Recently, I found this problem:
Find all integer solutions of $x,y,z$ of the following system: $$\left\{\begin{matrix} 5x + (a + 2)y + (a + 2)z = a \\(2a+4)x+(a^2+3)y+(2a+2)z=3a-1 \\ (2a + 4)x + (2a + 2)y + (a^2 + 3)z = a + 1 \end{matrix}\right. $$
where $a\in Z$
I don't know about matrix calculations, so I would like to find the solutions with, for example, substitution or sum-difference method.
For example I have tried subtracting the third equation from the second, getting: $(a-1)^2(y-z)=2(a-1)$ but here I'm stuck.
Hint
That is such an awesome start. Now Either $a=1$ or not. If so, then the equations reduce to$$5x+3y+3z=1\\3x+2y+2z=1$$which is very easy to solve.
The case $a\ne 1$ is more interesting, for which$$(a-1)^2(y-z)=2(a-1)\implies (a-1)(y-z)=2$$therefore we have an answer only if $$a-1|2\to a\in\{-1,0,2,3\}$$under which we have $$y-z={2\over a-1}$$which gives us the answer by substitution.