# Combining the duality principle and the graphical method

I am trying to minimize this linear program by combining the duality principle and the graphical method:

$$\begin{array}{ll} \min & 2x_1+3x_2+3x_3+6x_4+4x_5\\ \text{Subject to:} & 2x_1+x_2-2x_3+3x_4-2x_5=-1\\ &x_1+3x_2+x_3+2x_4+x_5=1\\ & x_1,x_2,x_3,x_4,x_5\ge0 \end{array}$$

I can't seem to find an example of how to approach this, can anyone show me how I would go about this?

The dual problem is:

objective function

$\text{max} \ -y_1+y_2$

constraints

$2y_1+y_2 \leq 2$

$y_1+3y_2 \leq 3$

$-2y_1+y_2 \leq 3$

$3y_1+2y_2 \leq 6$

$-2y_1+y_2 \leq 4$

$y_1,y_2$ are unrestricted.

You have to solve every constraint for $y_2$. Then you can draw the constraints in a $y_1-y_2$-coordinate system. If you consider the inequality-signs you will get a solution space.

Now you set the objective function equal to zero and solve the equation for $y_2$. After you have drawn it in the coordinate system you have to move it parallel till you just touch the solution space.