Question

Giải hệ phương trình:

x³ + y³ = 1

x⁵ + y⁵ = x² + y²

Answer 1

Do \(x^3+y^3=1\) \(\Rightarrow x^5+y^5=\left(x^2+y^2\right)\left(x^3+y^3\right)\)

\(\Leftrightarrow x^5+y^5=x^5+y^5+x^2y^3+x^3y^2\)

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

\(\Rightarrow\left[{}\begin{matrix}xy=0\\x+y=0\end{matrix}\right.\)

Nếu \(x+y=0\Rightarrow x^3=-y^3\Rightarrow x^3+y^3=0\) ( mâu thuẫn)

Nếu \(xy=0\) \(\Rightarrow x^3+y^3=1\Rightarrow\left(x+y\right)^3=1\Rightarrow x+y=1\)

ta có\(\left\{{}\begin{matrix}xy=0\\x+y=1\end{matrix}\right.\) \(\Rightarrow\left(x,y\right)=\left\{\left(1,0\right);\left(0,1\right)\right\}\)