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I am to solve the system of equations below for the field mod 7. I am also to solve another system of equations that's the same system as below but with the right hand sides replaced with 2, 4, and 1 respectively. I was told that these two problems could be solved simultaneously using an agumented matrix but wouldn't this completely change the general solutions for $x,y,z$?

Thanks in advance!

$$x + 4y +5z=1$$ $$6y +4z=2$$ $$4x +3z =3$$

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    $\begingroup$ Are you familiar with Gauss elimination? $\endgroup$ – MereMortal47 Nov 18 '17 at 1:30
  • $\begingroup$ Remember that the integers mod $7$ is a field, and Gaussian elimination is valid for any field. $\endgroup$ – Riley Nov 18 '17 at 1:33
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If you're trying to solve multiple systems, the thing to do is express them as matrix equations $AX=B_1$ (and $AX=B_2$, etc.), where $X=\left(\begin{smallmatrix}x\\y\\z\end{smallmatrix}\right)$, compute $A^{-1}$, and then the solution will be $X=A^{-1}B_1$ (and $X=A^{-1}B_2$, etc.).

Here $$A=\left(\begin{matrix}1&4&5\\0&6&4\\4&0&3\end{matrix}\right)$$

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