*y = xy+2ax+2by (1)

y*x = yx + 2ay + 2bx (2)

We'll put (1) = (2):

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

We'll eliminate like terms:

2ax+2by = 2ay + 2bx

The coefficients of x from both sides have to be equal:

2a = 2b

We'll divide by 2:

a = b

If the law has the property (x*y)*z = x*(y*z), the law is associative:

(x*y)*z = x*(y*z)

(xy+2ax+2by)*z = x*(yz+2ay+2bz) (3)

But a = b and we'll re-write (3):

(xy+2ax+2ay)*z = x*(yz+2ay+2az)

(xy+2ax+2ay)z + 2a(xy+2ax+2ay) + 2z = x(yz+2ay+2az) + 2ax + 2(yz+2ay+2az)

We'll remove the brackets:

xyz + 2axz + 2ayz + 2axy + 4a^2x + 4a^2y + 2z = xyz + 2axy + 2axz + 2ax + 2yz + 4ay + 4az

We'll eliminate like terms:

xyz + 2axz + 2ayz + 2axy + 4a^2x + 4a^2y + 2z = xyz + 2axy + 2axz + 2ax + 2yz + 4ay + 4az

Since the law is associative, the correspondent coefficients from both sides:

2a = 2

We'll divide by 2:

a = 1

Since a = b, b =1, too.

The law of composition is determined and it's expression is:

x*y = xy+2x+2y