a, Khi m=2, hệ pt có dạng
\(\left\{{}\begin{matrix}x+2y=2\\2x-2y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3x=3\\2x-2y=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=1\\2\times1-2y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=\dfrac{1}{2}\end{matrix}\right.\)
Vậy hệ pt có nghiệm (1;1/2)
b, \(\left\{{}\begin{matrix}x+my=2\\mx-2y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2-my\\m\left(2-my\right)-2y=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=2-my\\2m-m^2y-2y-1=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=2-my\\\left(-m^2-2\right)y+2m-1=0\left(\cdot\right)\end{matrix}\right.\)
Hệ pt có nghiệm duy nhất khi pt (.) có nghiệm duy nhất
\(\Leftrightarrow-m^2-2\ne0\Leftrightarrow-m^2\ne2\Leftrightarrow m^2\ne-2\)(luôn đúng)
\(\forall m\) ( 1 ) , hê pt có dạng
\(\left\{{}\begin{matrix}x=2-my\\\left(-m^2-2\right)y=1-2m\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=2-my\\y=\dfrac{1-2m}{-m^2-2}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=2-\dfrac{m\left(1-2m\right)}{-m^2-2}\\y=\dfrac{1-2m}{-m^2-2}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{-2m^2-4-m+2m^2}{-m^2-2}\\y=\dfrac{1-2m}{-m^2-2}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{m+4}{m^2+2}\\y=\dfrac{2m-1}{m^2+2}\end{matrix}\right.\)
Để x>0 thì \(\dfrac{m+4}{m^2+2}>0\) mà m2+2 > 0 ( luôn đúng) \(\Rightarrow m+4>0\Leftrightarrow m>-4\left(2\right)\)
Để y<0 thì \(\dfrac{2m-1}{m^2+2}< 0\) mà m2+2 > 0 ( luôn đúng )
\(\Rightarrow2m-1< 0\Leftrightarrow m< \dfrac{1}{2}\left(3\right)\)
Từ (1),(2),(3) \(\Rightarrow\forall m\) thỏa mãn \(-4< m< \dfrac{1}{2}\) thì hệ pt đã cho có nghiệm duy nhất (x;y) sao cho x>0 , y< 0
