I have got to maximize $$Z=x+2y-3z+4w$$
subject to constraints $$x+y+2z+3w=12$$ $$y+2z+w=8$$
$x,y,z,w\geq 0$ .
The question asks to show without actually solving the lpp that it has an optimal solution. For that I added the two constraints to get $x+2y+4z+4w=20$ and from this I put the value of $x+2y+4w$ in objective function which became $Z=20-7z$.
Thereafter I concluded that Z could not be greater than $20$ i.e. Z is bounded above. Corresponding to this maximum value of Z we have z=0 and using this I can find solutions such as $(0,6,0,2)$ and $(4,8,0,0)$ which are feasible. So the optimal solutions exist. But I want to know what are the other methods of showing it. Can we use duality theory?