The binary operation * is defined on the set of R, of real numbers by \(x * y = 3x + 3y - xy, \forall x, y \in R\). Determine, in terms of x, the identity e...

The binary operation * is defined on the set of R, of real numbers by \(x * y = 3x + 3y - xy, \forall x, y \in R\). Determine, in terms of x, the identity element of the operation.

Answer Details

To find the identity element of an operation, we need to find an element that when combined with any other element using the operation, results in that same element. Let's assume that the identity element is "a", so we need to find the value of "a" that satisfies the following equation for any real number "x": x * a = a * x = x Substituting the definition of the operation, we get: x * a = 3x + 3a - ax = x Simplifying this equation by subtracting 3a from both sides, we get: x - 3a = ax Dividing both sides by x, we get: 1 - 3a/x = a Now, we have an expression for "a" in terms of "x". Let's check each option to see which one satisfies the equation for any value of "x". : a = 2x/(x-3), x ≠ 3 Substituting this into our equation, we get: 1 - 3(2x/(x-3))/x = 2x/(x-3) Simplifying this expression, we get: -3x/(x-3) = 0 This equation is not true for all values of "x", since it is only true when x = 0. Therefore, is not the identity element. : a = 2x/(x+3), x ≠ -3 Substituting this into our equation, we get: 1 - 3(2x/(x+3))/x = 2x/(x+3) Simplifying this expression, we get: 3x/(x+3) - 1 = 0 This equation is true for all values of "x", except for x = -3. Therefore, is the identity element. : a = 3x/(x-3), x ≠ 3 Substituting this into our equation, we get: 1 - 3(3x/(x-3))/x = 3x/(x-3) Simplifying this expression, we get: -6x/(x-3) = 0 This equation is not true for all values of "x", since it is only true when x = 0. Therefore, is not the identity element. : a = 3x/(x+3), x ≠ -3 Substituting this into our equation, we get: 1 - 3(3x/(x+3))/x = 3x/(x+3) Simplifying this expression, we get: 6x/(x+3) - 1 = 0 This equation is true for all values of "x", except for x = -3. Therefore, is the identity element. Therefore, the identity element of the operation is: a = 2x/(x+3), x ≠ -3.