If 2\(^{x + y}\) = 16 and 4\(^{x - y} = \frac{1}{32}\), find the values of x and y.
A. x = \(\frac{3}{4}\), y = \(\frac{11}{4}\) B. x = \(\frac{3}{4}\), y = \(\frac{13}{4}\) C. x = \(\frac{2}{3}\), y = \(\frac{4}{5}\) D. x = \(\frac{2}{3}\), y = \(\frac{13}{4}\)
Correct Answer: B
Explanation
2\(^{x + y}\) = 16 ; 4\(^{x - y}\) = \(\frac{1}{32}\). \(\implies 2^{x + y} = 2^4\) \(x + y = 4 ... (1)\) \(2^{2(x - y)} = 2^{-5} \) \(2^{2x - 2y} = 2^{-5}\) \(\implies 2x - 2y = -5 ... (2)\) Solving the equations (1) and (2) simultaneously, we have x = \(\frac{3}{4}\) and y = \(\frac{13}{4}\)
