\(\left\{{}\begin{matrix}mx+4y=m+2\\x+my=m\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}mx+4y=m+2\\x=m-my\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m\left(m-my\right)+4y-m-2=0\\x=m-my\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m^2-m^2y+4y-m-2=0\\x=m-my\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(4-m^2\right)y+m^2-m-2=0\left(.\right)\\x=m-my\end{matrix}\right.\)
+ hệ pt có nghiệm duy nhất khi pt (.) có nghiệm duy nhất \(\Rightarrow4-m^2\ne0\Leftrightarrow m^2\ne4\Leftrightarrow m\ne\pm2\)
Với \(m\ne\pm2\), ta có :
\(\left\{{}\begin{matrix}\left(4-m^2\right)y=-m^2+m+2\\x=m-my\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{\left(2-m\right)\left(m+1\right)}{\left(2-m\right)\left(2+m\right)}\\x=m-my\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{m+1}{m+2}\\x=m-my\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{m+1}{m+2}\\x=m-\dfrac{m\left(m+1\right)}{m+2}=\dfrac{m^2+2m-m^2-m}{m+2}=\dfrac{m}{m+2}\end{matrix}\right.\)
+ hệ pt có vô số nghiệm khi pt (.) có vô số nghiệm
\(\Rightarrow\left\{{}\begin{matrix}4-m^2=0\\m^2-m-2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}m=2\\m=-2\end{matrix}\right.\\\left(m-2\right)\left(m+1\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}m=2\\m=-2\end{matrix}\right.\\\left[{}\begin{matrix}m=2\\m=-1\end{matrix}\right.\end{matrix}\right.\)\(\Leftrightarrow m=2\)
+ hệ pt vô nghiệm khi pt (.) vô nghiệm
\(\Rightarrow\left\{{}\begin{matrix}4-m^2=0\\m^2-m-2\ne0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}m=2\\m=-2\end{matrix}\right.\\\left(m-2\right)\left(m+1\right)\ne0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}m=2\\m-2\end{matrix}\right.\\m\ne2\\m\ne-1\end{matrix}\right.\)\(\Leftrightarrow m=-2\)