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

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February 8th 2011, 06:39 PM
tn11631

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 $\leq$ 10
5x+y $\leq$ 12
x+5y $\leq$ 15
x $\geq$ 0, y $\geq$ 0

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.
February 8th 2011, 07:30 PM
CaptainBlack

Quote:

Originally Posted by tn11631

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 $\leq$ 10
5x+y $\leq$ 12
x+5y $\leq$ 15
x $\geq$ 0, y $\geq$ 0

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
February 8th 2011, 07:32 PM
CaptainBlack

Quote:

Originally Posted by tn11631

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 $\leq$ 10
5x+y $\leq$ 12
x+5y $\leq$ 15
x $\geq$ 0, y $\geq$ 0

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
February 8th 2011, 07:38 PM
tn11631

Quote:

Originally Posted by CaptainBlack

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 $\leq$ 15? 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 $\leq$ 15. Thanks !
February 8th 2011, 08:10 PM
CaptainBlack

Quote:

Originally Posted by tn11631

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 $\leq$ 15? 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 $\leq$ 15. Thanks !

$w=15-x-5y$

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

CB
February 8th 2011, 08:14 PM
tn11631

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