Doubt based on IMO Olympiad

Question

Can anyone please tell me the answer of this question?

Answer 1

Correct option is (A) 3, 1

Given: 3^x+y = 81

81 can be written as 3⁴.

⇒ 3^x+y = 3⁴

When Bases are same on both sides, Exponents should be equal.

⇒ x + y = 4    ........(1)

Given: 81^x-y = 3⁸

⇒ (3⁴)^x-y = 3⁸

⇒ 3^4(x-y) = 3⁸

⇒ 4(x - y) = 8

⇒ x - y = 2    ..........(2)

Adding Equations (1) and (2), we get

(x + y) + (x - y) = 4 + 2

⇒ x + y + x - y = 6

⇒ 2x = 6

⇒ x = 3

Substituting x = 3 in Equation (1), we get

3 + y = 4

⇒ y = 4 - 3

⇒ y = 1