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Unit 7.5
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Copyright © 2011 Pearson, Inc.
7.5Systems of
Inequalities in Two Variables
Copyright © 2011 Pearson, Inc. Slide 7.5 - 2
What you’ll learn about
Graph of an Inequality Systems of Inequalities Linear Programming
… and whyLinear programming is used in business and industry to maximize profits, minimize costs, and to help management make decisions.
Copyright © 2011 Pearson, Inc. Slide 7.5 - 3
Steps for Drawing the Graph of an Inequality in Two Variables
1. Draw the graph of the equation obtained by replacing the inequality sign by an equal sign. Use a dashed line if the inequality is < or>. Use a solid line if the inequality is ≤ or ≥.
2. Check a point in each of the two regions of the plane determined by the graph of the equation. If the point satisfies the inequality, then shade the region containing the point.
Copyright © 2011 Pearson, Inc. Slide 7.5 - 4
Example Graphing a Linear Inequality
Draw the graph of y 2x 4. State the boundary of the region.
Copyright © 2011 Pearson, Inc. Slide 7.5 - 5
Example Graphing a Linear Inequality
Because of " ," the graph of y 2x 4 is part of the
graph of the inequality. The point (0,0) satisfies the
inequality because 0 2(0) 4. Thus the graph of
y 2x 4 consists
of all of the points
on or below the line
y 2x 4.
Draw the graph of y 2x 4. State the boundary of the region.
Copyright © 2011 Pearson, Inc. Slide 7.5 - 6
Example Solving a System of Inequalities Graphically
Solve the system 2x 3y 4 and y x2 .
Copyright © 2011 Pearson, Inc. Slide 7.5 - 7
Example Solving a System of Inequalities Graphically
Graph both inequalities and find their intersection.
Solve the system 2x 3y 4 and y x2 .
Copyright © 2011 Pearson, Inc. Slide 7.5 - 8
Linear Programming
Sometimes decision making in management science
requires that we find a minimum or a maximum of
a linear function
f a1x
1 a
2x
2L a
nx
n
called an objective function, over a set of points.
Such a problem is a linear programming problem.
The feasible xy points or set of points is the solution
of inequalities, called constraints. The solution occurs
at one of the vertex points, or corner points,
along the boundary region.
Copyright © 2011 Pearson, Inc. Slide 7.5 - 9
Example Solving a Linear Programming Problem
Find the maximum and minimum values of the objective
function f 3x 4y, subject to the constraints given by
the system of inequalities.
3x 2y 12
2x 5y 19
x 0
y 0
Copyright © 2011 Pearson, Inc. Slide 7.5 - 10
Example Solving a Linear Programming Problem
The corner points are:
0, 0 0,
19
5
4,0 2,3
f 3x 4y
3x 2y 12
2x 5y 19
x 0
y 0
Copyright © 2011 Pearson, Inc. Slide 7.5 - 11
Example Solving a Linear Programming Problem
The following table evaluates f at the corner points
of the region.
The maximum value of f is 18 at (2, 3)
The minimum value is 0 at (0, 0).
Copyright © 2011 Pearson, Inc. Slide 7.5 - 12
Quick Review
Find the x- and y-intercepts of the line.
1. 3x 4y 24
2. x
20
y
301
Find the point of intersection of the two lines.
3. x y 3 and 2x y 5
4. x y 1 and y 3x 1
5. 7x 3y 10 and x y 1
Copyright © 2011 Pearson, Inc. Slide 7.5 - 13
Quick Review Solutions
Find the x- and y-intercepts of the line.
1. 3x 4y 24 (0,6) and (8,0)
2. x
20
y
301 (0,30) and (20,0)
Find the point of intersection of the two lines.
3. x y 3 and 2x y 5 (8/3,1/3)
4. x y 1 and y 3x 1 (0,1)
5. 7x 3y 10 and x y 1 (1.3,0.3)
Copyright © 2011 Pearson, Inc. Slide 7.5 - 14
Chapter Test
1. Given A 1 3
4 0
, B
2 1
4 3
.
Find (a) A B (b) A B (c) 2A, and (d) 3A 2B.
Find AB and BA, or state that a given product is not possible.
