Question

Solve for x and y in
(3/4)^x(2/3)^y=32/27

Answer 1

Rewrite using prime factors:
(3/4)^x(2/3)^y = 3^{x-y} 2^{\,y-2x} = 2^5 3^{-3}.

Equate exponents:
y - 2x = 5
x - y = -3.

Solve: from x - y = -3 ⇒ x = y - 3. Substitute:
y - 2(y - 3) = 5 ⇒ -y + 6 = 5 ⇒ y = 1, hence x = -2.

So x = -2, y = 1.