Solve a system of equations in $\mathbb{F}_8$

Question

Let's realize $K=\mathbb{F}_8$ as $\mathbb{Z}_2[A]/(A^3+A^2+1)$ and let $a$ be a class of an element in A. Solve in a field $K$ the following system of equations: $$ \begin{cases} (a+1)x+a^2y=a^2+a\\ (a^2+1)x+ay=a+1 \end{cases} $$ I need to grasp the basic concepts of how to solve this because I don't really understand how to start the solution. Any help would be helpful!

Answer 1

Multiplying the second equation by $a$ and subtracting the first one gives $$ 0=(a(a^2+1)-(a+1))x=(a^3-1)x. $$ Now since $a^3+a^2+1=0$ we have $a\neq 0$ and hence $a^3-1=a^2\neq 0$. Hence it follows that $x=0$ and $y=(a+1)/a$.