Xét hàm: $f(t)=2^t+t$.
Ta có: $f'(t)=2^t\ln2+1>0,\forall t>0$
Suy ra $f(t)$ đồng biến trên khoảng $(0;+\infty)$.
Xét hàm: $g(t)=2^t+\log_2t$.
Ta có: $g'(t)=2^t\ln2+\dfrac{1}{t\ln2}>0,\forall t>0$
Suy ra $g(t)$ đồng biến trên khoảng $(0;+\infty)$.
Ta có:
$\left\{\begin{array}{l}2^x+x=y+\log_2y\\\log_2x+y=5\end{array}\right.$
$\Leftrightarrow \left\{\begin{array}{l}f(x)=f(\log_2y)\\\log_2x+y=5\end{array}\right.$
$\Leftrightarrow \left\{\begin{array}{l}x=\log_2y\\\log_2x+y=5\end{array}\right.$
$\Leftrightarrow \left\{\begin{array}{l}y=2^x\\\log_2x+2^x=5\end{array}\right.$
$\Leftrightarrow \left\{\begin{array}{l}y=2^x\\g(x)=g(2)\end{array}\right.$
$\Leftrightarrow \left\{\begin{array}{l}x=2\\y=4\end{array}\right.$