A binary operation $\oplus$ om real numbers is defined by x $\oplus$ y = xy + x + y for two real numbers x and y. Find the value of 3 $\oplus$ - $\frac{2}{3}$.

1. Analyze the Question

We are given a binary operation defined as $x \oplus y = xy + x + y$. We need to evaluate this operation for the specific values $x = 3$ and $y = -\frac{2}{3}$.

2. Explain the Concept

A binary operation is simply a rule that tells you how to combine two numbers to get a third number. In this case, the rule is:

• Multiply the two numbers ($xy$)
• Add the first number ($x$)
• Add the second number ($y$)

To solve this, we substitute the given values into the formula and follow the BODMAS rule (multiplication before addition).

3. Show the Solution

Given: $x \oplus y = xy + x + y$ Substitute $x = 3$ and $y = -\frac{2}{3}$:

$3 \oplus (-\frac{2}{3}) = (3 \times -\frac{2}{3}) + 3 + (-\frac{2}{3})$

Step 1: Perform the multiplication $3 \times (-\frac{2}{3}) = -2$

Step 2: Substitute back into the expression $-2 + 3 - \frac{2}{3}$

Step 3: Simplify the addition $(-2 + 3) - \frac{2}{3}$ $= 1 - \frac{2}{3}$

Step 4: Subtract the fraction To subtract, convert $1$ to a fraction with a denominator of $3$: $\frac{3}{3} - \frac{2}{3} = \frac{1}{3}$

4. Identify the Answer

Comparing our result to the given options: A. $-\frac{1}{2}$ B. $\frac{1}{3}$ C. $-1$ D. $2$

The calculated value matches option B.

5. Tip

When dealing with binary operations involving fractions, always simplify the multiplication part first. It often results in a whole number that makes the subsequent addition or subtraction much easier to handle. Don't rush to find a common denominator until you have simplified the entire expression as much as possible!