Tan S.T. Finite Mathematics for the Managerial, Life, and Social Sciences
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152 2 SYSTEMS OF LINEAR EQUATIONS AND MATRICES
9. a. For the product AB of two matrices A and B to be
defined, the number of
_____
of A must be equal to the
number of
_____
of B.
b. If A is an m n matrix and B is an n p matrix, then
the size of AB is
_____
.
10. a. If the products and sums are defined for the matrices A,
B, and C, then the associative law states that (AB)C
_____
; the distributive law states that A(B C)
_____
.
b. If I is an identity matrix of size n, then IA A if A is
any matrix of size
_____
.
11. A matrix A is nonsingular if there exists a matrix A
1
such
that
_____
_____
I. If A
1
does not exist, then A is
said to be
_____
.
12. A system of n linear equations in n variables written in the
form AX B has a unique solution given by X
_____
if
A has an inverse.
Review Exercises
CHAPTER 2
In Exercises 1–4, perform the operations if possible.
1.
2.
3.
4.
In Exercises 5–8, find the values of the variables.
5.
6.
7.
8.
In Exercises 9–16, compute the expressions if possible,
given that
C ⴝ £
3 ⴚ12
164
213
§
B ⴝ £
213
ⴚ2 ⴚ1 ⴚ1
142
§
A ⴝ £
131
ⴚ213
402
§
c
x 31
0 y 2
d£
11
3 z
42
§ c
12 4
22
d
£
3 a 3
1 b
c 1 d
§ £
36
e 24
12
§
c
3 x
y 3
dc
1
2
d c
7
4
d
c
1 x
y 3
d c
z 2
3 w
d
c
132
123
d£
1
4
2
§
33214 £
21
10
21
§
c
12
34
d c
12
5 2
d
£
12
13
21
§ £
10
01
12
§
9. 2A 3B 10. 3A 2 B
11. 2(3A) 12. 2(3A 4B)
13. A(B C) 14. AB AC
15. A(BC) 16.
In Exercises 17–24, solve the system of linear equations
using the Gauss–Jordan elimination method.
17. 2x 3y 5 18. 3x 2y 3
3x 4y 12x 4y 14
19. x y 2z 5 20. 3x 2y 4z 16
3x 2y z 10 2x y 2z 1
2x 3y 2z 10 x 4y 8z 18
21. 3x 2y 4z 11
2x 4y
5z 4
x 2y z 10
22. x 2y 3z 4w 17
2x y 2z 3w 9
3x y 2z 4w 0
4x 2y 3z w 2
23. 3x 2y z 4 24. 2 x 3y z 10
x 3y 4z 33x 2y 2z 2
2
x 3y 5z 7 x 3y 4z 7
x 8y 9z 10 4 x y z 4
In Exercises 25–32, find the inverse of the matrix (if it
exists).
25. 26.
27. 28.
29. 30.
31. 32.
A £
213
124
312
§A £
12 4
31 2
106
§
A £
124
213
102
§A £
231
1 12
121
§
A c
24
1 2
dA c
34
22
d
A c
24
16
dA c
31
12
d
1
2
1CA CB 2
87533_02_ch2_p067-154 1/30/08 9:43 AM Page 152
REVIEW EXERCISES 153
In Exercises 33–36, compute the value of the expressions
if possible, given that
33. (A
1
B)
1
34. (ABC)
1
35. (2A C)
1
36. (A B)
1
In Exercises 37–40, write each system of linear equations
in the form AX ⴝ C. Find A
ⴚ1
and use the result to solve
the system.
37. 2x 3y 8 38. x 3y 1
x 2y 32x 4y 8
39. x 2y 4z 13 40. 2 x 3
y 4 z 17
2x 3y 2z 0 x 2y 4z 7
x 4y 6z 15 3x y 2z 14
41. G
ASOLINE
S
ALES
Gloria Newburg operates three self-
service gasoline stations in different parts of town. On a
certain day, station A sold 600 gal of premium, 800 gal of
super, 1000 gal of regular gasoline, and 700 gal of diesel
fuel; station B sold 700 gal of premium, 600 gal of super,
1200 gal of regular gasoline, and 400 gal of diesel fuel; sta-
tion C sold 900 gal of premium, 700 gal of super, 1400 gal
of regular gasoline, and 800 gal of diesel fuel. Assume that
the price of gasoline was $3.20/gal for premium, $2.98/gal
for super, and $2.80/gal for regular and that diesel fuel sold
for $3.10/gal. Use matrix algebra to find the total revenue
at each station.
