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Math Help - Linear Programming - Elimination Method help

  1. #1
    ljj
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    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!!
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  2. #2
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    Re: Linear Programming - Elimination Method help

    Quote Originally Posted by ljj View Post
    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?
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  3. #3
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    Re: Linear Programming - Elimination Method help

    Hello, ljj!

    \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.

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