Is this transformation of linear Diophanthine Equations always possible

Question

Given a set of $N$ linear Diophantine equations over $V$ variables such that:

1. Each variable appears 2 or 3 times
2. Each equation has at most 3 variables
3. $\frac12<\frac NV<1$

Can we always rewrite these equations such that each equation (with not necessarily integer constants) has at most 2 variables?

Answer 1

The answer to your question is no and a counter-example is given below.

Indeed, consider the following three equations in five variables:

$a-2b+c=0$

$2a-3b+d=0$

$3a-4b+e=0$

$c -2d+e=0$

The solutions to this system of equations are spanned by two particular solutions: $a=1,b=1,c=1,d=1,e=1$ and $a=1,b=2,c=3,d=4,e=5$

Now suppose you rewrite the equations and get one with only two variables. Since $(1,1,1,1,1)$ is a solution, it must be of the form $k.x=k.y$ where $x$ and $y$ stand for two different variables in the set $\{a,b,c,d,e\}$ and $k$ is any non zero constant. Obviously, this equation is not satisfied by $(1,2,3,4,5)$.