42. C
OMMON
S
TOCK
T
RANSACTIONS
Jack Spaulding bought
10,000 shares of stock X, 20,000 shares of stock Y, and
30,000 shares of stock Z at a unit price of $20, $30, and $50
per share, respectively. Six months later, the closing prices
of stocks X, Y, and Z were $22, $35, and $51 per share,
respectively. Jack made no other stock transactions during
the period in question. Compare the value of Jack’s stock
holdings at the time of purchase and 6 months later.
43. I
NVESTMENTS
William’s and Michael’s stock holdings are
given in the following table:
BAC GM IBM TRW
William 800 1200 400 1500
Michael 600 1400 600 2000
The prices (in dollars per share) of the stocks of BAC, GM,
IBM, and TRW at the close of the stock market on a cer-
tain day are $50.26, $31.00, $103.07 and $38.67, respec-
tively.
a. Write a 2 4 matrix A giving the stock holdings of
William and Michael.
b. Write a 4 1 matrix B giving the closing prices of the
stocks of BAC, GM, IBM, and TRW.
c. Use matrix multiplication to find the total value of the
stock holdings of William and Michael at the market
close.
44. I
NVESTMENT
P
ORTFOLIOS
The following table gives the
number of shares of certain corporations held by Olivia
C ⴝ c
11
ⴚ12
dB ⴝ c
31
42
dA ⴝ c
12
ⴚ12
d
and Max in their stock portfolios at the beginning of the
September and at the beginning of October:
September
IBM Google Boeing GM
Olivia 800 500 1200 1500
Max 500 600 2000 800
October
IBM Google Boeing GM
Olivia 900 600 1000 1200
Max 700 500 2100 900
a. Write matrices A and B giving the stock portfolios of
Olivia and Max at the beginning of September and at
the beginning of October, respectively.
b. Find a matrix C reflecting the change in the stock port-
folios of Olivia and Max between the beginning of Sep-
tember and the beginning of October.
45. M
ACHINE
S
CHEDULING
Desmond Jewelry wishes to pro-
duce three types of pendants: type A, type B, and type C.
To manufacture a type-A pendant requires 2 min on
machines I and II and 3 min on machine III. A type-B pen-
dant requires 2 min on machine I, 3 min on machine II, and
4 min on machine III. A type-C pendant requires 3 min on
machine I, 4 min on machine II, and 3 min on machine III.
There are hr available on machine I, hr available on
machine II, and 5 hr available on machine III. How many
pendants of each type should Desmond make in order to
use all the available time?
46. P
ETROLEUM
P
RODUCTION
Wildcat Oil Company has two re-
fineries, one located in Houston and the other in Tulsa. The
Houston refinery ships 60% of its petroleum to a Chicago
distributor and 40% of its petroleum to a Los Angeles dis-
tributor. The Tulsa refinery ships 30% of its petroleum to
the Chicago distributor and 70% of its petroleum to the Los
Angeles distributor. Assume that, over the year, the
Chicago distributor received 240,000 gal of petroleum and
the Los Angeles distributor received 460,000 gal of petro-
leum. Find the amount of petroleum produced at each of
Wildcat’s refineries.
47. I
NPUT
–O
UTPUT
M
ATRICES
The input–output matrix associ-
ated with an economy based on agriculture (A) and manu-
facturing (M) is given by
AM
A
M
a. Determine the amount of agricultural products con-
sumed in the production of $200 million worth of man-
ufactured goods.
b. Determine the dollar amount of manufactured goods
required to produce $300 million worth of all goods in
the economy.
c. Which sector consumes the greater amount of agricul-
tural products in the production of 1 unit of goods in
that sector? The lesser?
c
0.2 0.15
0.1 0.15
d
4
1
2
3
1
2
87533_02_ch2_p067-154 1/30/08 9:43 AM Page 153
154 2 SYSTEMS OF LINEAR EQUATIONS AND MATRICES
48. I
NPUT
–O
UTPUT
M
ATRICES
The matrix
is an input–output matrix associated with an economy and
the matrix
(units in millions of dollars) is a demand vector. Find the
final outputs of each industry such that the demands of
industry and the consumer sector are met.
