# Determine the neutral element of the law of composition x*y = 3xy + 6(x+y) + 10

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x*y = 3xy + 6(x+y) + 10

We know that:

x*e = e*x = x

==> x*e = 3*x*e + 6(x+e) + 10

==> x = 3xe + 6x + 6e + 10

==> -6e - 3xe = 10 + 6x - x

==> -3e(2+ x) = 10 + 5x

==: -3e(2+x) = 5(2+x)

==> -3e = 5

**==> e= -5/3**

We have to determine the neutral element for x*y = 3xy + 6(x+y) + 10

To find this, let the neutral element be N, now x*N = e*N = x

From the given expression:

x*N = 3xN + 6(x+N) + 10

As this is also equal to x

=>3xN + 6(x+N) + 10 = x

=> 3xN + 6x + 6N + 10 = x

=> 3xN + 6N = x - 6x - 10

=> 3xN + 6N = -5x - 10

=> 3N ( x + 2) = -5x - 10

=> 3N ( x + 2) = -5 (x + 2)

=> 3N = -5

=> N = -5/3

**So the neutral element is -5 /3.**

To determine the neutral element we'll write it's property:

x*e = e*x = x

Now, we'll apply the law of composition for

x*e = 3xe + 6(x+e) + 10

But x*e = x

3xe + 6(x+e) + 10 = x

We'll remove the brackets:

3xe + 6x + 6e + 10 = x

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

3xe + 6e = x - 6x - 10

We'll factorize by 3e to the left side and we'll combine like terms to the right side:

3e(x + 2) = -5x - 10

We'll factorize by -5 to the right side:

3e(x + 2) = -5(x+2)

We'll divide by (x+2) and we'll get:

3e = -5

**The neutral element is e = -5/3.**

x*y =3xy+6(x+y)+10.

Let us say the neutral element is e.

An element e is neutral f x*e = ex = x.

Then x*e = 3xe+6(x+e)+10 = x ..........(1)

3xe + 6(x+e) +10 = x

3xe +6x+6e +10 = x

3(x+2)e = x-6x-10 = -5(x+2).

e = -5(x+2)/3(x+2) = -5/3.

Now let us consider e*x = 3ex+6(e+x) 10 = x..........(2)

Comparing (1) and (2) , Both LHS and RHS are same.

So e = -5/3 is the neutral element.