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Math Help - Linear Programming Problems? Find the corresponding values of the slack variables.

  1. #1
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    Linear Programming Problems? Find the corresponding values of the slack variables.

    So i'm not sure where this question would fit within the categories given, but it is a linear programming problem and it is as follows:

    Consider the linear programming problem

    Maximize z=2x+5y
    subject to
    2x+3y \leq10
    5x+y \leq12
    x+5y \leq 15
    x \geq0, y \geq0

    Part (a): verify that x=[1] (this is a vector) is a feasible solution. For part (a) i
    [2]
    had no problems verifying it but I need it for part (b)

    Part (b) For the feasible solution in a, find the corresponding values of the slack variables.

    I started setting it up in canonical form but there are three constraints but two items in the vector given in (a) so i'm stuck. I had started adding in the slack variables as in: 2x+3y+u=10 etc.. but I don't know how to do this one when there are three constraints and two items in the vector.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by tn11631 View Post
    So i'm not sure where this question would fit within the categories given, but it is a linear programming problem and it is as follows:

    Consider the linear programming problem

    Maximize z=2x+5y
    subject to
    2x+3y \leq10
    5x+y \leq12
    x+5y \leq 15
    x \geq0, y \geq0

    Part (a): verify that x=[1] (this is a vector) is a feasible solution. For part (a) i
    [2]
    had no problems verifying it but I need it for part (b)

    Part (b) For the feasible solution in a, find the corresponding values of the slack variables.

    I started setting it up in canonical form but there are three constraints but two items in the vector given in (a) so i'm stuck. I had started adding in the slack variables as in: 2x+3y+u=10 etc.. but I don't know how to do this one when there are three constraints and two items in the vector.
    You have one slack variable for each constraint, write it in terms of x and y (rearrange the constraint equation with the slack variable on the left and every thing else on the right). Now substitute x=1, y=2 into each of these equations...

    CB
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by tn11631 View Post
    So i'm not sure where this question would fit within the categories given, but it is a linear programming problem and it is as follows:

    Consider the linear programming problem

    Maximize z=2x+5y
    subject to
    2x+3y \leq10
    5x+y \leq12
    x+5y \leq 15
    x \geq0, y \geq0

    Part (a): verify that x=[1] (this is a vector) is a feasible solution. For part (a) i
    [2]
    had no problems verifying it but I need it for part (b)

    Part (b) For the feasible solution in a, find the corresponding values of the slack variables.

    I started setting it up in canonical form but there are three constraints but two items in the vector given in (a) so i'm stuck. I had started adding in the slack variables as in: 2x+3y+u=10 etc.. but I don't know how to do this one when there are three constraints and two items in the vector.
    You have one equation for each inequality constraint each with a different slack variable. Solve for the slack variable in each and put x=1, y=2 to get the value of the slack variable corresponding to each constraint.

    CB
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    You have one equation for each inequality constraint each with a different slack variable. Solve for the slack variable in each and put x=1, y=2 to get the value of the slack variable corresponding to each constraint.

    CB
    Thats what I was doing on my paper but then I got confused because I had u=10-2x-3y and v=12-5x-y but then what about x+5y \leq15? I don't know what to do there because there are only two variables x and y so there should be two slack variables u, and v but then im still left with x+5y \leq15. Thanks !
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by tn11631 View Post
    Thats what I was doing on my paper but then I got confused because I had u=10-2x-3y and v=12-5x-y but then what about x+5y \leq15? I don't know what to do there because there are only two variables x and y so there should be two slack variables u, and v but then im still left with x+5y \leq15. Thanks !
    w=15-x-5y

    so at the given point w=15-1-10=4

    CB
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  6. #6
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    Ohh wow don't I feel dumb for missing that maybe too much math for one night. Thanks so much!
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