An operation is defined on the set of real numbers, R by x*y = x+y+3xy. If * is commutative, find the value x is an element of R for which no inverse exists
if y is x inverse, then x*y = 1
y = (1-x)/(1+3x)
so x = -1/3 has no inverse
I really don't understand
To find the real number x for which no inverse exists under the operation *, we need to consider the definition of inverse.
In general, an inverse of an element x under an operation * is another element y such that x * y = y * x = identity element, denoted as e.
Since * is commutative (x * y = y * x), we can write the operation as x * y = x + y + 3xy.
To find the inverse, let's set up the equation: x * y = e.
Using the definition of the identity element, we set e = 0 (i.e., the number that doesn't change the value of another number when the operation is applied).
So, we have x * y = x + y + 3xy = 0.
Rearranging the equation, we get: 3xy + x + y = 0.
Now, we can try to solve for y in terms of x.
First, let's rewrite the equation as: (3x + 1)y + x = 0.
Now, we have a linear equation of the form Ay + B = 0, where A = (3x + 1) and B = x.
For the inverse to exist, the coefficient A should not be equal to zero.
So, 3x + 1 ≠ 0.
Solving for x, we get: 3x ≠ -1, x ≠ -1/3.
Therefore, for x = -1/3, no inverse exists under the operation * on the set of real numbers (R).
To find the value of x for which no inverse exists in this operation, we need to determine if there exists an element y such that x*y = y*x = 0.
Let's set up the equation x * y = x + y + 3xy and solve for y:
x * y = x + y + 3xy
x*y - y - 3xy = x
y(x - 1 - 3x) = x
y(-2x - 1) = x
Now we have two cases to consider:
Case 1: y = 0
If y = 0, the equation becomes 0 = x, which means x = 0. In this case, x = 0 has an inverse since any number multiplied by 0 gives 0. Hence, it is not the answer.
Case 2: -2x - 1 = 0
In this case, we have -2x = 1, which gives us x = -1/2. This means that for x = -1/2, y(-2(-1/2) - 1) = -1/2 is not possible to solve. Therefore, x = -1/2 is the value for which no inverse exists in this operation.
In conclusion, the value of x that does not have an inverse in the defined operation * is x = -1/2.