For an objective function $Z=x+y$ with constraints $2x+2y\le 5$ and $4x-3y\ge 3$ ($x,y\ge 0$), at what point(s) will $Z$ achieve its maximum value?

Corner points of the feasible region:Intersection of $2x+2y=5$ with $Y$ -axis: $x=0⟹y=2.5$. Point $A(0,2.5)$.Intersection of $4x-3y=3$ with $X$ -axis: $y=0⟹x=0.75$. Point $B(0.75,0)$.Intersection of $2x+2y=5$ and $4x-3y=3$:Multiply eq 1 by 1.5: $3x+3y=7.5$.Add to eq 2: $7x=10.5⟹x=1.5$.$2(1.5)+2y=5⟹3+2y=5⟹y=1$. Point $C(1.5,1)$.Evaluate $Z=x+y$:$Z(A)=0+2.5=2.5$$Z(B)=0.75+0=0.75$$Z(C)=1.5+1=2.5$ Since the maximum value $2.5$ occurs at two corner points $A$ and $C$, the maximum occurs at every point on the line segment joining $A$ and $C$.