I have an equation
$$
Ax + By + Cz + dK = N.
$$
It is known that $z = xy$ and so the equation can be restated as
$$
Ax + By + Cxy + dK= N.
$$
$A, B, C, K$ and $N$ have a common factor $K$,
and so dividing by $K$ gives a equivalent scalar equation.
$$
ax + by + cxy + d = n,
$$
where $a, b, c, d$ and $n$ are respectively equal to
$$
a=\frac{A}{K},\; b=\frac{B}{K},\; c=\frac{C}{K},\; d=\frac{dK}{K}\;\text{ and }\; n=\frac{N}{K}.
$$
Therefore given these two equations supplied with variables $A, B, C, a, b, c, D$ and constants $K, N$ and $n$, is it possible to solve for $x, y$ and, $z = xy$.
Given the constraint $z = xy$, It is known that there can only be one unique Natural Number solution.
I have searched Google and various math based websites to gain a thorough understanding of solving for unknowns and have learned that Gaussian Elimination is the way to proceed when you have a full system of unique equations Yet, given that I have only 2 equations and am searching for essentially 3 but truly 2 unknowns, is using this method still applicable?
Can someone please assist me in understanding this further? Many thanks.
$$cxy+ax+by=e$$ where $e=n-d$. $$c^2xy+acx+bcy=ce$$ $$(cx+b)(cy+a)=ab+ce$$ So you have to find all the ways of factoring $ab+ce$ and then for each one, say, $ab+ce=mn$, you have to see whether $cx+b=m$ and $cy+a=n$ lead you to integer values for $x$ and $y$.