D £
4
12
16
§
A £
0.1 0.3 0.4
0.3 0.1 0.2
0.2 0.1 0.3
§
49. I
NPUT
–O
UTPUT
M
ATRICES
The input–output matrix associ-
ated with an economy based on agriculture (A) and manu-
facturing (M) is given by
AM
A
M
a. Find the gross output of goods needed to satisfy a
consumer demand for $100 million worth of agricul-
tural products and $80 million worth of manufactured
goods.
b. Find the value of agricultural products and manufac-
tured goods consumed in the internal process of pro-
duction in order to meet this gross output.
c
0.2 0.15
0.1 0.15
d
Before Moving On . . .
CHAPTER 2
1. Solve the following system of linear equations, using the
Gauss–Jordan elimination method:
2. Find the solution(s), if it exists, of the system of linear
equations whose augmented matrix in reduced form fol-
lows.
a. b.
c. d.
e.
3. Solve each system of linear equations using the Gauss–
Jordan elimination method.
a. b.
4x y 2
3x y 2z 13x y 5
x 2y 4z 2x 2y 3
c
101
01 2
`
2
3
d
≥
1000
0100
0010
0001
∞
0
0
0
0
¥£
100
013
000
†
2
1
0
§
£
100
010
000
†
3
0
1
§£
100
010
001
†
2
3
1
§
3x 3y 3z 5
x 3y 2
z 2
2x y z 1
4. Let
Find (a) AB, (b) (A C
T
)B, and (c) C
T
B AB
T
.
5. Find A
1
if
6. Solve the system
by first writing it in the matrix form AX B and then find-
ing A
1
.
3x y z 0
2x y z 1
2x z 4
A £
212
0 13
110
§
C £
2 2
11
34
§
B £
1 12
311
210
§A c
1 24
301
d
87533_02_ch2_p067-154 1/30/08 9:43 AM Page 154
How many souvenirs should Ace
Novelty make in order to maximize
its profit? The company produces
two types of souvenirs, each of
which requires a certain amount of
time on each of 2 different
machines. Each machine can be
operated for only a certain number
of hours per day. In Example 1, page
164, we show how this production
problem can be formulated as a
linear programming problem, and in
Example 1, page 175, we solve this
linear programming problem.
M
ANY PRACTICAL PROBLEMS involve maximizing or
minimizing a function subject to certain constraints.
For example, we may wish to maximize a profit function subject
to certain limitations on the amount of material and labor
available. Maximization or minimization problems that can be
formulated in terms of a linear objective function and constraints
in the form of linear inequalities are called linear programming
problems. In this chapter, we look at linear programming
problems involving two variables. These problems are amenable
to geometric analysis, and the method of solution introduced
here will shed much light on the basic nature of a linear
programming problem.
LINEAR
PROGRAMMING:
A GEOMETRIC
APPROACH
3
© Christa Knijff/Alamy
87533_03_ch3_p155-200 1/30/08 9:47 AM Page 155
156 3 LINEAR PROGRAMMING: A GEOMETRIC APPROACH
Graphing Linear Inequalities
In Chapter 1, we saw that a linear equation in two variables x and y
a
,
b
not both equal to zero
has a solution set that may be exhibited graphically as points on a straight line in the
xy-plane. We now show that there is also a simple graphical representation for linear
inequalities in two variables:
Before turning to a general procedure for graphing such inequalities, let’s con-
sider a specific example. Suppose we wish to graph
2x ⫹ 3y ⬍ 6 (1)
We first graph the equation 2x ⫹ 3y ⫽ 6, which is obtained by replacing the given
inequality “⬍” with an equality “⫽” (Figure 1).
