Maximize $z=4x_1+3x_2$ subject to constraints:$2x_1+x_2\le 1000$$x_1+x_2\le 800$$x_1\le 400$$x_2\le 700$$x_1,x_2\ge 0$ Find the optimum units of $x_1$ and $x_2$ to be used.

We evaluate the objective function $z=4x_1+3x_2$ at the corner points of the feasible region formed by the constraints.The intersection points/corner points are:$(0,0)⟹z=0$$(400,0)⟹z=4(400)+3(0)=1600$$(400,200)$ [Intersection of $x_1=400$ and $2x_1+x_2=1000$ ] $⟹z=4(400)+3(200)=1600+600=2200$$(200,600)$ [Intersection of $2x_1+x_2=1000$ and $x_1+x_2=800$ ] $⟹z=4(200)+3(600)=800+1800=2600$$(100,700)$ [Intersection of $x_1+x_2=800$ and $x_2=700$ ] $⟹z=4(100)+3(700)=400+2100=2500$$(0,700)⟹z=3(700)=2100$ The maximum value of $z$ is $2600$ at $x_1=200,x_2=600$.