how to solve system of linear equations of XOR operation? how can i solve this set of equations ? to get values of $x,y,z,w$ ?
$$\begin{aligned} 1=x \oplus y \oplus z \end{aligned}$$
$$\begin{aligned}1=x \oplus y \oplus w \end{aligned}$$
$$\begin{aligned}0=x \oplus w \oplus z \end{aligned}$$
$$\begin{aligned}1=w \oplus y \oplus z \end{aligned}$$
this is not a real example, the variables don't have to make sense, i just want to know the method.

I'll try it the 'artisanal way'.
These four rules should be enough :
$\begin{align}
&x\oplus 0=x\\
&x\oplus 1=\overline{x}\\
&x\oplus x=0\\
&x\oplus \overline{x}=1\\
\end{align}$ 
$\tag{1}1=x\oplus y\oplus z$
$\tag{2}1=x\oplus y\oplus w$
$\tag{3}0=x\oplus w\oplus z$
$\tag{4}1=w\oplus y\oplus z$  
Applying $x\oplus$ on the three first equations and $w\oplus$ on the last gives :
$\tag{5}\overline{x}=y\oplus z$
$\tag{6}\overline{x}=y\oplus w$
$\tag{7}x= w\oplus z$
$\tag{8}\overline{w}=y\oplus z$  
$(6)\oplus(7)\ $ gives $\ \tag{6'}1=y\oplus z$
so that from $(8)$ $w=0$
from $(7)$ and $(6)$ $\ x=z$ and $\ \overline{x}=y$
from $(5)$ $\ \overline{x}=1$ and the final conclusion :
$$(w,x,y,z)=(0,0,1,0)$$
as found (earlier) by martini...
