   # In Exercises 1-8, use the simplex method to solve each linear programming problem. Maximize Subject to P = x + 2 y + 3 z 2 x + y + z ≤ 14 3 x + 2 y + 4 z ≤ 24 2 x + 5 y − 2 z ≤ 10 x ≥ 0 , y ≥ 0 ### Finite Mathematics for the Manager...

12th Edition
Soo T. Tan
Publisher: Cengage Learning
ISBN: 9781337405782

#### Solutions

Chapter
Section ### Finite Mathematics for the Manager...

12th Edition
Soo T. Tan
Publisher: Cengage Learning
ISBN: 9781337405782
Chapter 4.CRE, Problem 6CRE
Textbook Problem
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## In Exercises 1-8, use the simplex method to solve each linear programming problem. MaximizeSubject to P = x + 2 y + 3 z 2 x + y + z ≤ 14 3 x + 2 y + 4 z ≤ 24 2 x + 5 y − 2 z ≤ 10 x ≥ 0 , y ≥ 0

To determine

To find:

The solution of the linear programming problem by using the simplex method.

### Explanation of Solution

Given:

The given conditions are,

2x+y+z143x+2y+4z242x+5y2z10x0,y0,z0

Concept use:

(1) If all the entries are nonnegative, the optimal solution has been reached.

(2) If there is one or more negative entries, the optimal solution has been not been reached.

Calculation:

Consider the given equations.

P=x+2y+3z

Introduce the slack variables, u, v and rewrite the objective function in the standard form that gives the system of linear equation as follows,

2x+y+z=143x+2y+4z=242x+5y2z=10x2y3z+P=0

The initial simplex table is as follows,

 x y z u v w P Constant 2 1 1 1 0 0 0 14 3 2 4 0 1 0 0 24 2 5 −2 0 0 1 0 10 −1 −2 −3 0 0 0 1 0

Some of the elements in the last row are negative, so it is not in final form. Select the pivot row, pivot element and column for further simplification.

 x y z u v w P Constant 2 1 1 1 0 0 0 14 3 2 (4) 0 1 0 0 24 2 5 −2 0 0 1 0 10 −1 −2 −3 0 0 0 1 0

Apply the row operation R2R24.

 x y z u v w P Constant 2 1 1 1 0 0 0 14 34 12 (1) 0 14 0 0 6 2 5 −2 0 0 1 0 10 −1 −2 −3 0 0 0 1 0

Apply the row operation R1R1R2, R3R3+2R2 and R4R4+3R2

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