Observe that this line divides the xy-plane into two half-planes: an upper half-
plane and a lower half-plane. Let’s show that the upper half-plane is the graph of the
linear inequality
2x ⫹ 3y ⬎ 6 (2)
whereas the lower half-plane is the graph of the linear inequality
2x ⫹ 3y ⬍ 6 (3)
To see this, let’s write Inequalities (2) and (3) in the equivalent forms
(4)
and
(5)y ⬍⫺
2
3
x ⫹ 2
y ⬎⫺
2
3
x ⫹ 2
x
Upper half-plane
P(x, y)
2x + 3y = 6
Lower half-plane
L
y
x
5
−5
−5
5
2
3
Q x, − x + 2
ax ⫹ by ⫹ c ⱖ 0ax ⫹ by ⫹ c ⬎ 0
ax ⫹ by ⫹ c ⱕ 0ax ⫹ by ⫹ c ⬍ 0
ax ⫹ by ⫹ c ⫽ 0
FIGURE 1
A straight line divides the xy-plane into
two half-planes.
3.1 Graphing Systems of Linear Inequalities in Two Variables
87533_03_ch3_p155-200 1/30/08 9:48 AM Page 156
The equation of the line itself is
(6)
Now pick any point P(x, y) lying above the line L. Let Q be the point lying on
L and directly below P (see Figure 1). Since Q lies on L, its coordinates must satisfy
Equation (6). In other words, Q has representation QÓx, ⫺
ᎏ
2
3
ᎏ
x ⫹ 2Ô. Comparing the
y-coordinates of P and Q and recalling that P lies above Q, so that its y-coordinate
must be larger than that of Q, we have
But this inequality is just Inequality (4) or, equivalently, Inequality (2). Similarly, we
can show that every point lying below L must satisfy Inequality (5) and there-
fore (3).
This analysis shows that the lower half-plane provides a solution to our problem
(Figure 2). (By convention, we draw the line as a dashed line to show that the points
on L do not belong to the solution set.) Observe that the two half-planes in question
are disjoint; that is, they do not have any points in common.
Alternatively, there is a simpler method for determining the half-plane that pro-
vides the solution to the problem. To determine the required half-plane, let’s pick any
point lying in one of the half-planes. For simplicity, pick the origin (0, 0), which lies
in the lower half-plane. Substituting x ⫽ 0 and y ⫽ 0 (the coordinates of this point)
into the given Inequality (1), we find
2(0) ⫹ 3(0) ⬍ 6
or 0 ⬍ 6, which is certainly true. This tells us that the required half-plane is the one
containing the test point—namely, the lower half-plane.
Next, let’s see what happens if we choose the point (2, 3), which lies in the upper
half-plane. Substituting x ⫽ 2 and y ⫽ 3 into the given inequality, we find
2(2) ⫹ 3(3) ⬍ 6
or 13 ⬍ 6, which is false. This tells us that the upper half-plane is not the required half-
plane, as expected. Note, too, that no point P(x, y) lying on the line constitutes a solu-
tion to our problem, given the strict inequality ⬍.
This discussion suggests the following procedure for graphing a linear inequality
in two variables.
x
2x + 3y = 6
y
5
5
−5
2x + 3y < 6
y ⬎⫺
2
3
x ⫹ 2
y ⫽⫺
2
3
x ⫹ 2
3.1 GRAPHING SYSTEMS OF LINEAR INEQUALITIES IN TWO VARIABLES 157
FIGURE 2
The set of points lying below the dashed
line satisfies the given inequality.
87533_03_ch3_p155-200 1/30/08 9:48 AM Page 157
EXAMPLE 1
Determine the solution set for the inequality 2x ⫹ 3y ⱖ 6.
Solution
Replacing the inequality ⱖ with an equality ⫽, we obtain the equation
2x ⫹ 3y ⫽ 6, whose graph is the straight line shown in Figure 3.
Instead of a dashed line as before, we use a solid line to show that all points on the
line are also solutions to the inequality. Picking the origin as our test point, we find
2(0) ⫹ 3(0) ⱖ 6, or 0 ⱖ 6, which is false. So we conclude that the solution set is
made up of the half-plane not containing the origin, including (in this case) the line
given by 2x ⫹ 3y ⫽ 6.
