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

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x*y = xy - 3(x+y) + 12

We know that:

x*x' = e

x*e = x

==> x*x' = xx' - 3(x+x') +12

==> e = xx' - 3x - 3x' + 12 ............(1)

==> x*e = ex -3(x + e) + 12

==> x = ex - 3x - 3e + 12

==> 3e - ex = -3x -x + 12

==> e( 3 - x)= -4x + 12

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

**==> e = 4 **

Now substiute in (1):

==> e = xx' - 3x - 3x' + 12

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

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

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

**==> x' = (-3x+8)/(3-x)**

To find the invertible element for: x*y = xy - 3(x+y) + 12 we first need the neutral element.

Now let the neutrla element be N

=> xN - 3( x+ N) + 12 = x

=> xN - 3x - 3N +12 = x

=> xN - 3N = x+ 3x - 12

=> N ( x-3) = 4 ( x- 3)

=> N = 4

Now for the invertible element I

x * I = N = 4

=> xI - 3(x + I) + 12 = 4

=> xI - 3x - 3I + 12 = 4

=> I ( x-3) = 4 - 12 + 3x

=> I (x-3) = 3x - 8

=> I = (3x -8) / (x - 3)

**Therefore the invertible element is (3x - 8) / (x-3).**

x*y =xy-3(x+y) +12.

Before finding the ivertible element we find the identity element (or neutral element).

Let e be the neural element.

Then x*e = x.

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

xe -3x -3e +12 = x

(x-3)e = 3x+x-12 = 4(x-3)

e = 4(x-3)/(x-3) = 4...........(1)

Therefore x is the neural element.

Now now x has an invertible element if x*y = e =4 , as e = 4 for the composition as found at (1).

Or under the compostion, xy -3(x+y)+12 = 4

xy -3x-3y +12 = 4

y(x-3) = 3x-12+4 = 3x -8.

y = (3x-8)/(x-3) is the invertible of x.

Similarly invertible element of y = (3y-8)/(y-3) .

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).**