<s>[INST] Solving systems of equation with unknown How do i solve this system? 
$$
\left\{ 
\begin{array}{c}
ax+y+z=1 \\ 
x+ay+z=a \\ 
x+y+az=a^2
\end{array}
\right. 
$$
Ive reduced to this form. How should i continue to get infinitely many solutions, no solution and unique?
$$ \left[
    \begin{array}{ccc|c}
      1&a&1&a\\
      0&(a^2)-1&a-1&(a^2)-1\\
      0&(a-1)(a-2)&0&a-1
    \end{array}
\right] $$
 [/INST] hint:$$\left\{ 
\begin{array}{c}
ax+y+z=1 \\ 
x+ay+z=a \\ 
x+y+az=a^2
\end{array}
\right.$$sum of the equations :
$$x(a+2)+y(a+2)+z(a+2)=1+a+a2\\x+y+z=\frac{1+a+a^2}{a+2}\\$$now $$ax+y+z=1-(x+y+z=\frac{1+a+a^2}{a+2})\\(a-1)x=1-\frac{1+a+a^2}{a+2}\\(a-1)=\frac{a+2-1-a-a^2}{a+2}\\x=\frac{1-a^2}{(a+2)(a-1)}$$
</s>