# Linear Programming - Elimination Method help

• Nov 8th 2011, 10:09 AM
ljj
Linear Programming - Elimination Method help
Hi: This may be slightly more advanced than basic algebra but I think it fits.

I have the following equation:

Maximize Z = 60x + 90y

Subject to:

7x+7y < or = 350
160x+80y > or = 4000
y < or = 25
x > or = 20

This is a maximization / linear programming example where I'd find the feasable region and then determine the optimum solution.

My first step was to sub 0 in for x and y in both equations, getting 25,0 and 50,0 and 0,50 for my coordinates. The constraints of y<or =20 and x > or =0 give me the feasable region but I am having trouble using the substution method to solve for the optimum solution.

Every example I have seen and looked up have been simple numbers where you can either add down, subtract, or multiple by a small amount to get a variable to cancel out. In this situation, 7 does not go into 80 or 160 evenly.

I tried using the LCM and came up with 80, but when I run that number through I end up with 560x = 0, so I know I must be doing something wrong.

How do you handle numbers that do not cancel out evenly?

Thanks!!
• Nov 8th 2011, 10:58 AM
TheEmptySet
Re: Linear Programming - Elimination Method help
Quote:

Originally Posted by ljj
Hi: This may be slightly more advanced than basic algebra but I think it fits.

I have the following equation:

Maximize Z = 60x + 90y

Subject to:

7x+7y < or = 350
160x+80y > or = 4000
y < or = 25
x > or = 20

This is a maximization / linear programming example where I'd find the feasable region and then determine the optimum solution.

My first step was to sub 0 in for x and y in both equations, getting 25,0 and 50,0 and 0,50 for my coordinates. The constraints of y<or =20 and x > or =0 give me the feasable region but I am having trouble using the substution method to solve for the optimum solution.

Every example I have seen and looked up have been simple numbers where you can either add down, subtract, or multiple by a small amount to get a variable to cancel out. In this situation, 7 does not go into 80 or 160 evenly.

I tried using the LCM and came up with 80, but when I run that number through I end up with 560x = 0, so I know I must be doing something wrong.

How do you handle numbers that do not cancel out evenly?

Thanks!!

Hint: 7 divides 350! The first equation reduces to

$x+y \le 50$

Now can you use it for elimination?
• Nov 9th 2011, 08:55 AM
Soroban
Re: Linear Programming - Elimination Method help
Hello, ljj!

Quote:

$\text{Maximize: }z \:=\: 60x + 90y$

$\text{Subject to: }\:\begin{Bmatrix}x+y \:\le \:50 & [1] \\ 2x+y \:\ge\:50 & [2] \\ x \ge 20 & [3] \\ y \le 25 & [4]\end{Bmatrix}$

The line of [1] has intercepts (50, 0) and (0, 50).
Draw the line and shade the region below the line.

The line of [2] has intercepts (25,0) and (0,50).
Draw the line and shade the region above the line.

The line of [3] is the vertical line: $x \,=\,20.$
Draw the line and shade the region to the right of the line.

The line of [4] is the horizontal line: $y \,=\,25.$
Draw the line and shade the region below the line.

The critical region looks like this:
Code:

      |       *       |**       | * *  |       |  *  * |       |  *  *       |    * D| *  C     - + - - * o---o       |      *|:::::*       |      Eo:::::::*       |      |*::::::::*   - - + - - - + o---------o - -       |        A        B
The vertices are: . $\begin{Bmatrix}A:& (25,0) \\ B: & (50,0) \\ C: & (25,25) \\ D: ^& (20,25) \\ E: & (20,10) \end{Bmatrix}$

Test the vertices in the z-function and determine the maximum z.