Given the two equations:

I. $ x^2 = 81 $

II. $ (2y + 3)^2 = 49 $

Determine the correct relationship between $ x $ and $ y $:

Question

Given the two equations:

I. $ x^2 = 81 $

II. $ (2y + 3)^2 = 49 $

Determine the correct relationship between $ x $ and $ y $:

Accepted Answer

**Solution:** 1. Solve equation I: $ $x^2 = 81 $$ - Taking square root on both sides, we get: $$$ x = \pm 9$ $$ 2. Solve equation II: $ $(2y + 3)^2 = 49$ $ - Taking square root on both sides: $$$ 2y + 3 = \pm 7$ $$ - Case 1: $ 2y + 3 = 7 $ $$ $2y = 4 \implies y = 2 $$$ - Case 2: $ 2y + 3 = -7 $ $$ $2y = -10 \implies y = -5 $$$ 3. Possible values: - $ x = 9 $ or $ x = -9 $ - $ y = 2 $ or $ y = -5 $ 4. Compare $ x $ and $ y $ for all combinations: - $ x = 9, y = 2 $ : $ x > y $ - $ x = 9, y = -5 $ : $ x > y $ - $ x = -9, y = 2 $ : $ x < y $ - $ x = -9, y = -5 $ : $ x < y $ Since $ x $ can be both greater than or less than $ y $ depending on values chosen, no definite relation exists. **Answer:** $ x = y $ or no relation between $ x $ & $ y $ (Option d)

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