<p>Let \(\log_3 x = A\) and \(\log_3 y = B\). If \(\dfrac{A}{2}+\dfrac{B}{3}=\dfrac{7}{2}\) and \(\dfrac{A}{3}+\dfrac{B}{2}=\dfrac{2}{3}\), then which of the following are correct?</p>
Step-by-Step Solution
Key Concept: Convert the linear system in A and B into a standard algebraic equation, then use logarithm properties (x = 3^A, y = 3^B) to find x and y values, checking which statements hold true.
<p><strong>Step 1: Set up the system of linear equations</strong></p><p>Given: A/2 + B/3 = 7/2 ... (1)</p><p>A/3 + B/2 = 2/3 ... (2)</p><p><strong>Step 2: Solve for A and B</strong></p><p>Multiply (1) by 6: 3A + 2B = 21 ... (1')</p><p>Multiply (2) by 6: 2A + 3B = 4 ... (2')</p><p>From (1'): 3A + 2B = 21</p><p>From (2'): 2A + 3B = 4</p><p>Multiply (1') by 3: 9A + 6B = 63</p><p>Multiply (2') by 2: 4A + 6B = 8</p><p>Subtract: 5A = 55 → <strong>A = 11</strong></p><p>Substitute in (2'): 2(11) + 3B = 4 → 22 + 3B = 4 → <strong>B = -6</strong></p><p><strong>Step 3: Convert back to x and y</strong></p><p>Since log₃ x = A = 11 → <strong>x = 3¹¹</strong></p><p>Since log₃ y = B = -6 → <strong>y = 3⁻⁶ = 1/3⁶ = 1/729</strong></p><p><strong>Step 4: Verify answer options</strong></p><p>Option A: x = 3¹¹ ✓ (CORRECT)</p><p>Option D: y = 1/729 ✓ (CORRECT)</p><p>∴ Answer: A, D</p>
Correct Answer: A, D