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