a: Thay m=3 vào hệ, ta được:
\(\left\{{}\begin{matrix}3x+2y=1\\3x+\left(3+1\right)y=-1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}3x+2y=1\\3x+4y=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3x+4y-3x-2y=-1-1\\3x+2y=1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}2y=-2\\3x=1-2y=1-\left(-2\right)=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=-1\\x=1\end{matrix}\right.\)
b:
để hệ có vô số nghiệm thì \(\dfrac{m}{3}=\dfrac{2}{m+1}=\dfrac{1}{-1}\)
=>\(\left\{{}\begin{matrix}m^2+m=6\\m+1=-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(m+3\right)\left(m-2\right)=0\\m=-3\end{matrix}\right.\)
=>m=-3
Để hệ vô nghiệm thì \(\dfrac{m}{3}=\dfrac{2}{m+1}\ne\dfrac{1}{-1}=-1\)
=>\(\left\{{}\begin{matrix}m^2+m=6\\m+1\ne-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(m+3\right)\left(m-2\right)=0\\m\ne-3\end{matrix}\right.\)
=>m=2
Để hệ có nghiệm duy nhất thì \(\dfrac{m}{3}\ne\dfrac{2}{m+1}\)
=>\(m^2+m\ne6\)
=>\(m^2+m-6\ne0\)
=>(m+3)(m-2)<>0
=>\(m\notin\left\{-3;2\right\}\)
\(\left\{{}\begin{matrix}mx+2y=1\\3x+\left(m+1\right)y=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3mx+6y=3\\3mx+\left(m^2+m\right)y=-m\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}3mx+\left(m^2+m\right)y-3mx-6y=-m-3\\mx+2y=1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}y\left(m+3\right)\left(m-2\right)=-\left(m+3\right)\\mx+2y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{-1}{m-2}\\mx=1-2y=1+\dfrac{2}{m-2}=\dfrac{m-2+2}{m-2}=\dfrac{m}{m-2}\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}y=-\dfrac{1}{m-2}\\x=\dfrac{1}{m-2}\end{matrix}\right.\)
c: Để hệ có nghiệm duy nhất là số nguyên thì \(\left\{{}\begin{matrix}m\in\left\{-3;2\right\}\\m-2\inƯC\left(1;-1\right)\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}m\in\left\{-3;2\right\}\\m-2\in\left\{1;-1\right\}\end{matrix}\right.\)
=>\(m\in\left\{3;1\right\}\)
