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CONTENTS v
7.7 Normal Approximation to the Binomial Distribution 359

Chapter Summary and Chapter Review Exercises 363
Chapter Project: An Unexpected Expected Value 368
PART THREE
8 Markov Processes 369

8.1 The Transition Matrix 369
8.2 Regular Stochastic Matrices 381
8.3 Absorbing Stochastic Matrices 389
Chapter Project: Doubly Stochastic Matrices 401
9 The Theory of Games 404

9.1 Games and Strategies 404
9.2 Mixed Strategies 410
9.3 Determining Optimal Mixed Strategies 417
Chapter Project: Simulating the Outcomes of Mixed-Strategy
Games 428
PART FOUR
10 The Mathematics of Finance 430

10.1 Interest 430
10.2 Annuities 440
10.3 Amortization of Loans 449
10.4 Personal Financial Decisions 458
10.5 A Unifying Equation 474
Chapter Project: Two Items of Interest 489
11 Logic 491
11.1 Introduction to Logic 491
11.2 Truth Tables 495
11.3 Implication 504
11.4 Logical Implication and Equivalence 510
11.5 Valid Argument 518
11.6 Predicate Calculus 525
11.7 Logic Circuits 533
Chapter Project: A Logic Puzzle 542
vi CONTENTS
12 Difference Equations and Mathematical Models (Online*)

12.1 Introduction to Difference Equations D1
12.2 Difference Equations and Interest D8
12.3 Graphing Difference Equations D13
12.4 Mathematics of Personal Finance D22
12.5 Modeling with Difference Equations D26
Chapter Summary and Chapter Review Exercises D30
Chapter Project: Connections to Markov Processes D33
Appendix A Areas Under the Standard Normal Curve A-1
Appendix B Using the TI-84 Plus Graphing Calculator A-2
Appendix C Spreadsheet Fundamentals A-6
Appendix D Wolfram|Alpha A-10
Learning Objectives (Online*) A-11
Selected Answers SA-1
Index of Applications IA-1
Index I-1
*www.pearsonhighered.com/mathstatsresources
Preface
T his work is the twelfth edition of our text for the finite mathematics course taught to
first- and second-year college students, especially those majoring in business and the
social and biological sciences. Finite mathematics courses exhibit tremendous diversity
with respect to both content and approach. Therefore, in developing this book, we
incorporated a wide range of topics from which an instructor may design a curriculum,
as well as a high degree of flexibility in the order in which the topics may be presented.
For the mathematics of finance, we even allow for flexibility in the approach of the pres-
entation.
The Series
This text is part of a highly successful series consisting of three texts: Finite Mathematics
& Its Applications, Calculus & Its Applications, and Calculus & Its Applications, Brief
Version. All three titles are available for purchase in a variety of formats, including as an
eBook within the MyMathLab online course.
fourteenth edition fourteenth edition
twelfth edition
Finite Calculus
& ITS APPLICATIONS
Calculus
& ITS APPLICATIONS
Mathematics
& ITS APPLICATIONS
BRIEF VERSION
Goldstein
Schneider Goldstein Goldstein
Siegel Lay Lay
Hair Schneider Schneider
Asmar Asmar
Topics Included
This edition has more material than can be covered in most one-semester courses. There-
fore, the instructor can structure the course to the students’ needs and interests. The
book divides naturally into four parts:
• Part One (Chapters 1–4) consists of linear mathematics: linear equations, matrices,
and linear programming.
• Part Two (Chapters 5–7) is devoted to counting, probability, and statistics.
• Part Three (Chapters 8 and 9) covers topics utilizing the ideas of the other parts.
• Part Four (Chapters 10–12) explores key topics from discrete mathematics that are
sometimes included in the modern finite mathematics curriculum.
Minimal Prerequisites
Because of great variation in student preparation, we keep formal prerequisites to a min-
imum. We assume only a first year of high school algebra, and we review, as needed,
those topics that are typically weak spots for students.
vii
viii PREFACE
New to This Edition

We welcome to this edition a new co-author, Steven Hair from Penn State University.
Steve has brought a fresh eye to the content and to the MyMathLab course that accom-
panies the text.
We are grateful for the many helpful suggestions made by reviewers and users of the
text. We incorporated many of these into this new edition. We also analyzed aggregated
student usage and performance data from MyMathLab for the previous edition of this
text. The results of this analysis helped improve the quality and quantity of exercises
that matter the most to instructors and students. Additionally, we made the following
improvements in this edition:
• Help-Text Added. We added blue “help text” next to steps within worked-out
examples to point out key algebraic and numerical transitions.
• Updated Technology. We changed the graphing calculator screen captures to the
more current TI-84 Plus CE format.The discussions of Excel now refer to Excel
2013 and Excel 2016.
• Additional Exercises and Updated Data. We have added or updated 440 exer-
cises and have updated the real-world data appearing in the examples and exercises.
The book now contains 3580 exercises and 370 worked-out examples.
• Technology Solutions. We added technology-based solutions to more examples to
provide flexibility for instructors who incorporate technology. For instance, the sec-
tion on the method of least-squares (1.4) now relies more on technology and less on
complicated calculations. In Section 7.6, several examples now demonstrate how to
compute the area under a normal curve using a graphing calculator, in addition to
the table-based method. In the finance chapter, many TI-84 Plus TVM Solver screen
captures accompany examples to confirm answers. Instructors have the option of
using TVM Solver for financial calculations instead of complicated formulas.
• Linear Inequalities Section Relocated. We moved this section from 1.2 (in the 11e)
to the beginning of the linear programming chapter (Ch. 3) in this edition. The
move places the topic in the chapter where it is used. Also, the move allows us to use
conventional names (such as slope-intercept form) in the section.
• Improved Coverage of Counting Material. In Chapter 5, we added several defini-
tions and discussions to aid student comprehension of counting problems. We
moved the definition of factorials to 5.4 and rewrote the permutation and combina-
tion formulas in 5.5 in terms of factorials. In 5.6, the complement rule for counting
is now formally defined, and we have added a discussion of when addition, subtrac-
tion, and multiplication is appropriate for solving counting problems.
• Section Added to the End of the Finance Chapter. Titled “A Unifying Equa-
tion,” this new section shows that the basic financial concepts can be described by a
difference equation of the form yn = a # yn - 1 + b, y0 given, and that many of the
calculations from the chapter can be obtained by solving this difference equation.
Examples and exercises show that this difference equation also can be used to solve
problems in the physical, biological, and social sciences. This section can be taught as
a standalone section without covering the preceding sections of the finance chapter.
• Revision of Logic Material. We substantially revised Chapter 11 on logic to better
meet student needs.We moved the definition of logical equivalence and De Morgan’s
laws from 11.4 to 11.2. By stating key ideas related to truth tables and implications
in terms of logical equivalence, students will be better equipped to understand these
concepts. To remove confusion between the inclusive and exclusive “or” statements,
we removed the word “either” from inclusive “or” statements in English. In 11.4, we
added the definition of the inverse of an implication. This is a key concept in the
topic of implications and logical arguments. To help students understand when a
logical argument is invalid, we expanded 11.5 to include more discussion of invalid
arguments. Additionally, we added the fallacies of the inverse and converse, and two
new examples where arguments are proven to be invalid.
PREFACE ix
• Difference Equation Chapter Moved Online. We moved former Chapter 11 online

(relabeling it Chapter 12 in the process). The chapter is available directly to students
at www.pearsonhighered.com/mathstatsresources and within MyMathLab. All
support materials for the chapter appear online within MyMathLab. Note: The new
section at the end of the finance chapter contains the fundamental concepts from the
difference equation chapter.
New to MyMathLab
Many improvements have been made to the overall functionality of MyMathLab
(MML) since the previous edition. However, beyond that, we have also invested in
increasing and improving the content specific to this text.
• Instructors now have more exercises than ever to choose from in assigning home-
work. There are approximately 2540 assignable exercises in MML.
• We heard from users that the Annotated Instructor Edition for the previous edition
required too much flipping of pages to find answers, so MML now contains a down-
loadable Instructor Answers document—with all answers in one place. (This aug-
ments the downloadable Instructor Solutions Manual, which contains all solutions.)
• Interactive Figures are now in HTML format (no plug-in required) and are sup-
ported by assignable exercises and tutorial videos.
• An Integrated Review version of the MML course contains pre-made quizzes to
assess the prerequisite skills needed for each chapter, plus personalized remediation
for any gaps in skills that are identified.
• New Setup & Solve exercises require students to show how they set up a problem as
well as the solution, better mirroring what is required of students on tests.
• StatCrunch, a fully functional statistics package, is provided to support the statistics
content in the course.
• MathTalk and StatTalk videos highlight applications of the content of the course to
business. The videos are supported by assignable exercises.
• Study skills modules help students with the life skills that can make the difference
between passing and failing.
• 110 new tutorial videos by Brian Rickard (University of Arkansas) were added to
support student learning.
• Tutorial videos involving graphing calculators are now included within MML exer-
cises to augment videos showing “by hand” methods. If you require graphing calcu-
lator usage for the course, your students will find these videos very helpful. (If you
do not use calculators, you can hide these videos from students.)
• Graphing Calculator and Excel Spreadsheet Manuals, specific to this course, are
now downloadable from MML.
Trusted Features
Though this edition has been improved in a variety of ways to reflect changing student
needs, we have maintained the popular overall approach that has helped students be
successful over the years.
Relevant and Varied Applications

