Linear programming using graphical method - 3 variables

Question

Here is the problem:

Maximize: $ F= 12x+12y+5z$

Subject to the constraints:

$2x+4y+3z\leq12$

$x+2y+z\leq8$

$ x,y,z \geq 0$

Using simplex method I got these answers:

$x,y,z(6,0,0), F=72; $

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I need to use the graphical method to solve this, but I have no idea how if it contains 3 variables.

Answer 1

Normally if one were to draw this by hand, one would change all the inequalities into equalities, then graph those planes, mark the direction where the inequalities imply, denote a feasible region, and then take note of where the intersections happen between the constraints in the feasible region.

Graphically, after doing all of this, the feasible region of the model would look like the following:

All the extreme points (shown in red in the following picture), are

• $\left(0,0,0\right) \longrightarrow z = 0$
• $\left(6,0,0\right) \longrightarrow z = 72$
• $\left(0,3,0\right) \longrightarrow z = 36$
• $\left(0,0,4\right) \longrightarrow z = 20$

Since we're looking to maximize $z$, the optimal solution for the model would be $(6,0,0)$ where $z=72$.