The feasible region for a $LPP$ is shown in the following figure. Evaluate $Z = 4x + y$ at each of the corner points of this region. Find the minimum value of $Z$,if it exists.

(N/A) The lines $x + 2y = 4$ and $x + y = 3$ intersect at $(2, 1)$. From the figure,it is an unbounded shaded region with the corner points $A(4, 0)$,$B(2, 1)$,and $C(0, 3)$.
We evaluate $Z = 4x + y$ at these corner points:

Corner points	Corresponding value of $Z$
$(4, 0)$	$Z = 4(4) + 0 = 16$
$(2, 1)$	$Z = 4(2) + 1 = 9$
$(0, 3)$	$Z = 4(0) + 3 = 3$

The smallest value obtained is $3$ at $(0, 3)$. Since the region is unbounded,we must check if $Z < 3$ has any common points with the feasible region.
Graphing the inequality $4x + y < 3$,we observe that the open half-plane $4x + y < 3$ has no common point with the feasible region. Therefore,the minimum value of $Z$ is $3$ at the point $(0, 3)$.