We provide realistic applications that illustrate the uses of finite mathematics in other
disciplines and everyday life. The variety of applications is evident in the Index of Appli-
cations at the end of the text. Wherever possible, we attempt to use applications to moti-
vate the mathematics. For example, the concept of linear programming is introduced in
Chapter 3 via a discussion of production options for a factory with labor limitations.
Plentiful Examples
The twelfth edition includes 370 worked examples. Furthermore, we include computa-
tional details to enhance comprehension by students whose basic skills are weak.
x PREFACE
nowing that students often refer back to examples for help, we built in fidelity between
K
exercises and examples. In addition, students are given Now Try exercise references
immediately following most examples to encourage them to check their understanding
of the given example.
Exercises to Meet All Student Needs

The 3580 exercises comprise about one-quarter of the book—the most important part of
the text, in our opinion. The exercises at the ends of the sections are typically arranged
in the order in which the text proceeds, so that homework assignments may be made
easily after only part of a section is discussed. Interesting applications and more chal-
lenging problems tend to be located near the ends of the exercise sets. Exercises have
odd-even pairing, when appropriate. Chapter Review Exercises are designed to prepare
students for end-of-chapter tests. Answers to the odd-numbered exercises, and all Chap-
ter Review Exercises, are included at the back of the book.
Check Your Understanding Problems

The Check Your Understanding problems are a popular and useful feature of the
book. They are carefully selected exercises located at the end of each section, just
before the exercise set. Complete solutions follow the exercise set. These problems pre-
pare students for the exercise sets beyond just covering simple examples. They give
students a chance to think about the skills they are about to apply and reflect on what
they’ve learned.
Use of Technology
We incorporated technology usage into the text in ways that provide you with flexibility,
knowing that the course can vary quite a bit based on how technology is incorporated.
Our basic approach in the text is to assume minimal use of technology and clearly label
the opportunities to make it a greater part of the course. Many of the sections contain
Incorporating Technology features that show how to use Texas Instruments graphing
calculators, Excel spreadsheets, and Wolfram|Alpha. In addition, the text contains
appendixes on the use of these technologies. Each type of technology is clearly labeled
with an icon:
(Graphing Calculator),
(Spreadsheet),
(Wolfram|Alpha)
In our discussions of graphing calculators, we specifically refer to the TI-84 Plus
models, since these are the most popular graphing calculators. New to this edition,
screen shots display the new color versions of the TI-84. Spreadsheets refer to Micro-
soft Excel 2016. The web application discussed is Wolfram|Alpha, which is an excep-
tionally fine and versatile product that is available online or on mobile devices for free
or at low cost. We feel that Wolfram|Alpha is a powerful tool for learning and exploring
mathematics, which is why we chose to include activities that use it. We hope that by
modeling appropriate use of this technology, students will come to appreciate the appli-
cation for its true worth.
End-of-Chapter Study Aids

Near the end of each chapter is a set of problems entitled Fundamental Concept Check
Exercises that help students recall key ideas of the chapter and focus on the relevance of
these concepts as well as prepare for exams. Each chapter also contains a two-column
grid giving a section-by-section summary of key terms and concepts with examples.
Finally, each chapter has Chapter Review Exercises that provide more practice and
preparation for chapter-level exams.
PREFACE xi
Chapter Projects
Each chapter ends with an extended project that can be used as an in-class or out-of-
class group project or special assignment. These projects develop interesting applica-
tions or enhance key concepts of the chapters.
Technology and Supplements

MyMathLab® Online Course (access code required)
Built around Pearson’s best-selling content, MyMathLab is an online homework, tutorial, and
assessment program designed to work with this text to engage students and improve results.
MyMathLab can be successfully implemented in any classroom environment—lab-based, hybrid,
fully online, or traditional. By addressing instructor and student needs, MyMathLab
improves student learning.
Used by more than 37 million students worldwide, MyMathLab delivers consistent, measur-
able gains in student learning outcomes, retention, and subsequent course success. Visit www.
mymathlab.com/results to learn more.
Preparedness
One of the biggest challenges in Finite Mathematics courses is making sure students are ade-
quately prepared with the prerequisite skills needed to successfully complete their course work.
Pearson offers a variety of content and course options to support students with just-in-time reme-
diation and key-concept review.
• Integrated Review Courses can be used for just-in-time prerequisite review. These courses
provide additional content on review topics, along with pre-made, assignable skill-check quiz-
zes, personalized homework assignments, and videos integrated throughout the course.
Motivation
Students are motivated to succeed when they’re engaged in the learning experience and under-
stand the relevance and power of mathematics. MyMathLab’s online homework offers stu-
dents immediate feedback and tutorial assistance that motivates them to do more, which
means they retain more knowledge and improve their test scores.
• Exercises with immediate feedback—over 2540 assignable exercises—are based on the text-
book exercises, and regenerate algorithmically to give students unlimited opportunity for
practice and mastery. MyMathLab provides helpful feedback when students enter incorrect
answers and includes optional learning aids including Help Me Solve This, View an Example,
videos, and an eText.
xii PREFACE
• Setup and Solve Exercises ask students to first describe how they will set up and ap-
proach the problem. This reinforces students’ conceptual understanding of the process
they are applying and promotes long-term retention of the skill.
• MathTalk and StatTalk videos connect the math to the real world (particularly busi-
ness). The videos include assignable exercises to gauge students’ understanding of video
content.
• Learning Catalytics™ is a student re-
sponse tool that uses students’ smart-
phones, tablets, or laptops to engage
them in more interactive tasks and
thinking. Learning Catalytics fosters
student engagement and peer-to-peer
learning with real-time analytics.
Learning and Teaching Tools

• Interactive Figures illustrate key concepts and allow manipulation for use as teaching and
learning tools. MyMathLab includes assignable exercises that require use of figures and
instructional videos that explain the concept behind each figure.
• Instructional videos—238 example-based videos—are available as learning aids within

exercises and for self-study. The Guide to Video-Based Assignments makes it easy to assign
videos for homework by showing which MyMathLab exercises correspond to each video.
PREFACE xiii
• Graphing Calculator videos are available to augment “by hand” methods, allowing you to
match the help that students receive to how graphing calculators are used in the course. Videos
are available within select exercises and in the Multimedia Library.
• Complete eText is available to students through their MyMathLab courses for the lifetime
of the edition, giving students unlimited access to the eText within any course using that edi-
tion of the textbook.
• StatCrunch, a fully functional statistics package, is provided to support the statistics content
in the course.
• Skills for Success Modules help students with the life skills that can make the difference
between passing and failing. Topics include “Time Management” and “Stress Management.”
• Excel Spreadsheet Manual, specifically written for this course.
• Graphing Calculator Manual, specifically written for this course.
• PowerPoint Presentations are available for download for each section of the book.
• Accessibility and achievement go hand in hand. MyMathLab is compatible with the JAWS
screen reader, and enables multiple-choice and free-response problem types to be read and
interacted with via keyboard controls and math notation input. MyMathLab also works with
screen enlargers, including ZoomText, MAGic, and SuperNova. And, all MyMathLab videos
have closed-captioning. More information is available at http://mymathlab.com/accessibility.
• A comprehensive gradebook with enhanced reporting functionality allows you to efficiently
manage your course.
• The Reporting Dashboard provides insight to view, analyze, and report learning outcomes.
Student performance data is presented at the class, section, and program levels in an accessi-
ble, visual manner so you’ll have the information you need to keep your students on track.
• Item Analysis tracks class-wide understanding of particular exercises so you can refine
your class lectures or adjust the course/department syllabus. Just-in-time teaching has
never been easier!
MyMathLab comes from an experienced partner with educational expertise and an eye on the
future. Whether you are just getting started with MyMathLab, or have a question along the way,
we’re here to help you learn about our technologies and how to incorporate them into your course.
To learn more about how MyMathLab helps students succeed, visit www.mymathlab.com or con-
tact your Pearson rep.
MathXL® is the homework and assessment engine that runs MyMathLab. (MyMathLab is
MathXL plus a learning management system.) MathXL access codes are also an option.
Student Solutions Manual

