Solve : $2^{x}+2^{y}=12 ; x+y=5, x,y\in \mathbb{R}$

Question

How to solve $2^{x}+2^{y}=12 ; x+y=5, x,y\in \mathbb{R}$?

Attempt:
$2^{x}+2^{y}=12\implies 2^{5-y}+2^y=12\implies 2^y=4,8\implies y=2,3$

Answer 1

$$2^x+2^y=2^{5-y}+2^y=\frac{2^5}{2^y}+2^y=12 $$

$$2^5+2^{2y}=12\cdot2^y \Longleftrightarrow 2^{2y}-12\cdot2^y+2^5=0 $$

$$2^{2y}-12\cdot2^y+2^5=(2^y-2^3)(2^y-2^2) = 0$$

$$2^y=2^3\Longleftrightarrow y=3 $$

$$2^y=2^2 \Longleftrightarrow y=2 $$