For which value(s) of parameter m is there a solution for this system Imagine a system with one parameter $m$:
\begin{cases}
mx + y = m\\
mx + 2y = 1\\
2x + my = m + 1
\end{cases}
Now the question is: when does this system of equations have a solution?
I know how to do it with the Gaussian method, but how can I do this without the Gaussian method, let's say with Cramer's rule?

Compute the values of $x$ and $y$ dependent on $m$ for the following system, then solve $2x + my = m + 1$ (the last equation) to find the values of parameter $m$ for $x$ and $y$:
\begin{cases}
mx + y = m\\
mx + 2y = 1\\
\end{cases}
So, 
\begin{cases}
2mx + 2y =2 m\\ 
mx + 2y = 1\\
\end{cases}
Subtracting two equations, will have:
$$mx=2m-1$$


*

*If $m \neq 0$, we may divide by $m$ and get $x = (2m-1)/m$ and $y =
   1-m$.

*If $m = 0$, the system has no solution.
Putting $x$ and $y$ in the last equation ($m\neq 0$), we'll have:
$$m^3-3m+2=0 $$ 
$$(m^3-1)-3m+3=0$$
$$(m-1)(m^2+m+1)-3(m-1)=0$$
$$(m-1)(m^2+m-2)=0$$
Thus the values of parameter $m$ are  $m=1$ or $m=-2$.