ISBN-10: 0-134-46344-7 | ISBN-13: 978-0-134-46344-5
Contains fully worked-out solutions to odd-numbered exercises. Available in print and
downloadable from within MyMathLab.
Instructor Answers / Instructor Solutions Manual (downloadable)

ISBN-10: 0-134-46343-9 | ISBN-13: 978-0-134-46343-8
The Instructor Answers document contains a list of answers to all student edition exercises. The
Instructor Solutions Manual contains solutions to all student edition exercises. Downloadable
from the Pearson Instructor Resource Center www.pearsonhighered.com/irc, or from within
MyMathLab.
xiv PREFACE
TestGen (downloadable)
ISBN-10: 0-134-46346-3 | ISBN-13: 978-0-134-46346-9
TestGen enables instructors to build, edit, print, and administer tests using a bank of questions
developed to cover all objectives in the text. TestGen is algorithmically based, allowing you to
create multiple but equivalent versions of the same question or test. Instructors can also modify
testbank questions or add new questions. The software and testbank are available to qualified
instructors for download and installation from Pearson’s online catalog www.pearsonhighered.
com and from within MyMathLab.
PowerPoints
ISBN-10: 0-134-46407-9 | ISBN-13: 978-0-134-46407-7
Contains classroom presentation slides for this textbook featuring lecture content, worked-out
examples, and key graphics from the text. Available to qualified instructors within MyMathLab
or through the Pearson Instructor Resource Center www.pearsonhighered.com/irc.
Acknowledgments
While writing this book, we have received assistance from many people, and our heartfelt
thanks go out to them all. Especially, we should like to thank the following reviewers,
who took the time and energy to share their ideas, preferences, and often their enthusi-
asm, with us during this revision:
Jeff Dodd, Jacksonville State University
Timothy M. Doyle, University of Illinois at Chicago
Sami M. Hamid, University of North Florida
R. Warren Lemerich, Laramie County Community College
Antonio Morgan, Robert Morris University
Arthur J. Rosenthal, Salem State University
Mary E. Rudis, Great Bay Community College
Richard Smatt, Mount Washington College
Paul J. Welsh, Pima Community College
The following faculty members provided direction on the development of the
MyMathLab course for this edition:
Mark A. Crawford, Jr., Waubonsee Community College
Cymra Haskell, University of Southern California
Ryan Andrew Hass, Oregon State University
Melissa Hedlund, Christopher Newport University
R. Warren Lemerich, Laramie County Community College
Sara Talley Lenhart, Christopher Newport University
Enyinda Onunwor, Stark State College
Lynda Zenati, Robert Morris University
We wish to thank the many people at Pearson who have contributed to the success
of this book. We appreciate the efforts of the production, design, manufacturing, mar-
keting, and sales departments. We are grateful to Lisa Collette for her thorough proof-
reading and John Morin and Rhea Meyerholtz for their careful and thorough checking
for accuracy. Our sincere thanks goes to Erica O’Leary for her assistance throughout the
revision of the book. Content Producer Patty Bergin did a fantastic job keeping the
book on schedule. The authors wish to extend special thanks to editor Jeff Weidenaar.
If you have any comments or suggestions, we would like to hear from you. We hope
you enjoy using this book as much as we have enjoyed writing it.
Larry J. Goldstein Martha J. Siegel

larrygoldstein@predictiveanalyticsshop.com msiegel@towson.edu
David I. Schneider Steven M. Hair

dis@math.umd.edu smh384@psu.edu
chapter
1
Linear Equations
and Straight Lines
1.1 Coordinate Systems and Graphs 1.3 The Intersection Point of a Pair of Lines
1.2 The Slope of a Straight Line 1.4 The Method of Least Squares
M any applications considered later in this text involve linear equations and their geo-
metric counterparts—straight lines. So let us begin by studying the basic facts
about these two important notions.
1.1 Coordinate Systems and Graphs

Often, we can display numerical data by using a Cartesian coordinate system on either
a line or a plane. We construct a Cartesian coordinate system on a line by choosing an
arbitrary point O (the origin) on the line and a unit of distance along the line. We then
assign to each point on the line a number that reflects its directed distance from the ori-
gin. Positive numbers refer to points on the right of the origin, negative numbers to
points on the left. In Fig. 1, we have drawn a Cartesian coordinate system on the line
and have labeled a number of points with their corresponding numbers. Each point on
y the line corresponds to a number (positive, negative, or zero).
y-axis
b (a, b) 232 1
2
15
8
origin 22 21 0 1 2 3
x Figure 1
O a
x-axis
In a similar fashion, we can construct a Cartesian coordinate system to numerically
locate points on a plane. Each point of the plane is identified by a pair of numbers (a, b).
See Fig. 2. To reach the point (a, b), begin at the origin, move a units in the x direction
Figure 2 (to the right if a is positive, to the left if a is negative), and then move b units in the y
1
2 chapter 1 Linear Equations and Straight Lines
direction (up if b is positive, down if b is negative). The numbers a and b are called,
respectively, the x- and y-coordinates of the point.
EXAMPLE 1 Plotting Points Plot the following points:

(a) (2, 1) (b) ( -1, 3) (c) ( -2, -1) (d) (0, -3)
SOLUTION y
(21, 3) (2, 1)
x
(22, 21) (0, 23)
Now Try Exercise 1
An equation in x and y is satisfied by the point (a, b) if the equation is true when x
is replaced by a and y is replaced by b. This collection of points is usually a curve of
some sort and is called the graph of the equation.
EXAMPLE 2 Solution of an Equation Are the following points on the graph of the equation
8x - 4y = 4?
(a) (3, 5) (b) (5, 17)
SOLUTION (a) 8x - 4y = 4 Given equation
8#3 - 4#5 = 4
?
x = 3, y = 5
?
24 - 20 = 4 Multiply.
4=4 Subtract.
Since the equation is satisfied, the point (3, 5) is on the graph of the equation.
(b) 8x - 4y = 4 Given equation

8 # 5 - 4 # 17 = 4
?
x = 5, y = 17
?
40 - 68 = 4 Multiply.
?
-28 = 4 Subtract.
The equation is not satisfied, so the point (5, 17) is not on the graph of the
equation. Now Try Exercises 11 and 13
Linear Equations
A linear equation is an equation whose graph is a straight line. Figure 3 shows four
examples of linear equations, along with their graphs and some points on their graphs.
y y y y
5 5 5 10
(0, 3)
(6, 3) (0, 6)
(3, 0) (0, 0) (3, 0)
x x x x
25 5 25 5 210 10 25 5
25 25 25 210
x53 y53 y 5 12 x y 5 22 x 1 6
Figure 3 Four linear equations and their graphs
1.1 Coordinate Systems and Graphs 3
Intercepts
The intercepts of a line are the points where the line crosses the x- and y-axes. These
points have 0 for at least one of their coordinates. For the graph of y = -2x + 6 in
Fig. 3, the x-intercept is the point (3, 0) and the y-intercept is the point (0, 6).* The
y-intercept of a line having an equation of the form y = mx + b is the point (0, b),
since setting x equal to 0 gives y the value b. The x-intercept is the point having the
solution of the equation 0 = mx + b as the first coordinate and 0 as the second
coordinate.
Table 1 shows how to draw the graphs of the four types of linear equations
shown in Fig. 3. The equations y = b and y = mx are actually special cases of
y = mx + b.
Table 1 Graphs of Linear Equations
Equation Description of Graph How to Draw Graph

x=a Vertical line through the point Plot (a, 0) and draw the vertical line
(a, 0) through the point.
y=b Horizontal line through the Plot (0, b) and draw the horizontal line
point (0, b) through the point.
y = mx Line through the origin Draw the line through the origin and
any other point on the graph.
y = mx + b; Line having two different Draw the line through any two points
m ≠ 0, b ≠ 0 intercepts (often the two intercepts) of the line.
General Form of a Linear Equation Any equation whose graph is a straight line can
be written in the general form
cx + dy = e
where c, d, and e are constants and c and d are not both zero.
An equation in general form having d ≠ 0 (that is, an equation in which y

