a/ \(\left\{{}\begin{matrix}x+xy^2=1-y^2\\y+x^2y=1-x^2\end{matrix}\right.\)
\(\Rightarrow x-y+xy^2-x^2y=x^2-y^2\)
\(\Leftrightarrow\left(x-y\right)-xy\left(x-y\right)=\left(x-y\right)\left(x+y\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x=y\\1-xy=x+y\end{matrix}\right.\)
TH1: \(x=y\Rightarrow x+x^3=1-x^2\Leftrightarrow x^3+x^2+x-1=0????\)
TH2: \(1-xy=x+y\Leftrightarrow1-x=y\left(x+1\right)\Rightarrow y=\frac{1-x}{1+x}\)
\(\Rightarrow\frac{1-x}{1+x}=\frac{1-x^2}{1+x^2}\Leftrightarrow\left(1-x\right)\left(1+x^2\right)=\left(1-x^2\right)\left(1+x\right)\)
\(\Rightarrow...\)
b/ ĐKXĐ: \(x;y\ge0\)
Đặt \(\left\{{}\begin{matrix}\sqrt{x}=a\\\sqrt{y}=b\end{matrix}\right.\)
\(\left\{{}\begin{matrix}a^4+a=2b^2\\b^4+b=2a^2\end{matrix}\right.\) \(\Rightarrow a^4-b^4+a-b=2\left(b^2-a^2\right)\)
\(\Leftrightarrow\left(a-b\right)\left(a+b\right)\left(a^2+b^2\right)+\left(a-b\right)+\left(a-b\right)\left(2a+2b\right)=0\)
\(\Leftrightarrow\left(a-b\right)\left[\left(a+b\right)\left(a^2+b^2\right)+1+2a+2b\right]=0\)
\(\Rightarrow a-b=0\Rightarrow a=b\)
\(\Rightarrow a^4+a=2a^2\)
\(\Leftrightarrow a\left(a^3-2a+1\right)=0\)
\(\Leftrightarrow a\left(a-1\right)\left(a^2+a-1\right)=0\)
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