# Thread: Solving Linear Program Geometrically

1. ## Solving Linear Program Geometrically

Maximize: $\displaystyle 2x-3y$

Subject to:

$\displaystyle y-x \leq 5$
$\displaystyle x+y \leq 11$
$\displaystyle x \leq 4$
$\displaystyle x,y \geq 0$

I had to miss class and this is what we learned....if anyone knows of a good website with how-to or example problems that would be greatly helpful.

If not, a little help to start me off would be great and what I need to do.

2. A good way to visualize this problem is to note that the maximum value of $\displaystyle z = 2x - 3y$ cannot be attained at a point inside the polygonal domain; for at that point $\displaystyle 2x - 3y$ would have to curve back downward in all directions, and no plane can do this.

The maximum value must therefore be attained at a point on the boundary of the domain. Now, as you move across the boundary lines, $\displaystyle z$ either increases or decreases (or stays the same) continually at the same rate due to its planar nature. Thus, we will certainly find the peak value of $\displaystyle z$ at a vertex of the domain, as we can move in the increasing direction of the boundary until we find it.

The problem then becomes to find the intersections of the lines

$\displaystyle \begin{array}{rcl} y-x & = & 5 \\ x+y & = & 11 \\ x & = & 4 \\ x,y & = & 0 \end{array}$

that belong to the boundary of the domain (not all intersections will), and find the one at which $\displaystyle z$ is greatest.

Hope this helps.

3. Originally Posted by toop
Maximize: $\displaystyle 2x-3y$

Subject to:

$\displaystyle y-x \leq 5$
$\displaystyle x+y \leq 11$
$\displaystyle x \leq 4$
$\displaystyle x,y \geq 0$

I had to miss class and this is what we learned....if anyone knows of a good website with how-to or example problems that would be greatly helpful.

If not, a little help to start me off would be great and what I need to do.