# Graphical analysis and Visualization for LPP

$$Q4.$$Use graphical analysis to find the optimal solution(s) of the LPP

a)

Maximize $$3x+2y$$

subject to $$3x-4y\le11$$ $$x,y\ge0$$

$$b)$$

Maximize $$5x-7y$$

subject to $$x+y\le18$$ $$x+3y\ge-6$$ $$x,y\ge0$$

$$Q5.$$ Visualizing canonical form problems

a) In $$\mathbb{R}^3$$ draw the plane $$P:x_1+x_2+x_3=1$$ inside $$P$$ draw the set where $$x_1,x_2,x_3\ge0.$$ Also inside $$P$$ draw the line where $$x_1=0,x_2=0,$$ and $$x_3=0.$$

b) In $$\mathbb{R}^3$$ draw the plane $$P:3x-4y+s=11$$.Inside $$P$$ draw the set where $$x,y,s\ge0$$ and the line $$x=0,y=0,$$ and $$s=0$$.

$$Q4.$$

a)

The darkest region contains all the feasible solution, and the blue line is the objective function, as $$z=3x+2y$$ increasing, the blue line keeps moving up, but some parts of the line will always in the feasible region, therefore it doesn't have max feasible solution in this case.

b)

The darkest region contains all the feasible solution, and the blue line is the objective function, as $$z=5x-7y$$ increasing, the blue line keeps moving down, that the blue point is the max feasible solution in this case.$$x=18,y=0$$

$$Q5.$$

$$a)$$

This is how $$P$$ looks like, The graph didn't show the parts that go though the $$x_1,x_2$$ plane, , but the plane does go though, basicly $$P$$ is a plane that has intersections $$(1,0,0),(0,1,0)$$ and $$(0,0,1)$$ on $$x,y,z$$-axis

And the blue triangle is the parts greater then zero, in another words $$x_1,x_2,x_3\ge0$$ in $$P$$

The red line is union of $$x_1,x_2,x_3=0$$ in $$P$$

$$b)$$

This is $$P$$, The graph didn't show the parts that go though the $$x,y$$ plane, but the plane does go though.

This is $$x,y,s\ge0$$ in $$P$$, which is not bounded.

The red line is union of $$x,y,s=0$$ in $$P$$