Question

giai he phuong trinh\(\left\{{}\begin{matrix}x^3+y^3=1\\x^5+y^5=x^2+y^2\end{matrix}\right.\)

Answer 1

\(\left\{{}\begin{matrix}x^3+y^3=^{ }1\left(1\right)\\x^5+y^5=x^2+y^2\left(2\right)\end{matrix}\right.\)

(2)\(\Leftrightarrow x^5-x^2+y^5-y^2=0\)

\(\Leftrightarrow x^2\left(x^3-1\right)+y^2\left(y^3-1\right)=0\)

\(\Leftrightarrow x^2\left(-y\right)^3+y^2\left(-x\right)^3=0\)

\(\Leftrightarrow x^2y^3+y^2x^3=0\)

\(\Leftrightarrow x^2y^2\left(x+y\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\Rightarrow y=1\\y=0\Rightarrow x=1\\x=-y\left(loại\right)\end{matrix}\right.\)