3.4 Day 1 Linear Programming 2010 October 27, 2010
3.4 Linear Programming
DAY 1
Objectives: • To find maximum and minimum values
• To solve problems with linear programming
Aug 14­9:04 PM
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3.4 Day 1 Linear Programming 2010 October 27, 2010
Check Skills You'll Need:
Solve each system of equations.
1. y = ­3x + 3
y = 2x ­ 7
2. x + 2y = 5
x ­ y = ­1
3. 4x + 3y = 7
2x ­ 5y = ­3
Aug 14­9:31 PM
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3.4 Day 1 Linear Programming 2010 October 27, 2010
Definitions:
Linear programming is a technique that identifies the minimum or maximum value of some quantity.
This quantity is measured with an objective function.
Limits on variables in the objective function are constraints, written as linear inequalities.
The constraints form a system of inequalities and the
feasible region contains all the points that satisfy those constraints.
Aug 14­9:44 PM
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3.4 Day 1 Linear Programming 2010 October 27, 2010
Given the objective function P = 3x + 2y and the following constraints, find the values of x and y that maximize P.
Constraints: y > 3/2x ­ 3
y < ­x + 7
x > 0, y > 0
Step 1: Graph the constraints. Step 2: Find the coordinates
for each vertex.
10
9
8
7
6
5
4
3
2
1
A (0, 0)
B (2, 0)
C (4, 3)
D (0, 7)
Step 3 ­>
1 2 3 4 5 6 7 8 9 10
Sep 1­5:57 PM
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3.4 Day 1 Linear Programming 2010 October 27, 2010
Given the objective function P = 3x + 2y and the following constraints, find the values of x and y that maximize P.
Step 3: Evaluate P at each vertex.
P = 3x + 2y
A (0, 0)
P = 3(0) + 2(0) = 0
B (2, 0)
P = 3(2) + 2(0) = 6
C (4, 3)
P = 3(4) + 2(3) = 18
D (0, 7)
P = 3(0) + 2(7) = 14
Therefore, when x = 4 and y = 3, P has its maximum value of 18.
Sep 1­5:57 PM
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3.4 Day 1 Linear Programming 2010 October 27, 2010
Homework: page 142 (1 ­ 9)
Sep 1­5:57 PM
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