appears) can be solved for y. The resulting equation will have the form of one of the
last three equations in Table 1. An equation in which y does not appear can be
solved for x and the resulting equation will have the form of the first equation in
Table 1.
EXAMPLE 3 Graph of an Equation Write the equation x - 2y = 4 in one of the forms shown in
Table 1 and draw its graph.
SOLUTION Since y appears in the equation, solve for y.
x - 2y = 4 Given equation
-2y = -x + 4 Subtract x from both sides.
1
y= 2x -2 Divide both sides by -2.
Since the equation y = 12 x - 2 has the form of the last equation in Table 1, it can be
graphed by finding its two intercepts and drawing the straight line through them.
*Intercepts are sometimes defined as numbers, such as x-intercept 3 and y-intercept 6. In this text, we define
them as pairs of numbers, such as (3, 0) and (0, 6).
y The y-intercept is the point (0, -2) since setting x equal to 0 gives y the value -2.
5 The x-intercept is found by setting y equal to 0 and solving for x.
y = 12 x - 2 Given equation
(4, 0) 1
0= 2x -2 Set y equal to 0.
x
1
25 5 2= 2x Add 2 to both sides.
(0, 22)
x=4 Multiply both sides by 2. Rewrite.
25 Therefore, the x-intercept is the point (4, 0).
Figure 4 Graph of x - 2y = 4 The graph in Fig. 4 was obtained by plotting the intercepts (4, 0) and (0, -2) and
drawing the straight line through them. Now Try Exercise 27
EXAMPLE 4 Graph of an Equation Write the equation -2x + 3y = 0 in one of the forms shown in
Table 1 and draw its graph.
SOLUTION Since y appears in the equation, solve for y.

y -2x + 3y = 0 Given equation
5
3y = 2x Add 2x to both sides.
(6, 4)
2
y= 3x Divide both sides by 3.
x
210 (0, 0) 10
Because the graph of the equation y = 23x passes through the origin, the point (0, 0)
is both the x-intercept and the y-intercept of the graph. In order to draw the graph, we
must locate another point on the graph. Let’s choose x = 6. Then y = 23 # 6 = 4. There-
25 fore, the point (6, 4) is on the graph. The graph in Fig. 5 was obtained by plotting the
points (0, 0) and (6, 4) and drawing the straight line through them.
Figure 5 Graph of -2x + 3y = 0 Now Try Exercise 19
The next example gives an application of linear equations.
EXAMPLE 5 Linear Depreciation For tax purposes, businesses must keep track of the current v alues
of each of their assets. A common mathematical model is to assume that the current
value y is related to the age x of the asset by a linear equation. A moving company buys
a 40-foot van with a useful lifetime of 5 years. After x months of use, the value y, in
dollars, of the van is estimated by the linear equation
y = 25,000 - 400x.
(a) Draw the graph of this linear equation.

(b) What is the value of the van after 5 years?
(c) When will the value of the van be $15,000?
(d) What economic interpretation can be given to the y-intercept of the graph?
SOLUTION (a) The y-intercept is (0, 25,000). To find the x-intercept, set y = 0 and solve for x.
0 = 25,000 - 400x Set y = 0.

y
(0, 25,000)
400x = 25,000 Add 400x to both sides.
x = 62.5 Divide both sides by 400.
dollars
The x-intercept is (62.5, 0). The graph of the linear equation is sketched in Fig. 6.
Note how the value decreases as the age of the van increases. The value of the
x (months)
van reaches 0 after 62.5 months. Note also that we have sketched only the por-
(62.5, 0) tion of the graph that has physical meaning—namely, the portion for x between
Figure 6 0 and 62.5.
(b) After 5 years (or 60 months), the value of the van is
y = 25,000 - 400(60) = 25,000 - 24,000 = 1000.
Since the useful life of the van is 5 years, this value represents the salvage value of
the van.
(c) Set the value of y to 15,000, and solve for x.
15,000 = 25,000 - 400x Set y = 15,000.

400x + 15,000 = 25,000 Add 400x to both sides.
400x = 10,000 Subtract 15,000 from both sides.
x = 25 Divide both sides by 400.
The value of the van will be $15,000 after 25 months.

(d) The y-intercept corresponds to the value of the van at x = 0 months—that is, the
initial value of the van, $25,000. Now Try Exercise 41
INCORPORATING Appendix B contains instructions for TI-84 Plus calculators. (For the specifics of
TECHNOLOGY other calculators, consult the guidebook for the calculator.) The appendix shows
how to obtain the graph of a linear equation of the form y = mx + b, find coordinates
of points on the line, and determine intercepts. Vertical lines can be drawn with the
Vertical command from the draw menu. To draw the vertical line x = k, go to the
home screen, press 2nd [draw] 4 to display the word V ertical, type in the value of k,
and press ENTER .
Appendix D contains an introduction to Wolfram | Alpha.
Straight lines can be drawn with instructions of the following forms:
plot ax + by = c; plot y = ax + b; plot x = a
If a phrase of the form for x from x1 to x2 is appended to the instruction, only the portion
of the line having x-values from x1 to x2 will be drawn.
An equation of the form ax + by = c, with b ≠ 0, can be converted to the form
y = mx + b with the instruction solve ax + by = c for y.
The intercepts of an equation can be found with an instruction of the form intercepts
[equation]. An expression in x can be evaluated at x = a with an instruction of the form
evaluate [expression] at x = a. For instance, the instruction
evaluate 2500 − 400x at x = 5
gives the result 500.
Check Your Understanding 1.1 Solutions can be found following the section exercises.
1. Plot the point (500, 200). 2. Is the point (4, -7) on the graph of the linear equation
2x - 3y = 1? Is the point (5, 3)?
EXERCISES 1.1
In Exercises 1–8, plot the given point.
1. (2, 3) 2. ( -1, 4) 5. ( -2, 1) 6. ( -1, - 52 )
3. (0, -2) 4. (2, 0) 7. ( -20, 40) 8. (25, 30)
y 38. Which of the following equations is graphed in Fig. 9?

Q (a) x + y = 3 (b) y = x - 1 (c) 2y = x + 3
1 y
x
1 (5, 4)
P
Figure 7 (1, 2)
9. What are the coordinates of the point Q in Fig. 7? x
10. What are the coordinates of the point P in Fig. 7?
In Exercises 11–14, determine whether the point is on the graph of Figure 9

