Two equations & three unknowns (in $\mathbb{Z}$)

Question

I just want to know this system-equation has answer $(x,y,z)$ in Integers Set or not?

$a_1x+b_1y+c_1z=d_1$
$a_2x+b_2y+c_2z=d_2$

(in Real Number Set, we just need to check this two plate (plane) are parallel or not, but in in Integer Set ... ????)

Answer 1

Of cource you assumed that $a_i,b_i,c_i,d_i$ are integers. There are many ways to see this problem. For instance if you set z=0 and if (the integers) $a_i,b_i$ are such that $a_1b_2-a_2b_1=1$ then you will always have an integer solution $(x_0=D_x,y_0=D_y,0).$ Another case is $d_1=d_2=0,$ then you always have $(0,0,0).$ In general, since you have three unknows you will set an integer value to the one of them, for instance to $z$ and then the criterion is the determinant $D=a_1b_2-a_2b_1$ to divide both $D_x,D_y.$ A better criterion is given by the previous suggested paper http://math.udel.edu/~lazebnik/papers/dior1.pdf in Proposition 3. A more geometric approach is to find the line (if the re is any) which is in the intersection of the two planes and see if there is any integer point on it.