\(\left\{{}\begin{matrix}x+y+xy=-1\left(1\right)\\x^2+y^2-xy=7\end{matrix}\right.\)\(\Rightarrow x^2+y^2+x+y=6\)

\(\Leftrightarrow\left(x+y\right)^2-2xy+x+y=6\)

\(\Leftrightarrow xy=\frac{\left(x+y\right)^2+x+y-6}{2}\)

Thay vào (1):\(2x+2y+\left(x+y\right)^2+x+y-6=-2\)

\(\Rightarrow\left[{}\begin{matrix}x+y=1\\x+y=-4\end{matrix}\right.\)\(\Rightarrow\left[{}\begin{matrix}xy=-2\\xy=3\end{matrix}\right.\)

Vậy x,y là nghiệm của pt:\(\left[{}\begin{matrix}X^2-X-2=0\\X^2+4X+3=0\end{matrix}\right.\)

Đến đây tự tìm x,y.

a, Cộng vế theo vế hai phương trình ta được:

\(x^2+y^2+2xy+x+y=2\)

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

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

\(\Leftrightarrow\left[{}\begin{matrix}x+y=1\\x+y=-2\end{matrix}\right.\)

TH1: \(x+y=1\)

\(pt\left(2\right)\Leftrightarrow xy+1=-1\Leftrightarrow xy=-2\)

Ta có hệ: \(\left\{{}\begin{matrix}x+y=1\\xy=-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x+y=1\\xy=-2\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=-1\\y=2\end{matrix}\right.\\\left\{{}\begin{matrix}x=2\\y=-1\end{matrix}\right.\end{matrix}\right.\)

TH2: \(x+y=-2\)

\(pt\left(2\right)\Leftrightarrow xy-2=-1\Leftrightarrow xy=1\)

Ta có hệ: \(\left\{{}\begin{matrix}x+y=-2\\xy=1\end{matrix}\right.\Leftrightarrow x=y=-1\)

b, \(\left\{{}\begin{matrix}x^3-y^3=7\left(x-y\right)\\x^2+y^2=x+y+2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(x-y\right)\left(x^2+y^2+xy-7\right)=0\\x^2+y^2=x+y+2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x=y\\x^2+y^2+xy=7\end{matrix}\right.\\x^2+y^2=x+y+2\end{matrix}\right.\)

TH1: \(\left\{{}\begin{matrix}x=y\\x^2+y^2=x+y+2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=y\\x^2-x-1=0\end{matrix}\right.\)

\(\Leftrightarrow x=y=\dfrac{1\pm\sqrt{5}}{2}\)

TH2: \(\left\{{}\begin{matrix}x^2+y^2+xy=7\\x^2+y^2=x+y+2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(x+y\right)^2-xy=7\\\left(x+y\right)^2-2xy-x-y=2\end{matrix}\right.\)

Đặt \(x+y=u;xy=v\)

Hệ trở thành: \(\left\{{}\begin{matrix}u^2-v=7\\u^2-2v-u=2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}v=u^2-7\\u^2-2\left(u^2-7\right)-u=2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}v=u^2-7\\u^2+u-12=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}v=u^2-7\\\left[{}\begin{matrix}u=3\\u=-4\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}v=2\\u=3\end{matrix}\right.\\\left\{{}\begin{matrix}v=9\\u=-4\end{matrix}\right.\end{matrix}\right.\)

Với \(\left\{{}\begin{matrix}v=2\\u=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}xy=2\\x+y=3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=2\\y=1\end{matrix}\right.\\\left\{{}\begin{matrix}x=1\\y=2\end{matrix}\right.\end{matrix}\right.\)

Với \(\left\{{}\begin{matrix}v=9\\u=-4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}xy=9\\x+y=-4\end{matrix}\right.\left(vn\right)\)

\(\left\{{}\begin{matrix}\left(x-y\right)^2+xy=3\left(x-y\right)\\\left(x-y\right)^2+3xy=7\left(x-y\right)^3\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}3\left(x-y\right)^2+3xy=9\left(x-y\right)\\\left(x-y\right)^2+3xy=7\left(x-y\right)^3\end{matrix}\right.\)

\(\Rightarrow7\left(x-y\right)^3-9\left(x-y\right)=-2\left(x-y\right)^2\)

\(\Leftrightarrow7\left(x-y\right)^3+2\left(x-y\right)^2-9\left(x-y\right)=0\)

\(\Leftrightarrow\left(x-y\right)\left(7\left(x-y\right)^2+2\left(x-y\right)-9\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x-y=0\\x-y=1\\x-y=\dfrac{-9}{7}\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}y=x\\y=x-1\\y=x+\dfrac{9}{7}\end{matrix}\right.\)

TH1: \(y=x\) thay vaò pt đầu:

\(x^2-x^2+x^2=3\left(x-x\right)\Rightarrow x^2=0\Rightarrow\left\{{}\begin{matrix}x=0\\y=0\end{matrix}\right.\)

TH2: \(y=x-1\) thay vào pt đầu:

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

\(\Rightarrow\left[{}\begin{matrix}x=2\Rightarrow y=1\\x=-1\Rightarrow y=-2\end{matrix}\right.\)

TH3: \(y=x+\dfrac{9}{7}\):

\(x^2-x\left(x+\dfrac{9}{7}\right)+\left(x+\dfrac{9}{7}\right)^2=\dfrac{-27}{7}\Leftrightarrow x^2+\dfrac{9}{7}x+\dfrac{270}{49}=0\) (vô nghiệm)

Vậy hệ đã cho có 3 cặp nghiệm:

\(\left(x;y\right)=\left(0;0\right);\left(2;1\right);\left(-1;-2\right)\)

mình chịu lun

Cộng vế với vế:

\(x^2+2xy+y^2+x+y=6\)

\(\Leftrightarrow\left(x+y\right)^2+\left(x+y\right)-6=0\)

\(\Rightarrow\left[{}\begin{matrix}x+y=-3\\x+y=2\end{matrix}\right.\)

TH1: \(\left\{{}\begin{matrix}x+y=-3\\xy=5\end{matrix}\right.\)

Theo Viet đảo, x và y là nghiệm của:

\(t^2+3t+5=0\) (vô nghiệm)

TH2: \(\left\{{}\begin{matrix}x+y=2\\xy=0\end{matrix}\right.\)

Theo Viet đảo, x và y là nghiệm:

\(t^2-2t=0\Rightarrow\left[{}\begin{matrix}t=0\\t=2\end{matrix}\right.\)

\(\Rightarrow\left(x;y\right)=\left(2;0\right);\left(0;2\right)\)

chiu ban oi