What is the invertible element of the law of composition? x*y = xy - 3(x+y) + 12

To determine the invertible element, we'll have to
determine first the neutral elemnt. Let's write the property of the invertible element
to see why:

x * x' = x'*x =
e

So, it is necessary to calculate the neutral
element.

We'll write the property of the neutral
element:

x*e = x

 x*e = xe -
3(x+e) + 12

But x*e = x

xe -
3(x+e) + 12 = x

We'll remove the
brackets:

xe - 3x - 3e + 12 =
x

We'll combine the elements that contain
e:

e(x-3) - 3x + 12 = x

We'll
subtract -3x+12 both sides:

e(x-3) = x + 3x -
12

e(x-3) = 4x - 12

We'll
factorize by 4 to the right side:

e(x-3) =
4(x-3)

We'll divide by
(x-3):

The neutral element is e =
4.

Now, we can calculate the invertible
element:

x * x' = e

xx' -
3(x+x') + 12 = 4

We'll remove the
brackets:

xx' - 3x - 3x' + 12 =
4

We'll isolate the elements that contain x' to the left
side:

xx' - 3x' = 3x - 12 +
4

We'll factorize by x' to the left side and we'll combine
like terms to the right side:

x'(x-3) = 3x -
8

We'll divide by (x-3):

x' =
(3x - 8) / (x-3)

The invertible element is x'
= (3x - 8) / (x-3).