I encountered a set of linear equations with modulo in only $2$ variables
$$(a_{11}x + a_{12}y) \mod 8 = b_1$$ $$(a_{21}x + a_{22}y) \mod 8 = b_2$$
In the case of simple equations without modulo, the solution of $Ax=b$ is $x=A^{-1}b$, but how to handle this modulo? Any clue or pointers will be very helpful.
You can do Gauss-Jordan elimination performing operations modulo $8$. Just be mindful that "division" is only defined for odd integers (you are really multiplying by the inverse modulo $8$).
You can apply Cramer's rule if the determinant is odd (so invertible mod 8). For an introduction to linear algebra over commutative rings see Wm. C. Brown: Matrices over commutative rings.
