Giải hệ phương trình
1. \(\left\{{}\begin{matrix}x^2+y^2+2x+2y=\left(x+2\right)\left(y+2\right)\\\left(\frac{x}{y+2}\right)^2+\left(\frac{y}{x+2}\right)^2=1\end{matrix}\right.\)
2. \(\left\{{}\begin{matrix}x^2-2xy-6=6y+2x\\\frac{3x^2}{y+1}=4-x\end{matrix}\right.\)
3.\(\left\{{}\begin{matrix}x^2-y=y^2-x\\x^2-x=y+3\end{matrix}\right.\)
4.\(\left\{{}\begin{matrix}x+y+\frac{1}{x}+\frac{1}{y}=\frac{9}{2}\\xy+\frac{1}{xy}+\frac{x}{y}+\frac{y}{x}=5\end{matrix}\right.\)
6.\(\left\{{}\begin{matrix}x^3\left(x-y\right)+x^2y^2=1\\x^2\left(xy+3\right)-3xy=3\end{matrix}\right.\)
7.\(\left\{{}\begin{matrix}x^2+3y-6x=0\\9x^2-6xy^2+y^4-3y+9=0\end{matrix}\right.\)
8.\(\left\{{}\begin{matrix}x^2+y^2+xy=1\\x+y-xy=2y^2-x^2\end{matrix}\right.\)
9.\(\left\{{}\begin{matrix}8x^3-y=y^3-2x\\x^2+y^2=x+2y\end{matrix}\right.\)
10.\(\left\{{}\begin{matrix}2x^2-3xy+y^2+x-y=0\\x^2+x+1=y^2\end{matrix}\right.\)
11.\(\left\{{}\begin{matrix}\left(x^2+y^2\right)\left(x+y+2\right)=4\left(y+2\right)\\x^2+y^2+\left(y+2\right)\left(x+y+2\right)=4\left(y+2\right)\end{matrix}\right.\)
12. \(\left\{{}\begin{matrix}x^2+7=4y^2+4y\\x^2+3xy+2y^2+x+y=0\end{matrix}\right.\)
13. \(\left\{{}\begin{matrix}x^2+y^2=5\\x^3+2y^3+\left(x-5\right)^2+\left(y+5\right)^2=55\end{matrix}\right.\)
14. \(\left\{{}\begin{matrix}\frac{1}{x^2}+\frac{1}{y^2}=3+x^2y^2\\\frac{1}{x^3}+\frac{1}{y^3}+3=x^3y^3\end{matrix}\right.\)
15.\(\left\{{}\begin{matrix}x^2+y^2+4x+2y=3\\x^2+7y^2-4xy+6y=13\end{matrix}\right.\)
16. \(\left\{{}\begin{matrix}x^2-5xy+x-5y^2=42\\7xy+6y^2+42=x\end{matrix}\right.\)
17.\(\left\{{}\begin{matrix}x^2+xy+y^2=13\\x^4+x^2y^2+y^4=91\end{matrix}\right.\)
18.\(\left\{{}\begin{matrix}x^2=\left(2-y\right)\left(2+y\right)\\2x^3=\left(x+y\right)\left(4-xy\right)\end{matrix}\right.\)
Đây là các bài hệ trong đề thi chuyên toán mong mọi người giúp vì mình bận quá nên không thể làm hết được ạ
1) \(\left\{{}\begin{matrix}x+y=1\\x^3+y^3=x^2+y^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=1\\\left(x+y\right)\left(x^2-xy+y^2\right)-\left(x^2+y^2\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=1\\x^2-xy+y^2-x^2-y^2=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=1\\xy=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=1\\\left[{}\begin{matrix}x=0\\y=0\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x+y=1\\x=0\end{matrix}\right.\\\left\{{}\begin{matrix}x+y=1\\y=0\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=0\\y=1\end{matrix}\right.\\\left\{{}\begin{matrix}x=1\\y=0\end{matrix}\right.\end{matrix}\right.\)
2) \(\left\{{}\begin{matrix}x^2+y^2=5\\x^4-x^2y^2+y^4=13\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+y^2=5\\\left(x^2+y^2\right)^2-3x^2y^2=13\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+y^2=5\\\left(5\right)^2-3x^2y^2=13\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+y^2=5\\-3x^2y^2=-12\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2=5-y^2\\x^2y^2=4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2=5-y^2\\\left(5-y^2\right)y^2=4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2=5-y^2\\-y^4+5y^2-4=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2=5-y^2\\-y^4+5y^2-4=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2=5-y^2\\\left[{}\begin{matrix}y^2=1\\y^2=4\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x^2=5-y^2\\y^2=1\end{matrix}\right.\\\left\{{}\begin{matrix}x^2=5-y^2\\y^2=4\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x^2=4\\y^2=1\end{matrix}\right.\\\left\{{}\begin{matrix}x^2=1\\y^2=4\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}\left[{}\begin{matrix}x=2\\x=-2\end{matrix}\right.\\\left[{}\begin{matrix}y=1\\y=-1\end{matrix}\right.\end{matrix}\right.\\\left\{{}\begin{matrix}\left[{}\begin{matrix}x=1\\x=-1\end{matrix}\right.\\\left[{}\begin{matrix}y=2\\y=-2\end{matrix}\right.\end{matrix}\right.\end{matrix}\right.\)
vậy \(S=\left\{\left(2;1\right),\left(2;-1\right),\left(-2;1\right),\left(-2;-1\right),\left(1;2\right),\left(1;-2\right),\left(-1;2\right),\left(-1;-2\right)\right\}\)
4) \(\left\{{}\begin{matrix}x^5+y^5=1\\x^9+y^9=x^4+y^4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^5-1=-y^5\\x^9-x^4+y^9-y^4=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left\{{}\begin{matrix}x^5-1=-y^5\\y^5-1=-x^5\end{matrix}\right.\\x^4\left(x^5-1\right)+y^4\left(y^5-1\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^5+y^5=1\\x^4\left(-y^5\right)+y^4\left(-x^5\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^5+y^5=1\\-x^4y^4\left(x+y\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^5+y^5=1\\\left[{}\begin{matrix}x=0\\y=0\\x+y=0\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x^5+y^5=1\\x=0\end{matrix}\right.\\\left\{{}\begin{matrix}x^5+y^5=1\\y=0\end{matrix}\right.\\\left\{{}\begin{matrix}x^5+y^5=1\\x=-y\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=1\\y=0\end{matrix}\right.\\\left\{{}\begin{matrix}x=0\\y=1\end{matrix}\right.\\\left\{{}\begin{matrix}x=-y\\x^5-x^5=1\end{matrix}\right.\end{matrix}\right.\)
vậy \(S=\left\{\left(1;0\right),\left(0;1\right)\right\}\)