Can I eliminate variables in systems of diophantine equations?

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.

• If your "linear algebra" calculation calls for dividing the coefficients then you've left the realm of diophantine equations (integer solutions). For more information you could edit your question to show the particular exercise you worked on. – Ethan Bolker Oct 9 '17 at 14:49
• @EthanBolker Basically I was working on a case where $c_1=c_2=1$ and I subtracted one equation from the other. Am I allowed to do this? – David Oct 9 '17 at 14:54
• Short answer so far: yes – Ethan Bolker Oct 9 '17 at 14:56
• And in any case, when any associated coefficients that are equal, for example, $a_1=a_2,\;\;b_1=b_2\;\;$ or $\;c_1 = c_2$, you can rid reduce the variables in such an equation. – Namaste Oct 9 '17 at 15:05
• From analytic geometry view the system of equations with three variables is a line consisting infinitely many point of which some of them may have integer coordinates(integer solutions of system). when you eliminate one variable you reduce the question to two variables, in fact you find the coordinates of projection of points which is entirely different to coordinates of points. – sirous Dec 6 '17 at 10:16