<s>[INST] How many solutions does this system of equations have? I have $2$ equations :

$$-\frac{a}{2}x-\frac{b}{2}+cx+d=x+2$$
  $$-2ax-2b+cx+d=2x+1$$
  with $a, b, c, x \neq 0$  

We have to find all possible solutions of $a, b, c, d$ that make the equations true for all $x$.
I found one solution
$a=-\frac{2}{3}$, 
$b=\frac{2}{3}$, 
$c=\frac{2}{3}$, 
$d=\frac{7}{3}$
But, I wonder if there are any other solutions to this system of equations? 
 [/INST] Write the equations as:
$$
\begin{cases}
\begin{align}
(a-2c+2)x+b-2d+4 = 0 \\
(2a-c+2)x+2b-d+1 = 0
\end{align}
\end{cases}
$$
A polynomial is identical $0$ (i.e. for all $x$) iff all its coefficients are $0\,$, which gives the system to solve:
$$
\begin{cases}
\begin{align}
a - 2c + 2 = 0 \\
2a - c + 2 = 0 \\[7px]
b - 2d + 4 = 0 \\
2b -d + 1 = 0
\end{align}
\end{cases}
$$
The system has the unique solution as posted $\;a=-\frac{2}{3}$, $b=\frac{2}{3}$, $c=\frac{2}{3}$, $d=\frac{7}{3}\,$.
</s>