Correct Option
The given equation is x - y = 8. We need to determine which of the provided statements must be true for all possible values of x and y satisfying this equation.
Upon evaluating each statement, it is found that none of them are universally true, as demonstrated by counter-examples:
- Statement 1: "Both x and y must be positive for any value of x and y." This is false. For instance, if x = 5, then y = 5 - 8 = -3. Here, x is positive, but y is negative. Also, if x = -1, then y = -1 - 8 = -9, where both x and y are negative.
- Statement 2: "If x is positive, y must be negative for any value of x and y." This is false. For example, if x = 10, then y = 10 - 8 = 2. Here, both x and y are positive.
- Statement 3: "If x is negative, y must be positive for any value of x and y." This is false. For instance, if x = -1, then y = -1 - 8 = -9. Here, both x and y are negative.
Since statements 1, 2, and 3 are not necessarily true, the correct option is that neither 1 nor 2 nor 3 must be true.
Incorrect Options
Statement 1: Both x and y must be positive for any value of x and y.
This statement is incorrect. Consider the case where x = 5. From the equation x - y = 8, we get y = 5 - 8 = -3. Here, x is positive, but y is negative. Therefore, it is not mandatory for both x and y to be positive.
Statement 2: If x is positive, y must be negative for any value of x and y.
This statement is incorrect. Let x = 10. According to the equation, y = 10 - 8 = 2. In this instance, x is positive and y is also positive. This contradicts the assertion that y must be negative when x is positive.
Statement 3: If x is negative, y must be positive for any value of x and y.
This statement is incorrect. Let x = -1. From the equation, y = -1 - 8 = -9. Here, both x and y are negative. This disproves the claim that y must be positive when x is negative.