Sequential and simultaneous aggregation of Diophantine equations

We consider the problem of aggregation for two systems, the system
$$ \sum_{j=1}^n a_{ij}x_j=b_i,\qquad i=1,2, $$
where $a_{ij}$ and $b_i$ are integers, and the system
$$ f_i(x)=b_i,\qquad i=1,2,\ldots,m, $$
where the functions $f_i(x)$ take integer non-negative values, $x=(x_1,\ldots,x_n)$.
An equivalent equation is constructed as a non-negative linear combination of equations of the given system by two groups of methods: the equations are aggregated either sequentially taken two at a time or simultaneously. We suggest new methods within these two groups to construct a single equation which is equivalent to the given system of equations in the sense that it has the same set of integer non-negative solutions. In comparison with the known methods of aggregation the methods suggested in this paper lead to equivalent equations with lesser coefficients.