the equation -2x + 13 y = -1.
39. Heating Water The temperature of water in a heating tea ket-
11. (1, 3) 12. (2, 6) 13. ( 12, 3 ) 14. ( 13, -1 ) tle rises according to the equation y = 30x + 72, where y is
In Exercises 15–18, each linear equation is in the form y = mx + b. the temperature (in degrees Fahrenheit) x minutes after the
Identify m and b. kettle was put on the burner.
(a) What physical interpretation can be given to the y-intercept
15. y = 5x + 8 16. y = -2x - 6 of the graph?
17. y = 3 18. y = 23 x (b) What will the temperature of the water be after 3 minutes?
(c) After how many minutes will the water be at its boiling
In Exercises 19–22, write each linear equation in the form point of 212°?
y = mx + b or x = a.
40. Life Expectancy The average life expectancy y of a person
19. 14x + 7y = 21 20. x - y = 3
born x years after 1960 can be approximated by the linear
21. 3x = 5 22. - 12 x + 23 y = 10 equation y = 16x + 70.
(a) What interpretation can be given to the y-intercept of the
In Exercises 23–26, find the x-intercept and the y-intercept of each line.
graph?
23. y = -4x + 8 24. y = 5 (b) In what year did people born that year have an average
25. x = 7 26. y = -8x life expectancy of 75 years?
(c) What is the average life expectancy of people born in
In Exercises 27–34, graph the given linear equation. 1999?
5
27. y = 13 x - 1 28. y = 2x 29. y = 2 41. Cigarette Consumption The worldwide consumption of ciga-
30. x = 0 31. 3x + 4y = 24 32. x + y = 3 rettes has been increasing steadily in recent years. The number
of trillions of cigarettes, y, purchased x years after 1960, is
33. x = - 52 34. 1
2x - 1
3y = -1 estimated by the linear equation y = .075x + 2.5.
35. Which of the following equations describe the same line as the (a) Draw the graph of this linear equation.
equation 2x + 3y = 6? (b) What interpretation can be given to the y-intercept of the
(a) 4x + 6y = 12 (b) y = - 23 x + 2 (c) x = 3 - 32 y graph?
2 (c) When were there 4 trillion cigarettes sold?
(d) 6 - 2x - y = 0 (e) y = 2 - 3 x (f) x + y = 1
(d) If this trend continues, how many cigarettes will be sold
36. Which of the following equations describe the same line as the
in the year 2024?
equation 12 x - 5y = 1?
(a) 2x - 15 y = 1 (b) x = 5y + 2 42. Ecotourism Income In a certain developing country, ecotour-
ism income has been increasing in recent years. The income y
(c) 2 - 5x + 10y = 0 (d) y = .1(x - 2)
(in thousands of dollars) x years after 2000 can be modeled
(e) 10y - x = -2 (f) 1 + .5x = 2 + 5y
by y = 1.15x + 14.
37. Each of the lines L1, L2, and L3 in Fig. 8 is the graph of one (a) Draw the graph of this linear equation.
of the equations (a), (b), and (c). Match each of the equations (b) What interpretation can be given to the y-intercept of
with its corresponding line. this graph?
(a) x + y = 3 (b) 2x - y = -2 (c) x = 3y + 3 (c) When was there $20,000 in ecotourism income?
(d) If this trend continues, how much ecotourism income will
y there be in 2022?
L1 43. Insurance Rates Yearly car insurance rates have been increas-
(0, 3) ing steadily in the last few years. The rate y (in dollars) for a
(0, 2) small car x years after 1999 can be modeled by y = 23x + 756.
L2 (a) Draw the graph of this linear equation.
(3, 0) (b) What interpretation can be given to the y-intercept of
(21, 0)
x this graph?
L3 (c) What was the yearly rate in 2007?
(0, 21)
(d) If this trend continues, when will the yearly rate be
Figure 8 $1308?
44. Simple Interest If $1000 is deposited at 3% simple inter- 51. Find an equation of the line having y-intercept (0, 5) and
est, the balance y after x years will be given by the equation x-intercept (4, 0).
y = 30x + 1000. 52. Find an equation of the line having x-intercept (5, 0) and par-
(a) Draw the graph of this linear equation. allel to the y-axis.
(b) Find the balance after two years.
(c) When will the balance reach $1180? 53. What is the equation of the x-axis?
45. College Freshmen The percentage, y, of college freshmen 54. Can a line other than the x-axis have more than one
who entered college intending to major in general biology x-intercept?
increased steadily from the year 2000 to the year 2014 and can 55. What is the general form of the equation of a line that is par-
be approximated by the linear equation y = .2x + 4.1 where x allel to the y-axis?
represents the number of years since 2000. Thus, x = 0 rep-
56. What is the general form of the equation of a line that is par-
resents 2000, x = 1 represents 2001, and so on. (Source: The
allel to the x-axis?
American Freshman: National Norms.)
(a) What interpretation can be given to the y-intercept of the In Exercises 57–60, find a general form of the given equation.
graph of the equation?
57. y = 2x + 3 58. y = 3x - 4
(b) In 2014, approximately what percent of college freshmen
5
intended to major in general biology? 59. y = - 23x -5 60. y = 4x - 6
(c) In what year did approximately 5.5% of college freshmen 61. Show that the straight line with x-intercept (a, 0) and y-intercept
intend to major in general biology? (0, b), where a and b are not zero, has bx + ay = ab as a general
46. College Freshmen The percentage, y, of college freshmen who form of its equation.
smoke cigarettes decreased steadily from the year 2004 to the 62. Use the result of Exercise 61 to find a general form of the
year 2014 and can be approximated by the linear equation equation of the line having x-intercept (5, 0) and y-intercept
y = -.46x + 6.32 where x represents the number of years (0, 6).
since 2004. Thus, x = 0 represents 2004, x = 1 represents
2005, and so on. (Source: The American Freshman: National In Exercises 63–70, give the equation of a line having the stated
Norms.) property. Note: There are many answers to each exercise.
(a) What interpretation can be given to the y-intercept of the 63. x-intercept (9, 0)
graph of the equation?
64. y-intercept (0, 10)
(b) In 2014, approximately what percent of college freshmen
smoked? 65. passes through the point ( -2, 5)
(c) In what year did approximately 2.6% of college freshmen 66. passes through the point (3, -3)
smoke?
67. crosses the positive part of the y-axis
47. College Tuition Average tuition (including room and board)
68. passes through the origin
for all institutions of higher learning in year x can be
approximated by y = 461x + 16,800 dollars, where x = 0 69. crosses the negative part of the x-axis
corresponds to 2004, x = 1 corresponds to 2005, and so on. 70. crosses the positive part of the x-axis
(Source: U.S. National Center of Education Statistics.)
71. The lines with equations y = 23x - 2 and y = -4x + c have
(a) Approximately what was the average tuition in 2011?
the same x-intercept. What is the value of c?
(b) Assuming that the formula continues to hold, when will
the average tuition exceed $25,000? 72. The lines with equations 6x - 3y = 9 and y = 4x + b have
the same y-intercept. What is the value of b?
48. Bachelor’s Degrees The number of bachelor’s degrees con-
ferred in mathematics and statistics in year x can be approxi- TECHNOLOGY EXERCISES
mated by y = 667x + 12,403, where x = 0 corresponds to
2003, x = 1 corresponds to 2004, and so on. (Source: U.S. In Exercises 73–76, (a) graph the line, (b) use the utility to deter-
National Center of Education Statistics.) mine the two intercepts, (c) use the utility to find the y-coordinate
(a) Approximately how many bachelor’s degrees in mathe- of the point on the line with x-coordinate 2.
matics and statistics were awarded in 2007? 73. y = -3x + 6 74. y = .25x - 2
(b) Assuming that the model continues to hold, approxi-
75. 3y - 2x = 9 76. 2y + 5x = 8
mately when will the number of bachelor’s degrees in
mathematics and statistics awarded exceed 25,000? In Exercises 77 and 78, determine an appropriate window, and
49. Find an equation of the line having x-intercept (16, 0) and graph the line.
y-intercept (0, 8). 77. 2y + x = 100
50. Find an equation of the line having x-intercept (.6, 0) and 78. x - 3y = 60
y-intercept (0, .9).
Solutions to Check Your Understanding 1.1

1. Because the numbers are large, make each hatchmark corre- the origin, moving 500 units to the right and 200 units up
spond to 100. Then the point (500, 200) is found by starting at (Fig. 10 on the next page).
y 2. 2x - 3y = 1 Given equation
?
2(4) - 3( -7) = 1 x = 4, y = -7
?
29 = 1 False
(500, 200)
Since the equation is not satisfied, (4, -7) is not on the graph.
100
x 2x - 3y = 1 Given equation
100 ?
2(5) - 3(3) = 1 x = 5, y = 3
1=1 True
Figure 10 Since the equation is satisfied, (5, 3) is on the graph.
1.2 The Slope of a Straight Line

In this section, we consider only lines whose equations can be written in the form
y = mx + b. Geometrically, this means that we will consider only nonvertical lines.
Slope is not defined for vertical lines.
DEFINITION Given a nonvertical line L with equation y = mx + b, the number m is

called the slope of L. That is, the slope is the coefficient of x in the equation of the
line. The equation is called the slope–intercept form of the equation of the line.
EXAMPLE 1 Finding the Slope of a Line from its Equation Find the slopes of the lines having the
following equations:
(a) y = 2x + 1 (b) y = - 34 x + 2 (c) y = 3 (d) -8x + 2y = 4
SOLUTION (a) m = 2.
(b) m = - 34 .
(c) When we write the equation in the form y = 0 # x + 3, we see that m = 0.
(d) First, write the equation in slope–intercept form.
-8x + 2y = 4 Given equation

2y = 8x + 4 Add 8x to both sides.
y = 4x + 2 Divide both sides by 2.
Thus, m = 4. Now Try Exercise 1
The definition of the slope is given in terms of an equation of the line. There is an
alternative equivalent definition of slope.
DEFINITION Alternative Definition of Slope Let L be a line passing through the