2. A 1 2
3 1
4 3
, B
2 3 1
2 1 0
1 2 3
3. A 1 4 , B 5 3
2 1
Copyright © 2011 Pearson, Inc. Slide 7.5 - 15
Chapter Test
4. Find the inverse matrix if it has one.
1 0 1
2 1 1
1 1 1
5. Find the reduced row echelon form of the matrix
2 1 1 1
3 1 2 1
5 2 2 3
Copyright © 2011 Pearson, Inc. Slide 7.5 - 16
Chapter Test
6. Use Gaussian elimination to solve the system of equations.
x z w 2
x y z 3
3x 2y 3z w 8
7. Solve the system of equations by finding the reduced
row echelon form of the augmented matrix.
x 2y 2z w 8
2x 7 y 7z 2w 25
x 3y 3z w 11
Copyright © 2011 Pearson, Inc. Slide 7.5 - 17
Chapter Test
8. Find the partial fraction decomposition of 3x 2
x2 3x 4.
9.Find the minimum and maximum, if they exist, of the
objective function f , subject to the constraints.
Objective function: f 7x 6y
Constraints: 7x 7 y 100
2x 5y 50
x 0, y 0
Copyright © 2011 Pearson, Inc. Slide 7.5 - 18
Chapter Test
10. A stockbroker sold a customer 200 shares of stock A,
400 shares of stock B, 600 shares of stock C, and 250
shares of stock D. The price per share of A, B, C, and D
are $80, $120, $200, and $300, respectively.
(a) Write a 14 matrix N representing the number or
share of each stock the customer bought.
(b) Write a 14 matrix P representing the price per
share of eachstock.
(c) Write a matrix product that gives the total cost of the
stocks that the customer bought.
Copyright © 2011 Pearson, Inc. Slide 7.5 - 19
Chapter Test Solutions
1. Given A 1 3
4 0
, B
2 1
4 3
. Find (a) A B
1 2
8 3
(b) A B 3 4
0 3
(c) 2A,
2 6
8 0
(d) 3A 2B.
7 11
4 6
Find AB and BA, or state that a given product is not possible.
2. A 1 2
3 1
4 3
, B
2 3 1
2 1 0
1 2 3
not possible;
15 4
1 3
5 13
3. A 1 4 , B 5 3
2 1
3 7 ; not possible
Copyright © 2011 Pearson, Inc. Slide 7.5 - 20
Chapter Test Solutions
4. Find the inverse matrix if it has one.
1 0 1
2 1 1
1 1 1
0.4 0.2 0.2
0.2 0.4 0.6
0.6 0.2 0.2
5. Find the reduced row echelon form of the matrix
2 1 1 1
3 1 2 1
5 2 2 3
1 0 0 1
0 1 0 2
0 0 1 3
Copyright © 2011 Pearson, Inc. Slide 7.5 - 21
Chapter Test Solutions
6. Use Gaussian elimination to solve the system of equations.
x z w 2
x y z 3
3x 2y 3z w 8 z w 2, w1, z, w
7. Solve the system of equations by finding the reduced row echelon
form of the augmented matrix.
x 2y 2z w 8
2x 7 y 7z 2w 25
x 3y 3z w 11 w 2, z 3, z, w
Copyright © 2011 Pearson, Inc. Slide 7.5 - 22
Chapter Test Solutions
8. Find the partial fraction decomposition of 3x 2
x2 3x 4.
1
x 1
2
x 49. Find the minimum and maximum, if they exist, of the
objective function f , subject to the constraints.
Objective function: f 7x 6y
Constraints: 7x 7 y 100
2x 5y 50
x 0, y 0
minimum is 106 at (10,6); no maximum
Copyright © 2011 Pearson, Inc. Slide 7.5 - 23
Chapter Test Solutions
10. A stockbroker sold a customer 200 shares of stock A, 400 shares
of stock B, 600 shares of stock C, and 250 shares of stock D. The
price per share of A, B, C, and D are $80, $120, $200, and $300,
res
pectively.
(a) Write a 1 4 matrix representing the number or share of each
stock the customer bought.
(b) Write a 1 4 matrix representing the price per share of each
stock.
200 400 600 250
$80 $120 $200 $30
N
P
(c) Write a matrix product that gives the total cost of the stocks that
the customer bought.
0
$259, 0 00TNP
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