The equation $ax+bu+cs=ayz+bvw+crt$ also has no integers solutions for any integers $a,b,c$

Question

Assuming that the system of equations:

$$x=yz$$ $$u=vw$$ $$s=rt$$

has no integers solutions. How I can prove that the equation $$ax+bu+cs=ayz+bvw+crt$$ also has no integers solutions for any integers $a,b,c$

Answer 1

$$ax+bu+cs=ayz+bvw+crt\\ a(x-yz)+b(u-vw)+c(s-rt)=0$$

Plugging in any specific integers for $x,y,z,u,v,w,r,s,t$ yields specific integer values $x-yz,u-vw,s-rt$. It really doesn't matter what these values are, they just become coefficients for $ae+bf+cg=0$ which is always solvable in integers (e.g., $a=b=c=0$). So the implication in the question is false as a starting point (i.e., solutions need not exist for the original "system" in order for the equation to have solutions).

After that, note that the original "system" is somewhat meaningless as there are no constraints given on the listed variables aside from "this system doesn't have solutions".