3.4. Consider the linear program z = -5x1 – 7x2 subject to -3x1 + 2x2 < 30 -2x1 + x2 < 12 minimize X1, X2 > 0. 115 (i) Draw a graph of the feasible region. (ii) Determine the extreme points of the feasible region. (iii) Determine two linearly independent directions of unboundedness. (iv) Convert the linear program to standard form and determine the basic feasible solutions and two linearly independent directions of unboundedness for this version of the problem. Verify that the directions of unboundedness satisfy Ad O and d > 0. LO
3.4. Consider the linear program z = -5x1 – 7x2 subject to -3x1 + 2x2 < 30 -2x1 + x2 < 12 minimize X1, X2 > 0. 115 (i) Draw a graph of the feasible region. (ii) Determine the extreme points of the feasible region. (iii) Determine two linearly independent directions of unboundedness. (iv) Convert the linear program to standard form and determine the basic feasible solutions and two linearly independent directions of unboundedness for this version of the problem. Verify that the directions of unboundedness satisfy Ad O and d > 0. LO
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Linear Programming Problem
**Problem Statement:**
Consider the linear program:
Minimize:
\[ z = -5x_1 - 7x_2 \]
Subject to:
\[ -3x_1 + 2x_2 \leq 30 \]
\[ -2x_1 + x_2 \leq 12 \]
\[ x_1, x_2 \geq 0 \]
**Tasks:**
1. **Graph of the Feasible Region:**
Draw a graph illustrating the feasible region determined by the constraints. The feasible region is the area where all the inequalities are satisfied simultaneously.
2. **Extreme Points of the Feasible Region:**
Identify the extreme points (vertices) of the feasible region. These are the points where the constraints intersect.
3. **Linearly Independent Directions of Unboundedness:**
Determine two linearly independent directions of unboundedness in the feasible region.
4. **Convert to Standard Form:**
Convert the linear program into standard form. Identify the basic feasible solutions and two linearly independent directions of unboundedness for this version of the problem. Verify that the directions of unboundedness satisfy:
\[ Ad = 0 \]
\[ d \geq 0 \]
**Note:** Standard form typically involves expressing the constraints with equality by adding slack variables and ensuring non-negativity of all variables involved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbffa73e6-1761-4c9e-97cb-0c164f013989%2F5cff3371-41cd-4067-b8b6-82c95d7bda95%2Fae8caze_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Programming Problem
**Problem Statement:**
Consider the linear program:
Minimize:
\[ z = -5x_1 - 7x_2 \]
Subject to:
\[ -3x_1 + 2x_2 \leq 30 \]
\[ -2x_1 + x_2 \leq 12 \]
\[ x_1, x_2 \geq 0 \]
**Tasks:**
1. **Graph of the Feasible Region:**
Draw a graph illustrating the feasible region determined by the constraints. The feasible region is the area where all the inequalities are satisfied simultaneously.
2. **Extreme Points of the Feasible Region:**
Identify the extreme points (vertices) of the feasible region. These are the points where the constraints intersect.
3. **Linearly Independent Directions of Unboundedness:**
Determine two linearly independent directions of unboundedness in the feasible region.
4. **Convert to Standard Form:**
Convert the linear program into standard form. Identify the basic feasible solutions and two linearly independent directions of unboundedness for this version of the problem. Verify that the directions of unboundedness satisfy:
\[ Ad = 0 \]
\[ d \geq 0 \]
**Note:** Standard form typically involves expressing the constraints with equality by adding slack variables and ensuring non-negativity of all variables involved.
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