points (x1, y1) and (x2, y2), where x1 ≠ x2. Then, the slope of L is given by the formula
y2 - y1
m= . (1)
x2 - x1
That is, the slope is the difference in the y-coordinates divided by the difference in the
x-coordinates, with both differences formed in the same order. Note: x1 is pronounced
“x sub 1.”
Before proving this definition equivalent to the first one given, let us show how it
can be used.
1.2 The Slope of a Straight Line 9
EXAMPLE 2 Finding the Slope of a Line from Two Points Find the slope of the line passing through
the points (1, 3) and (4, 6).
SOLUTION We have
[difference in y@coordinates] 6 - 3 3
m= = = = 1.
[difference in x@coordinates] 4 - 1 3
Thus, m = 1. Note that if we reverse the order of the points and use formula (1) to com-
pute the slope, then we get
3-6 -3
m= = = 1,
1-4 -3
which is the same answer. The order of the points is immaterial. The important con-
cern is to make sure that the differences in the x- and y-coordinates are formed in the
same order. Now Try Exercise 7
The slope of a line does not depend on which pair of points we choose as (x1, y1)
and (x2, y2). Consider the line y = 4x - 3 and two points (1, 1) and (3, 9), which are on
the line. Using these two points, we calculate the slope to be
9-1 8
m= = = 4.
3-1 2
Now, let us choose two other points on the line—say, (2, 5) and ( -1, -7)—and use these
points to determine m. We obtain
-7 - 5 -12
m= = = 4.
-1 - 2 -3
The two pairs of points give the same slope.
Justification of Formula (1) Since (x1, y1) and (x2, y2) are both on the line, both
points satisfy the equation of the line, which has the form y = mx + b. Thus,
y2 = mx2 + b
y1 = mx1 + b.
Subtracting these two equations gives
y2 - y1 = mx2 - mx1 = m(x2 - x1).
Dividing by x2 - x1, we have
y2 - y1
m= ,
x2 - x1
which is formula (1). So the two definitions of slope lead to the same number.
Let us now study four of the most important properties of the slope of a straight
line. We begin with the steepness property, since it provides us with a geometric inter-
pretation for the number m.
Steepness Property Let the line L have slope m. If we start at any point on the line
and move 1 unit to the right, then we must move m units vertically in order to return
to the line (Fig. 1 on the next page). (Of course, if m is positive, then we move up; and
if m is negative, we move down.)
y y y
m
1 1 1
m
x x x
Figure 1 m positive m negative m50
EXAMPLE 3 Steepness Property of a Line Illustrate the steepness property for each of the lines.
(a) y = 2x + 1 (b) y = - 34 x + 2 (c) y = 3
SOLUTION (a) Here, m = 2. So starting from any point on the line, proceeding 1 unit to the right,
we must go 2 units up to return to the line (Fig. 2).
(b) Here, m = - 34 . So starting from any point on the line, proceeding 1 unit to the
right, we must go 34 unit down to return to the line (Fig. 3).
(c) Here, m = 0. So going 1 unit to the right requires going 0 units vertically to return
to the line (Fig. 4).
y y y
y 5 2x 1 1
y 5 2 34 x 1 2
2
1 y53
1
1 2 34
x x
Figure 2 Figure 3 Figure 4

Now Try Exercise 59
In the next example, we introduce a new method for graphing a linear equation.
This method relies on the steepness property and is often more efficient than finding
two points on the line (e.g., the two intercepts).
EXAMPLE 4 Using the Steepness Property to Graph a Line Use the steepness property to draw the
graph of y = 12 x + 32 .
SOLUTION The y-intercept is ( 0, 32 ) , as we read from the equation. We can find another point on the
line by using the steepness property. Start at ( 0, 32 ) . Go 1 unit to the right. Since the slope
is 12 , we must move vertically 12 unit to return to the line. But this locates a second point
on the line. So we draw the line through the two points. The entire procedure is illus-
trated in Fig. 5.
y y
(1, 2) (1, 2)
1 (0, 32 )
(0, 32 ) 2
1
x x
Figure 5
Now Try Exercise 13
Actually, to use the steepness property to graph an equation, all that is needed is
the slope plus any point (not necessarily the y-intercept).
EXAMPLE 5 Using the Steepness Property to Graph a Line Graph the line of slope -1, which
passes through the point (2, 2).
SOLUTION Start at (2, 2), move 1 unit to the right and then -1 unit vertically—that is, 1 unit down.
The line through (2, 2) and the resulting point is the desired line. (See Fig. 6.)
y
Slope measures the steepness of a line. That is, the slope of a line tells whether it is
1
rising or falling, and how fast. Specifically, lines of positive slope rise as we move from
(2, 2) 21 left to right. Lines of negative slope fall, and lines of zero slope stay level. The larger the
(3, 1) magnitude of the slope, the steeper the ascent or descent will be. These facts are directly
x
implied by the steepness property. (See Fig. 7.)
Figure 6 y y
m52 m 5 22
m51 m 5 21
m5 1 m 5 2 12
2
x x
Figure 7
Justification of the Steepness Property Consider a line with equation y = mx + b,

y and let (x1, y1) be any point on the line. If we start from this point and move 1 unit to
the right, the first coordinate of the new point will be x1 + 1, since the x-coordinate is
(x1 1 1, y2)
(x1, y1) increased by 1. Now, go far enough vertically to return to the line. Denote the y-coordinate
of this new point by y2. (See Fig. 8.) We must show that to get y2, we add m to y1. That
1
x is, y2 = y1 + m. By equation (1), we can compute m as
y 5 mx 1 b [difference in y@coordinates] y2 - y1
m= = = y2 - y1.
[difference in x@coordinates] 1
Figure 8 In other words, y2 = y1 + m, which is what we desired to show.
Often, the slopes of the straight lines that occur in applications have interesting and
significant interpretations. An application in the field of economics is illustrated in the
next example.
EXAMPLE 6 Slope of the Cost Line A manufacturer finds that the cost y of producing x units of a
certain commodity is given by the equation y = 2x + 5000. What interpretation can be
given to the slope of the graph of this equation?
SOLUTION Suppose that the firm is producing at a certain level and increases production by 1 unit.
That is, x is increased by 1 unit. By the steepness property, the value of y then increases
by 2, which is the slope of the line whose equation is y = 2x + 5000. Thus, each addi-
tional unit of production costs $2. The graph of y = 2x + 5000 is called a cost curve. It
relates the size of production to total cost. The graph is a straight line, and economists
call its slope the marginal cost of production. The y-coordinate of the y-intercept is
called the fixed cost. In this case, the fixed cost is $5000, and it includes costs such as
rent and insurance, which are incurred even if no units are produced.
Now Try Exercise 35
In applied problems having time as a variable, the letter t is often used in place of
the letter x. If so, straight lines have equations of the form y = mt + b and are graphed
on a ty-coordinate system.
EXAMPLE 7 Straight-Line Depreciation The federal government allows businesses an income tax
deduction for the decrease in value (or depreciation) of capital assets (such as buildings
and equipment). One method of calculating the depreciation is to take equal amounts
over the expected lifetime of the asset. This method is called straight-line depreciation.
Suppose that, for tax purposes, the value V of a piece of equipment t years after pur-
chase is figured according to the equation V = -100,000t + 700,000 and the expected
life of the piece of equipment is 5 years.
(a) How much did the piece of equipment originally cost?
(b) What is the annual deduction for depreciation?
(c) What is the salvage value of the piece of equipment? (That is, what is the value of
the piece of equipment after 5 years?)
SOLUTION (a) The original cost is the value of V at t = 0, namely
V = -100,000(0) + 700,000 = 700,000.
That is, the piece of equipment originally cost $700,000.

(b) By the steepness property, each increase of 1 in t causes a decrease in V of 100,000.
That is, the value is decreasing by $100,000 per year. So the depreciation deduction
is $100,000 each year.
(c) After 5 years, the value of V is given by
V = -100,000(5) + 700,000 = 200,000.
The salvage value is $200,000.
We have seen in Example 5 how to sketch a straight line when given its slope and
one point on it. Let us now see how to find the equation of the line from this data.
Point-Slope Equation The equation of the straight line passing through (x1, y1) and
having slope m is given by y - y1 = m(x - x1).
EXAMPLE 8 Finding the Equation of a Line from Its Slope and a Point on the Line Find the slope–
intercept equation of the line that passes through (2, 3) and has slope 12 .
SOLUTION Here, x1 = 2, y1 = 3, and m = 12 . So the point–slope equation is
y - 3 = 12 (x - 2)
y - 3 = 12 x - 1 Perform multiplication on right side.
1
y= 2x +2 Add 3 to both sides.
Now Try Exercise 49
EXAMPLE 9 Finding the Equation of a Line Find the slope–intercept equation of the line through
the points (3, 1) and (6, 0).
SOLUTION We can compute the slope from equation (1).
y2 - y1 1-0 1
m= = = - .
x2 - x1 3 - 6 3
Now, we can determine the equation from the point–slope equation with (x1, y1) = (3, 1)
and m = - 13 .
y - 1 = - 13 (x - 3) Point-slope equation
y - 1 = - 13 x + 1 Perform multiplication on right side.
y= - 13 x +2 Add 1 to both sides.
[Question: What would the equation be if we had chosen (x1, y1) = (6, 0)?]
Now Try Exercise 55
EXAMPLE 10 Sales Generated by Advertising For each dollar of monthly advertising expenditure,
a store experiences a 6-dollar increase in sales. Even without advertising, the store has
$30,000 in sales per month. Let x be the number of dollars of advertising expenditure
per month, and let y be the number of dollars in sales per month.
(a) Find the equation of the line that expresses the relationship between x and y.
(b) If the store spends $10,000 in advertising, what will be the sales for the month?
(c) How much would the store have to spend on advertising to attain $150,000 in sales
for the month?
SOLUTION (a) The steepness property tells us that the line has slope m = 6. Since x = 0 (no adver-
tising expenditure) yields y = $30,000, the y-intercept of the line is (0, 30,000).
Therefore, the slope–intercept equation of the line is
y = 6x + 30,000.
(b) If x = 10,000, then y = 6(10,000) + 30,000 = 90,000. Therefore, the sales for the
month will be $90,000.
(c) We are given that y = 150,000, and we must find the value of x for which
150,000 = 6x + 30,000.
Solving for x, we obtain 6x = 120,000, and hence, x = $20,000. To attain $150,000

in sales, the store should invest $20,000 in advertising. Now Try Exercise 45
Verification of the Point–Slope Equation Let (x, y) be any point on the line passing
through the point (x1, y1) and having slope m. Then, by equation (1), we have
y - y1
m= .
x - x1
Multiplying through by x - x1 gives
y - y1 = m(x - x1). (2)
Thus, every point (x, y) on the line satisfies equation (2). So (2) gives the equation of the
line passing through (x1, y1) and having slope m.
Perpendicular and Parallel Lines

