a: Khi m=-3 thì hệ phương trình sẽ là:

\(\left\{{}\begin{matrix}-3x+2y=1\\x-2\cdot\left(-3\right)\cdot y=-3-2\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}-3x+2y=1\\x+6y=-5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}-3x+2y=1\\3x+18y=-15\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}20y=-14\\x+6y=-5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=-\dfrac{7}{10}\\x=-5-6y=-5-6\cdot\dfrac{-7}{10}=\dfrac{42}{10}-5=-\dfrac{8}{10}=-\dfrac{4}{5}\end{matrix}\right.\)

b: \(\left\{{}\begin{matrix}mx+2y=1\\x-2my=m-2\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=2my+m-2\\m\left(2my+m-2\right)+2y=1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=2my+m-2\\2m^2\cdot y+m^2-2m+2y=1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=2my+m-2\\y\left(2m^2+2\right)=-m^2+2m+1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=\dfrac{-m^2+2m+1}{2m^2+2}\\x=2m\cdot\dfrac{-m^2+2m+1}{2m^2+2}+m-2\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=\dfrac{-m^2+2m+1}{2m^2+2}\\x=\dfrac{m\left(-m^2+2m+1\right)}{m^2+1}+m-2\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=\dfrac{-m^2+2m+1}{2m^2+2}\\x=\dfrac{-m^3+2m^2+m+\left(m-2\right)\left(m^2+1\right)}{m^2+1}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=\dfrac{-m^3+2m^2+m+m^3+m-2m^2-2}{m^2+1}=\dfrac{2m-2}{m^2+1}\\y=\dfrac{-m^2+2m+1}{2m^2+2}\end{matrix}\right.\)

x-2y=-1

=>\(\dfrac{2m-2}{m^2+1}-\dfrac{2\cdot\left(-m^2+2m+1\right)}{2m^2+2}=1\)

=>\(\dfrac{2m-2}{m^2+1}-\dfrac{-m^2+2m+1}{m^2+1}=1\)

=>\(\dfrac{2m-2+m^2-2m-1}{m^2+1}=1\)

=>\(m^2-3=m^2+1\)

=>-3=1(vô lý)

Lời giải:

a)

Khi $m=1$ thì HPT trở thành:\(\left\{\begin{matrix} x-y=2\\ x+y=1\end{matrix}\right.\Rightarrow \left\{\begin{matrix} 2x=2+1\\ 2y=1-2\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x=\frac{3}{2}\\ y=\frac{-1}{2}\end{matrix}\right.\)

b) 

HPT \(\Leftrightarrow \left\{\begin{matrix} mx-y=2\\ x=1-my\end{matrix}\right.\Rightarrow m(1-my)-y=2\)

\(\Leftrightarrow y(m^2+1)=m-2\Rightarrow y=\frac{m-2}{m^2+1}\)

\(x=1-my=1-\frac{m^2-2m}{m^2+1}=\frac{1+2m}{m^2+1}\)

Để $x+y=-1$

$\Leftrightarrow \frac{m-2}{m^2+1}+\frac{1+2m}{m^2+1}=-1$

$\Leftrightarrow \frac{3m-1}{m^2+1}=-1$

$\Rightarrow 3m-1=-m^2-1$

$\Leftrightarrow m^2+3m=0\Rightarrow m=0$ hoặc $m=-3$

\(\hept{\begin{cases}\left(m+5\right)x+3y=1\\mx+2y=-4\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}\left(2m+10\right)x+6y=2\\3mx+6y=-12\end{cases}}\Leftrightarrow\hept{\begin{cases}\left(10-m\right)x=14\\mx+2y=-4\end{cases}}\)

Để hệ phương trình có 1 nghiệm duy nhất thì phương trình (10 - m)x = 14 cũng có 1 nghiệm duy nhất.

Điều này xảy ra khi \(m\ne10\)

Khi đó, hệ có nghiệm duy nhất \(\hept{\begin{cases}x=\frac{14}{10-m}\\y=\frac{40+13m}{2m-20}\end{cases}}\) 

khó đó nha

a) Thay \(m=1\) vào hệ phương trình, ta được:

\(\left\{{}\begin{matrix}3x-y=1\\x+2y=5\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=2\end{matrix}\right.\)

  Vậy ...

b) HPT \(\Leftrightarrow\left\{{}\begin{matrix}6x-2y=4m-2\\x+2y=3m+2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}7x=7m\\y=2m-1-3x\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=m\\y=-m-1\end{matrix}\right.\)

Ta có: \(x^2+y^2=5\) 

\(\Rightarrow m^2+m^2+2m+1=5\) \(\Leftrightarrow m^2+m-2=0\) \(\Rightarrow\left[{}\begin{matrix}m=1\\m=-2\end{matrix}\right.\)

  Vậy ...

c) Hệ phương trình luôn có nghiệm duy nhất

Ta có: \(x-3y>0\)

\(\Rightarrow m-3\left(-m-1\right)>0\)

\(\Leftrightarrow4m+3>0\) \(\Leftrightarrow m>-\dfrac{3}{4}\)

  Vậy ...