# Thread: Solving Linear Program Geometrically

1. ## Solving Linear Program Geometrically

Maximize: $2x-3y$

Subject to:

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

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 $z = 2x - 3y$ cannot be attained at a point inside the polygonal domain; for at that point $2x - 3y$ 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, $z$ 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 $z$ 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

$
\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 $z$ is greatest.

Hope this helps.

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

Subject to:

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

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.