Let us have a homogeneous linear system of equations which is represented by matrix $A \in Mat_{3,4}(\mathbb{Z}_{7})$. Find the number of solutions depending on parameters $a,b \in \mathbb{Z}_{7}$ if $$ \begin{pmatrix} \overline{2} & \overline{-4} & -b & a+b\\ \overline{5} & \overline{a} & \overline{1} & \overline{2} \\ \overline{5}a & \overline{2} & \overline{-3} & a-\overline{3} \end{pmatrix} $$.
So what i think I understand is that the even though some of the elements $a$ and $b$ have a horisontal line on them, they are still the members of the field $\mathbb{Z}_{7}$ (the reason why it is a field is that $\mathbb{Z}_{n}$ is a field if $n$ is a prime number).
The rest of the approach to the following task still remains a mystery to me. Because I learn algebra by myself then any helpful tips, tricks and ideas are more than welcome.