\(\left\{{}\begin{matrix}\sqrt{x+y}+\sqrt{x-y}=4\left(1\right)\\x^2+y^2=128\left(2\right)\end{matrix}\right.\)
\(\left(1\right)\Rightarrow2x+2\sqrt{x^2-y^2}=16\Leftrightarrow x+\sqrt{x^2-y^2}=8\Leftrightarrow\sqrt{x^2-y^2}=8-x\)
\(\Leftrightarrow\left\{{}\begin{matrix}8-x\ge0\\x^2-y^2=\left(8-x\right)^2\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x\le8\\y^2=16x-64\left(3\right)\end{matrix}\right.\)
\(\left(2\right)\left(3\right)\Rightarrow x^2+16x-64-128=0\Leftrightarrow\left[{}\begin{matrix}x=8\left(tm\right)\\x=-24\left(tm\right)\end{matrix}\right.\)
\(x=8\Rightarrow y^2=16.8-64=64\left(tm\right)\Rightarrow y=\pm8\)
\(x=-24\Rightarrow y^2=16\left(-24\right)-64=-448< 0\left(ktm\right)\)
\(\Rightarrow\left(x;y\right)=\left\{\left(8;-8\right);\left(8;8\right)\right\}\)
