\(\left\{{}\begin{matrix}mx-y=2m\\4x-my=m+6\end{matrix}\right.\)
Để hpt vô nghiệm thì: \(\dfrac{m}{4}=\dfrac{-1}{-m}\ne\dfrac{2m}{m+6}\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{m}{4}=\dfrac{1}{m}\\\dfrac{1}{m}\ne\dfrac{2m}{m+6}\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}m^2=4\\2m^2-m-6\ne0\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}m=\pm2\\m\ne2;m\ne-\dfrac{3}{2}\end{matrix}\right.\\ \Leftrightarrow m=-2\)
Để hpt có vô số nghiệm thì:
\(\dfrac{m}{4}=\dfrac{-1}{-m}=\dfrac{2m}{m+6}\\ \Leftrightarrow\left\{{}\begin{matrix}\dfrac{m}{4}=\dfrac{1}{m}\\\dfrac{1}{m}=\dfrac{2m}{m+6}\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}m^2=4\\2m^2-m-6=0\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}m=\pm2\\m=-\dfrac{3}{2};m=2\end{matrix}\right.\\ \Leftrightarrow m=2\)
Để hpt có nghiệm thì \(\dfrac{m}{4}\ne\dfrac{1}{m}\Leftrightarrow m\ne\pm2\)
\(\left\{{}\begin{matrix}mx-y=2m\\4x-my=m+6\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}m^2x-my=2m^2\\4x-my=m+6\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}mx-y=2m\\\left(m^2-4\right)x=2m^2-m-6\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}m\cdot\dfrac{2m+3}{m+2}-y=2m\\x=\dfrac{2m^2-m-6}{m^2-4}=\dfrac{\left(2m+3\right)\left(m-2\right)}{\left(m+2\right)\left(m-2\right)}=\dfrac{2m+3}{m+2}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{2m^2+3m}{m+2}-2m\\x=\dfrac{2m+3}{m+2}\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{2m^2+3m-2m\left(m+2\right)}{m+2}\\x=\dfrac{2m+3}{m+2}\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{2m+3m-2m^2-4m}{m+2}=\dfrac{-m}{m+2}\\x=\dfrac{2m+3}{m+2}\end{matrix}\right.\)