EXAMPLE 2
Graph x ⱕ⫺1.
Solution
The graph of x ⫽⫺1 is the vertical line shown in Figure 4. Picking the
origin (0, 0) as a test point, we find 0 ⱕ⫺1, which is false. Therefore, the required
solution is the left half-plane, which does not contain the origin.
EXAMPLE 3
Graph x ⫺ 2y ⬎ 0.
Solution
We first graph the equation x ⫺ 2y ⫽ 0, or (Figure 5). Since the
origin lies on the line, we may not use it as a test point. (Why?) Let’s pick (1, 2)
as a test point. Substituting x ⫽ 1 and y ⫽ 2 into the given inequality, we find
1 ⫺ 2(2) ⬎ 0, or ⫺3 ⬎ 0, which is false. Therefore, the required solution is the half-
plane that does not contain the test point—namely, the lower half-plane.
y ⫽
1
2
x
y
x
–5 5
5
2x + 3y = 6
2x + 3y ⱖ 6
Procedure for Graphing Linear Inequalities
1. Draw the graph of the equation obtained for the given inequality by replac-
ing the inequality sign with an equal sign. Use a dashed or dotted line if the
problem involves a strict inequality, ⬍ or ⬎. Otherwise, use a solid line to
indicate that the line itself constitutes part of the solution.
2. Pick a test point lying in one of the half-planes determined by the line
sketched in step 1 and substitute the values of x and y into the given inequal-
ity. For simplicity, use the origin whenever possible.
3. If the inequality is satisfied, the graph of the solution to the inequality is the
half-plane containing the test point. Otherwise, the solution is the half-plane
not containing the test point.
158 3 LINEAR PROGRAMMING: A GEOMETRIC APPROACH
FIGURE 3
The set of points lying on the line and in
the upper half-plane satisfies the given
inequality.
y
x
x = –1
x ≤ –1
FIGURE 4
The set of points lying on the line
x ⫽⫺1 and in the left half-plane satisfies
the given inequality.
87533_03_ch3_p155-200 1/30/08 9:48 AM Page 158
Graphing Systems of Linear Inequalities
By the solution set of a system of linear inequalities in the two variables x and y we
mean the set of all points (x, y) satisfying each inequality of the system. The graphi-
cal solution of such a system may be obtained by graphing the solution set for each
inequality independently and then determining the region in common with each solu-
tion set.
EXAMPLE 4
Determine the solution set for the system
Solution
Proceeding as in the previous examples, you should have no difficulty
locating the half-planes determined by each of the linear inequalities that make up
the system. These half-planes are shown in Figure 6. The intersection of the two
half-planes is the shaded region. A point in this region is an element of the solution
set for the given system. The point P, the intersection of the two straight lines deter-
mined by the equations, is found by solving the simultaneous equations
x ⫺ y ⫽ 0
4x ⫹ 3y ⫽ 12
x ⫺ y ⱕ 0
4x ⫹ 3y ⱖ 12
y
x
x – 2y = 0
3.1 GRAPHING SYSTEMS OF LINEAR INEQUALITIES IN TWO VARIABLES 159
y
x
x – y = 0
3–3
4x + 3y = 12
12
12
7
7
,
P
FIGURE 6
The set of points in the shaded area sat-
isfies the system
4x ⫹ 3y ⱖ 12
x ⫺ y ⱕ 0
FIGURE 5
The set of points in the lower half-plane
satisfies x ⫺ 2y ⬎ 0.
FIGURE a FIGURE b
TI 83/84 screen Graph of the inequality 2x ⫹ 3y ⱖ 6
Exploring with
TECHNOLOGY
A graphing utility can be used to plot the graph of a linear inequality. For exam-
ple, to plot the solution set for Example 1, first rewrite the equation 2x ⫹ 3y ⫽
6 in the form . Next, enter this expression for Y
1
in the calculator and
move the cursor to the left of Y
1
. Then, press repeatedly and select the
icon that indicates the shading option desired (see Figure a). The required graph
follows (see Figure b).
