If $x - y = 8$, then which of the following must be true? I. Both $x$ and $y$ are positive. II. If $x$ is positive, $y$ must be positive. III. If $x$ is negative, $y$ must be negative.

### Concept & Logic This problem tests the constraints of linear equations and the relative behavior of positive and negative integers across an equality. Rearranging the equation to isolate one variable helps expose these constraints. ### Step-by-Step Solution * **Given:** $x - y = 8$. * **Deduction:** Rearrange the equation to isolate $y$: $y = x - 8$. * Let's evaluate the three statements: * **Statement I:** "Both $x$ and $y$ are positive." Let $x = 4$. Then $y = 4 - 8 = -4$. Here, $y$ is negative. Thus, Statement I does not *have* to be true. * **Statement II:** "If $x$ is positive, $y$ must be positive." Using the same example above, if $x = 4$ (positive), $y = -4$ (negative). Thus, Statement II is false. * **Statement III:** "If $x$ is negative, $y$ must be negative." If $x$ is any negative number (e.g., $x = -2$), then substituting it yields $y = -2 - 8 = -10$. Since subtracting $8$ from a negative number always results in a number further down the negative number line, $y$ will always be negative. Thus, Statement III is true. ### Exam Strategy & Shortcut **Extreme Value Testing:** Do not attempt abstract reasoning when simple number plugging takes seconds. Pick an extreme positive number ($x = 10 \implies y = 2$), a small positive number ($x = 2 \implies y = -6$), and a negative number ($x = -2 \implies y = -10$). Comparing these three quick test cases instantly eliminates Statements I and II, leaving only III. ### Common Pitfall Assuming that because the difference ($8$) is positive, both numbers must be positive. Remember that subtracting a negative number (e.g., $5 - (-3) = 8$) can also yield a positive difference. ### Final Answer Therefore, the correct answer is **III only**.