An operation* is defined by a set of x*y=x+y+3xy as an element of real numbers, find the identity element
If y is the identity element, then x*y = x
x+y+3xy = x
y+3xy = 0
y(3x+1) = 0
y=0
so 0 is the identity element
extra credit: what is the inverse of x?
To find the identity element for the given operation, we need to find a number, let's call it 'e', which satisfies the property that for any real number x, the operation x*e is equal to x.
In other words, we need to find a number 'e' such that:
x * e = x for all real numbers x.
Let's substitute the given definition of the operation into this equation and solve for 'e':
x * e = x + e + 3xe
We want this equation to hold for all real numbers x. To simplify the equation, let's group the 'x' terms together and the 'e' terms together:
x - 3xe + e = x
Now, let's factor out the common factor 'x' from the terms on the left side:
x(1 - 3e) + e = x
For this equation to hold for all real numbers x, the coefficients of 'x' and the constant term 'e' on the left side must be zero. Therefore, we have:
1 - 3e = 0
3e = 1
e = 1/3
So, the identity element for the given operation is 1/3.
To find the identity element for the given operation, we need to find a real number, let's call it "e," that will satisfy the condition:
x * e = x + e + 3xe
Let's find the identity element step-by-step.
Step 1: Substitute "e" into the operation definition.
x * e = x + e + 3xe
Step 2: Combine like terms on the right-hand side.
x * e = (1 + 3x)e + x
Step 3: Set the coefficient of "e" equal to zero.
1 + 3x = 0
Step 4: Solve for "x."
3x = -1
x = -1/3
Step 5: Substitute the value of "x" back into the equation to determine the identity element.
(-1/3) * e = (-1/3) + e + 3(-1/3)e
Simplifying further:
(-1/3) * e = (-1/3) + e - e/3
(-1/3) * e = (-1/3) + 2e/3
To satisfy the equation, the left-hand side must be equal to the right-hand side. This means that the coefficient of "e" on both sides should be equal:
-1/3 = 2e/3
Cross-multiplying:
-1 * 3 = 2e
-3 = 2e
e = -3/2
Therefore, the identity element for the given operation is -3/2.