The next property of slope relates the slopes of two perpendicular lines.
Perpendicular Property When two nonvertical lines are perpendicular, their slopes
are negative reciprocals of one another. That is, if two lines with nonzero slopes m
and n are perpendicular to one another, then
1
m= - .
n
Conversely, if two lines have slopes that are negative reciprocals of one another, they
are perpendicular.
A proof of the perpendicular property is outlined in Exercise 88. Let us show how it
can be used to help find equations of lines.
EXAMPLE 11 Perpendicular Lines Find an equation of the line perpendicular to the graph of
y = 2x - 5 and passing through (1, 2).
SOLUTION The slope of the graph of y = 2x - 5 is 2. By the perpendicular property, the slope of a
line perpendicular to it is - 12 . If a line has slope - 12 and passes through (1, 2), it has the
point-slope equation
y - 2 = - 12 (x - 1) or y = - 12 x + 52 .
Now Try Exercise 21
The final property of slope gives the relationship between slopes of parallel lines. A
proof is outlined in Exercise 87.
Parallel Property Parallel lines have the same slope. Conversely, if two different
lines have the same slope, they are parallel.
EXAMPLE 12 Parallel Lines Find an equation of the line through (2, 0) and parallel to the line whose
equation is y = 13 x - 11.
SOLUTION The slope of the line having equation y = 13 x - 11 is 13 . Therefore, any line parallel to it
also has slope 13 . Thus, the desired line passes through (2, 0) and has slope 13 , so its equa-
tion is
y - 0 = 13 (x - 2) or y = 13 x - 23.
Now Try Exercise 23
INCORPORATING A graphing calculator can find the equation of the line through two points. Refer to
TECHNOLOGY the graphing calculator discussion in the Incorporating Technology feature of
ection 1.4 and find the equation of the least-squares fit to the two points.
S
Excel can find the equation of the line through two points. Refer to the Excel
discussion in the Incorporating Technology feature of Section 1.4 and find the
equation of the least-squares fit to the two points.
The following instructions produce the equation of the line described.
line through (a, b) and (c, d )

line through (a, b) with slope m
line through (a, b) perpendicular to y = mx + b
line through (a, b) parallel to y = mx + b
Check Your Understanding 1.2 Solutions can be found following the section exercises.
Suppose that the revenue y from selling x units of a certain com- 2. The cost curve discussed in Example 6 intersects the revenue
modity is given by the formula y = 4x. (Revenue is the amount of curve at the point (2500, 10,000). What economic interpre-
money received from the sale of the commodity.) tation can be given to the value of the x-coordinate of the
1. What interpretation can be given to the slope of the graph of intersection point?
this equation?
EXERCISES 1.2
In Exercises 1–6, find the slope of the line having the given equation. 20. y
1. y = 23 x + 7 2. y = -4
L
3. y - 3 = 5(x + 4) 4. 7x + 5y = 10
x y x y
5. + =6 6. - =1
5 4 7 8
(1, 2)
In Exercises 7–10, plot each pair of points, draw the straight line (21, 12 )
through them, and find its slope. x
7. (3, 4), (7, 9)

8. ( -2, 1), (3,-3)
21. y
9. (0, 0), (5, 4) y 5 24x 1 10
10. (4, 17), ( -2, 17)
11. What is the slope of any line parallel to the y-axis?
12. Why doesn’t it make sense to talk about the slope of the line
between the two points (2, 3) and (2, -1)? (2, 2)
L
In Exercises 13–16, graph the given linear equation by beginning at x
the y-intercept, and moving 1 unit to the right and m units in the
y-direction.
L perpendicular to y 5 24x 1 10
13. y = -2x + 1 14. y = 4x - 2
15. y = 3x 16. y = -2 22. y
In Exercises 17–24, find the equation of line L.

