If 27^x+1 = 9^x+3 = 3^y, find the values of x and y.

The correct answer is: For the given equation, x = 3 & y = 12.

Step-by-step solution:-
27^x+1 = 9^x+3 = 3^y
∴ (3³)^x+1 = (3²)^x+3 = 3^y
∴ 3^3x+3 = 3^2x+6 = 3^y .....................…................ [(bⁿ)^m = b^nm]
∴ 3x+3 = 2x+6 = y …............................ (Exponents of the same base in an equation are equal) .................... (Equation i)
Now, From Equation i, we get-
∴ 3x + 3 = 2x+6
∴ 3x - 2x = 6 - 3
∴ x = 3 ......................... (Equation ii)
Substituting Equation ii in Equation i, we get-
2x + 6 = y
∴ 2(3) + 6 = y
∴ 6 + 6 = y
∴ y =12
Final Answer:-
∴ For the given equation, x = 3 & y = 12.