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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.

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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 $8$).

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  • $\begingroup$ I'm confused - do you mean that a11, a12, a21, etc. all have to be odd? $\endgroup$ – EJoshuaS Dec 14 '16 at 20:52
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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.

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