Consider the system of diophantine equations bellow.
$\cases{a_1x+b_1y+c_1z=d_1 \\ a_2x+b_2y+c_2z=d_2}$
Am I allowed to eliminate variables like one would in linear algebra? I was doing an exercise the other day where I assumed this was allowed and the result I acquired did not seem to agree with the answer the authors gave. Perhaps I just expressed it differently but more than likely I messed up.