A linear programming problem has constraints $x+2y\le 8$ and $3x+2y\le 12$ with $x,y\ge 0$. What is the maximum value of $Z=x+y$?

Solution

1. The constraints are $x+2y\le 8$, $3x+2y\le 12$, and $x,y\ge 0$.
2. The vertices of the feasible region are found at the intersections of the boundary lines: $(0,0)$, $(4,0)$, $(0,4)$, and the intersection of $x+2y=8$ and $3x + 2y = 12$, which is $(2,3)$.
3. Evaluate the objective function $Z=x+y$ at each vertex: $Z(0,0)=0$, $Z(4,0)=4$, $Z(0,4)=4$, and $Z(2,3)=5$.
4. Comparing these values, the maximum value of $Z$ is $5$.
5. Hence the answer is (A).