If $\sqrt{(x-1)(y+2)} = 7$, $x$ and $y$ being positive whole numbers, then the values of $x$ and $y$ respectively are

Concept & Logic Squaring both sides of a radical equation transforms it into a standard algebraic equation. By evaluating the integer factors of the resulting constant, we can easily test the given whole-number options without complex algebraic isolation. Step-by-Step Solution * Given equation: $\sqrt{(x-1)(y+2)} = 7$ * Square both sides to eliminate the square root: $(x-1)(y+2) = 49$ * Since $x$ and $y$ are positive whole numbers, the terms $(x-1)$ and $(y+2)$ must be positive integer pairs that multiply to $49$. The prime factorization of $49$ is $7 \times 7$ (or $1 \times 49$). * Test Option (a) where $x = 8$ and $y = 5$: * Substitute the values into the factors: $(8 - 1)(5 + 2) = (7)(7) = 49$. * This perfectly matches our equation, making it the correct pair. Exam Strategy & Shortcut For equations with multiple variables and explicitly given options, Option Elimination (Back-solving) is the fastest approach. Simply plug the pairs $(x, y)$ from the options into the original equation $\sqrt{(x-1)(y+2)} = 7$. Testing (a) yields $\sqrt{(8-1)(5+2)} = \sqrt{7 \times 7} = \sqrt{49} = 7$. The problem is solved in seconds. Common Pitfall Students often try to algebraically isolate $x$ or $y$ (e.g., writing $x = \frac{49}{y+2} + 1$), which wastes time and complicates the process when the variables are restricted to whole numbers and options are readily provided to test. Final Answer Therefore, the correct answer is 8, 5.