\(\left\{{}\begin{matrix}x-\left(\sqrt{2}+1\right)y=2\\\left(\sqrt{2}+1\right)x-2y=3+4\sqrt{2}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(\sqrt{2}+1\right)x-\left(\sqrt{2}+1\right)^2y=2\left(\sqrt{2}+1\right)\\\left(\sqrt{2}+1\right)x-2y=3+4\sqrt{2}\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}-2y+\left(\sqrt{2}+1\right)^2y=3+4\sqrt{2}-2\sqrt{2}-2\\x-\left(\sqrt{2}+1\right)y=2\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}-2y+\left(3+2\sqrt{2}\right)y=1+2\sqrt{2}\\x-\left(\sqrt{2}+1\right)y=2\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}y\left(-2+3+2\sqrt{2}\right)=1+2\sqrt{2}\\x-\left(\sqrt{2}+1\right)y=2\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}\left(1+2\sqrt{2}\right)y=1+2\sqrt{2}\\x-\left(\sqrt{2}+1\right)y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{1+2\sqrt{2}}{1+2\sqrt{2}}=1\\x-\left(\sqrt{2}+1\right)=2\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}y=1\\x=2+\sqrt{2}+1=3+\sqrt{2}\end{matrix}\right.\)