ENTER
y ⫽ 2 ⫺
2
3
x
Plot1 Plot2 Plot3
Y
1
= 2–(2/3)X
\Y
2
=
\Y
3
=
\Y
4
=
\Y
5
=
\Y
6
=
\Y
7
=
87533_03_ch3_p155-200 1/30/08 9:48 AM Page 159
EXAMPLE 5
Sketch the solution set for the system
Solution
The first inequality in the system defines the right half-plane—all
points to the right of the y-axis plus all points lying on the y-axis itself. The second
inequality in the system defines the upper half-plane, including the x-axis. The
half-planes defined by the third and fourth inequalities are indicated by arrows in
Figure 7. Thus, the required region—the intersection of the four half-planes defined
by the four inequalities in the given system of linear inequalities—is the shaded
region. The point P is found by solving the simultaneous equations x ⫹ y ⫺ 6 ⫽ 0
and 2x ⫹ y ⫺ 8 ⫽ 0.
The solution set found in Example 5 is an example of a bounded set. Observe that the
set can be enclosed by a circle. For example, if you draw a circle of radius 10 with
center at the origin, you will see that the set lies entirely inside the circle. On the other
hand, the solution set of Example 4 cannot be enclosed by a circle and is said to be
unbounded.
EXAMPLE 6
Determine the graphical solution set for the following system of lin-
ear inequalities:
Solution
The required solution set is the unbounded region shown in Figure 8.
y ⱖ 0
x ⱖ 0
x ⫹ 2y ⱖ 40
2 x ⫹ y ⱖ 50
Bounded and Unbounded Solution Sets
The solution set of a system of linear inequalities is bounded if it can be
enclosed by a circle. Otherwise, it is
unbounded
.
y
x
2x + y – 8 = 0
10
510–5–10
5
P
(2, 4)
x + y – 6 = 0
2x ⫹ y ⫺ 8 ⱕ 0
x ⫹ y ⫺ 6 ⱕ 0
y ⱖ 0
x ⱖ 0
160 3 LINEAR PROGRAMMING: A GEOMETRIC APPROACH
FIGURE 7
The set of points in the shaded region,
including the x- and y-axes, satisfies the
given inequalities.
87533_03_ch3_p155-200 1/30/08 9:48 AM Page 160
3.1 GRAPHING SYSTEMS OF LINEAR INEQUALITIES IN TWO VARIABLES 161
FIGURE 8
The solution set is an unbounded region.
1. Determine graphically the solution set for the following
system of inequalities:
x ⱖ 0, y ⱖ 0
5x ⫹ 3y ⱕ 30
x ⫹ 2y ⱕ 10
2. Determine graphically the solution set for the following
system of inequalities:
Solutions to Self-Check Exercises 3.1 can be found on
page 163.
x ⱖ 2
x ⫺ 3y ⱕ 0
5x ⫹ 3y ⱖ 30
y
x
x + 2y = 40
5010
(20, 10)
–50
–10
2x + y = 50
50
10
3.1 Exercises
3.1 Self-Check Exercises
3.1 Concept Questions
In Exercises 1–10, find the graphical solution of each
inequality.
1. 4x ⫺ 8 ⬍ 0 2. 3y ⫹ 2 ⬎ 0
3. x ⫺ y ⱕ 0 4. 3x ⫹ 4y ⱕ⫺2
5. x ⱕ⫺3 6. y ⱖ⫺1
7. 2x ⫹ y ⱕ 4 8. ⫺3x ⫹ 6y ⱖ 12
9. 4x ⫺ 3y ⱕ⫺24 10. 5x ⫺ 3y ⱖ 15
In Exercises 11–18, write a system of linear inequalities
that describes the shaded region.
11.
y
x
y = 4
y = 2
x = 1 x = 5
1. a. What is the difference, geometrically, between the solu-
tion set of ax ⫹ by ⬍ c and the solution set of ax ⫹ by
ⱕ c?
b. Describe the set that is obtained by intersecting the solu-
tion set of ax ⫹ by ⱕ c with the solution set of ax ⫹
by ⱖ c.
2. a. What is the solution set of a system of linear inequal-
ities?
b. How do you find the solution of a system of linear
inequalities graphically?
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