The Correct Answer and Explanation is:
Based on the mathematical problem visible on the screen, the correct answer is:
The expression is undefined when x = y.
In mathematics, a core principle governs fractions and rational expressions: division by zero is undefined. This concept is fundamental because it is impossible to determine a meaningful value when a number is divided by zero. The expression shown, (x^2 – y^2) / (x – y), is a rational expression, which means it behaves like a fraction. Its value depends on the values of the variables x and y.
To determine when this specific expression is undefined, we must focus solely on its denominator, which is the part of the fraction below the division line. In this case, the denominator is (x – y). The entire expression becomes undefined at the exact point where its denominator equals zero.
We can find this point by setting up a simple equation: x – y = 0. Solving this equation for x by adding y to both sides gives us the condition x = y. This means that any time the value of x is identical to the value of y, the denominator becomes zero, and the expression is undefined. For instance, if x = 5 and y = 5, the denominator would be 5 – 5 = 0, making the operation invalid.
It is interesting to note that the numerator, x^2 – y^2, can be factored as a difference of squares into (x – y)(x + y). One might be tempted to simplify the expression by canceling the (x – y) term from both the numerator and denominator, leaving x + y. However, this simplification is only valid when x – y is not equal to zero. The original expression and its simplified form are not perfectly equivalent; the original has a restriction that x cannot equal y, a condition that the simplified form x + y does not have. This restriction is precisely what the question addresses.
