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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.2: Direct Methods For Solving Linear Systems

Problem 2CEXP

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Question

Consider the following LP:
max (2z1 + 5z2)
s.t.
3z1 + 2z2 < 18
21 < 5
%2 < 3
21, 22 >0
(a) Solve the LP with the graphical method.
(b) Place the model in standard form.
(c) Use a simplex algorithm in tableau form and solve the LP.

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Consider the following LP:min 5x1 + 4x2s.t. x1 + x2 ≥ 12x1 − x2 ≥ 13x2 ≤ 2x1, x2 ≥ 0.a. Write down the dual of this LP.b. We know that by strong duality, we can either solve this LP or its dual form to get the sameoptimal objective value. If you apply the simplex algorithm, which one would you rathersolve? Make your choice and apply the simplex algorithm to solve it.
Consider the following LP:min 5x1 + 4x2s.t. x1 + x2 ≥ 12x1 − x2 ≥ 13x2 ≤ 2x1, x2 ≥ 0.1. Write down the dual of this LP.2. We know that by strong duality, we can either solve this LP or its dual form to get the sameoptimal objective value. If you apply the simplex algorithm, which one would you rathersolve? Make your choice and apply the simplex algorithm to solve it.
Solve the following LP problem using simplex method:
Solve the following LP model using the simplex method.
Use the simplex algorithm to solve the following linear optimisation problem: maximisef(x1,x2,x3)=3x1+x2+2x3 subjectto: 3x1 +2x2 +x3 ≤8 x1 +2x2 +2x3 ≤4, x1,x2,x3 ≥0. You must address each of the 5 steps of the algorithm as presented in the course notes and videos in your answer.
How to solve this problem with the dual simplex algorithm: max 60x1 + 30x2 + 20x3subject to8x1 + 6x2 + x3 <= 484x1 + 2x2 + 1.5x3 <= 202x1 + 1.5x2 + 0.5x3 <= 8x1 + x2 + x3 <= 11 x1,x2,x3 >= 0
When using the simplex algorithm, an error was made in choosing the pivot row. Which of the following will be the result of this? The solution in the next tableau will: a. have a worse objective value b. be nonbasic c. be infeasible d. be unaffected
Solve the following linear optimization model using the Simplex Algorithm:Maximize z(x1, x2) = 3x1 + 2x2Subject to2x1 + 2x2 ≤ 82x1 + 1x2 ≤ 6x1, x2 ≥ 0
Consider the following system of equations: 4x1 − x2 − x3 = 5 −x1 + 4x2 − x3 = 10 −x1 − x2 + 4x3 = 15 Use the five-step Gaussian elimination algorithm to solve the system
How exactly does one go about utilizing numerical methods to solve a set of equations that have been arranged in a system? Provide your own explanation of the algorithm used in at least one of the methods.
Is A ivertible? If yes then write its algorithm and solve the system accordingly.
Question4: The simplex table obtained from an enterprise that produces three products under three constraints: machine hours, raw materials and meeting demand is given.Write down the primary and binary results and how the conclusion was reached.X₁ = ?X₂ = ?X₃ = ?S₁ = ?S₂ = ?S₃ = ?L₁ = ?L₂ = ?

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