L
17. y (5, 3)
1 y 5 13 x
(2, 3)
22 x
x L parallel to y 5 13 x
L
23. y
18. y
L y 5 2x 1 2
x
(3, 1) 1
1 2
x L
L parallel to y 5 2x 1 2
24. y
19. y L
x
(2, 21)
(1, 2)
(2, 0) y 5 212 x
x
L L perpendicular to y 5 212 x
In Exercises 25–28, give the slope–intercept form of the equation 34. Find the equation of the line passing through the point (1, 4)
of the line. and having y-intercept (0, 4).
y 35. Cost Curve A manufacturer has fixed costs (such as rent and
25.
insurance) of $2000 per month. The cost of producing each
5 unit of goods is $4. Give the linear equation for the cost of
producing x units per month.
36. Demand Curve The price p that must be set in order to sell q
x items is given by the equation p = -3q + 1200.
25 5 (a) Find and interpret the p-intercept of the graph of the
equation.
(b) Find and interpret the q-intercept of the graph of the
25 equation.
(c) Find and interpret the slope of the graph of the equation.
(d) What price must be set in order to sell 350 items?
y (e) What quantity will be sold if the price is $300?
26.
(f) Draw the graph of the equation.
5
37. Boiling Point of Water At sea level, water boils at a tempera-
ture of 212°F. As the altitude increases, the boiling point
of water decreases. For instance, at an altitude of 5000 feet,
x water boils at about 202.8°F.
25 5 (a) Find a linear equation giving the boiling point of water in
terms of altitude.
(b) At what temperature does water boil at the top of Mt.
25 Everest (altitude 29,029 feet)?
38. Cricket Chirps Biologists have found that the number of chirps
that crickets of a certain species make per minute is related to
27. y the temperature. The relationship is very close to linear. At
5 68°F, those crickets chirp about 124 times a minute. At 80°F,
they chirp about 172 times a minute.
(a) Find the linear equation relating Fahrenheit temperature
F and the number of chirps c.
x (b) If you count chirps for only 15 seconds, how can you
25 5 quickly estimate the temperature?
39. Cost Equation Suppose that the cost of making 20 cell phones
25 is $6800 and the cost of making 50 cell phones is $9500.
(a) Find the cost equation.
(b) What is the fixed cost?
(c) What is the marginal cost of production?
28. y (d) Draw the graph of the equation.
5
Exercises 40–42 are related.
40. Cost Equation Suppose that the total cost y of making x coats
x is given by the formula y = 40x + 2400.
25 5 (a) What is the cost of making 100 coats?
(b) How many coats can be made for $3600?
(c) Find and interpret the y-intercept of the graph of the
25 equation.
(d) Find and interpret the slope of the graph of the equation.
29. Find the equation of the line passing through the point (2, 3) 41. Revenue Equation Suppose that the total revenue y from the
and parallel to the x-axis. sale of x coats is given by the formula y = 100x.
30. Find the equation of the line passing through the point (2, 3) (a) What is the revenue if 300 coats are sold?
and parallel to the y-axis. (b) How many coats must be sold to have a revenue of $6000?
(c) Find and interpret the y-intercept of the graph of the
31. Find the y-intercept of the line passing through the point equation.
(5, 6) and having slope 35 . (d) Find and interpret the slope of the graph of the equation.
32. Find the y-intercept of the line passing through the points 42. Profit Equation Consider a coat factory with the cost and rev-
( -1, 3) and (4, 6). enue equations given in Exercises 40 and 41.
33. Find the equation of the line passing through (0, 4) and hav- (a) Find the equation giving the profit y resulting from mak-
ing undefined slope. ing and selling x coats.
(b) Find and interpret the y-intercept of the graph of the 55. (5, -3) and ( -1, 3) on line.
profit equation. 56. (2, 1) and (4, 2) on line.
(c) Find and interpret the x-intercept of the graph of the
profit equation. 57. (2, -1) and (3, -1) on line.
(d) Find and interpret the slope of the graph of the profit 58. (0, 0) and (1, -2) on line.
equation.
(e) How much profit will be made if 80 coats are sold? In each of Exercises 59–62, we specify a line by giving the slope
(f) How many coats must be sold to have a profit of $6000? and one point on the line. We give the first coordinate of some
(g) Draw the graph of the equation found in part (a). points on the line. Without deriving an equation of the line, find the
43. Heating Oil An apartment complex has a storage tank to second coordinate of each of the points.
hold its heating oil. The tank was filled on January 1, but no 59. Slope is 2, (1, 3) on line; (2, ); (0, ); ( -1, ).
more deliveries of oil will be made until sometime in March.
60. Slope is -3, (2, 2) on line; (3, ); (4, ); (1, ).
Let t denote the number of days after January 1, and let y
denote the number of gallons of fuel oil in the tank. Cur- 61. Slope is - 14 , ( -1, -1) on line; (0, ); (1, ); ( -2, ).
rent records show that y and t will be related by the equation 62. Slope is 13 , ( -5, 2) on line; ( -4, ); ( -3, ); ( -2, ).
y = 30,000 - 400t.
(a) Graph the equation y = 30,000 - 400t. 63. Each of the lines (A), (B), (C), and (D) in Fig. 9 is the graph
(b) How much oil will be in the tank on February 1? of one of the linear equations (a), (b), (c), and (d). Match
(c) How much oil will be in the tank on February 15? each line with its equation.
(d) Determine the y-intercept of the graph. Explain its sig-
nificance. y y
(e) Determine the t-intercept of the graph. Explain its sig- 2 2
nificance.
44. Cash Reserves A corporation receives payment for a large
contract on July 1, bringing its cash reserves to $2.3 million.
Let y denote its cash reserves (in millions) t days after July 1. x x
22 2 22 2
The corporation’s accountants estimate that y and t will be
related by the equation y = 2.3 - .15t.
(a) Graph the equation y = 2.3 - .15t.
22 22
(b) How much cash does the corporation have on the morn-
ing of July 16? sAd sBd
(c) Determine the y-intercept of the graph. Explain its
significance.
(d) Determine the t-intercept of the graph. Explain its
y y
significance.
(e) Determine the cash reserves on July 4. 2 2
(f) When will the cash reserves be $.8 million?
45. Weekly Pay A furniture salesperson earns $220 a week plus
10% commission on her sales. Let x denote her sales and y her x x
income for a week. 22 2 22 2
(a) Express y in terms of x.
(b) Determine her week’s income if she sells $2000 in mer-
chandise that week. 22 22
(c) How much must she sell in a week in order to earn $540?
sCd sDd
46. Weekly Pay A salesperson’s weekly pay depends on the vol-
ume of sales. If she sells x units of goods, then her pay is Figure 9
y = 5x + 60 dollars. Give an interpretation to the slope and
the y-intercept of this straight line. (a) x + y = 1 (b) x - y = 1
(c) x + y = -1 (d) x - y = -1
In Exercises 47–58, find an equation for each of the following lines.
47. Slope is - 12 ; y-intercept is (0, 0). 64. The table that follows gives several points on the line
Y1 = mx + b. Find m and b.
48. Slope is 3; y-intercept is (0, -1).
49. Slope is - 13 ; (6, -2) on line.
X Y1
50. Slope is 1; (1, 2) on line. 4.8 3.6
4.9 4.8
51. Slope is 12 ; (2, -3) on line. 5.0 6.0
5.1 7.2
52. Slope is -7; (5, 0) on line. 5.2 8.4
5.3 9.6
53. Slope is - 25 ; (0, 5) on line. 5.4 10.8
54. Slope is 0; (7, 4) on line. Y1 5 10.8
In Exercises 65–70, give an equation of a line with the stated prop- number of home health aide jobs increases linearly during
erty. Note: There are many answers to each exercise. that time, find the equation that relates the number of jobs,
65. rises as you move from left to right y, to the number of years after 2014, x. Use the equation
to estimate the number of home health aide jobs in 2018.
66. falls as you move from left to right (Source: Bureau of Labor Statistics, Occupational Projections
67. has slope 0 Data.)
68. slope not defined 77. Bachelor’s Degrees in Business According to the U.S.
69. parallel to the line 2x + 3y = 4 National Center of Education Statistics, 263,515 bach-
elor’s degrees in business were awarded in 2001 and 360,823
70. perpendicular to the line 5x + 6y = 7 were awarded in 2013. If the number of bachelor’s degrees
71. Temperature Conversion Celsius and Fahrenheit temperatures in business continues to grow linearly, how many bach-
are related by a linear equation. Use the fact that 0°C = 32°F elor’s degrees in business will be awarded in 2020? (Source:
and 100°C = 212°F to find an equation. National Center for Education Statistics, Digest of Educa-
tion Statistics.)
72. Dating of Artifacts An archaeologist dates a bone fragment
discovered at a depth of 4 feet as approximately 1500 b.c. and 78. Pizza Stores According to Pizza Marketing Quarterly, the
dates a pottery shard at a depth of 8 feet as approximately number of U.S. Domino’s Pizza stores grew from 4818 in 2001
2100 b.c. Assuming that there is a linear relationship between to 4986 in 2013. If the number of stores continues to grow
depths and dates at this archeological site, find the equation linearly, when will there be 5100 stores?
that relates depth to date. How deep should the archaeologist
79. Super Bowl Commercials The average cost of a 30-second
dig to look for relics from 3000 b.c.?
advertising slot during the Super Bowl increased linearly from
73. College Tuition The average college tuition and fees at four- $3.5 million in 2012 to $4.5 million in 2015. Find the equation
year public colleges increased from $3735 in 2001 to $8312 that relates the cost (in millions of dollars) of a 30-second
in 2013. (See Fig. 10.) Assuming that average tuition and fees slot, y, to the number of years after 2012, x. What was the
increased linearly with respect to time, find the equation that average cost in 2014?
relates the average tuition and fees, y, to the number of years
80. Straight-Line Depreciation A multi-function laser printer pur-
after 2001, x. What were the average tuition and fees in 2009?
chased for $3000 depreciates to a salvage value of $500 after
(Source: National Center for Education Statistics, Digest of
4 years. Find a linear equation that gives the depreciated value
Education Statistics.)
of the multi-function laser printer after x years.
$12000 8 81. Supply Curve Suppose that 5 million tons of apples will be
supplied at a price of $3000 per ton and 6 million tons of
6 apples will be supplied at a price of $3400 per ton. Find the
$9000
equation for the supply curve and draw its graph. Let the
units for q be millions of tons and the units for p be thou-
$6000 4
sands of dollars.
$3000 2 82. Demand Curve Suppose that 5 million tons of apples will be
demanded at a price of $3000 per ton and 4.5 million tons of
apples will be demanded at a price of $3100 per ton. Find the
$0 0
equation for the demand curve and draw its graph. Let the
2001 2013 2000 2013
units for q be millions of tons and the units for p be thou-
Figure 10 College Tuition Figure 11 College sands of dollars.
Enrollments (in millions)
83. Show that the points (1, 3), (2, 4), and (3, -1) are not on the
74. College Enrollments Two-year college enrollments increased same line.
from 5.9 million in 2000 to 7.0 million in 2013. (See Fig. 11.) 84. For what value of k will the three points (1, 5), (2, 7), and
Assuming that enrollments increased linearly with respect (3, k) be on the same line?
to time, find the equation that relates the enrollment, y, to
85. Find the value of a for which the line through the points (a, 1)
the number of years after 2000, x. When was the enrollment
and (2, -3.1) is parallel to the line through the points ( -1, 0)
6.5 million? (Source: National Center for Education Statistics,
and (3.8, 2.4)
Digest of Education Statistics.)
86. Rework Exercise 85, where the word parallel is replaced by
75. Gas Mileage A certain car gets 25 miles per gallon when the
the word perpendicular.
tires are properly inflated. For every pound of pressure that
the tires are underinflated, the gas mileage decreases by 12 mile 87. Prove the parallel property. [Hint: If y = mx + b and
per gallon. Find the equation that relates miles per gallon, y, to y = m′x + b′ are the equations of two lines, then the two
the amount that the tires are underinflated, x. Use the equa- lines have a point in common if and only if the equation
tion to calculate the gas mileage when the tires are underin- mx + b = m′x + b′ has a solution for x.]
flated by 8 pounds of pressure. 88. Prove the perpendicular property. [Hint: Without loss of
76. Home Health Aid Jobs According to the U.S. Department of generality, assume that both lines pass through the origin.
Labor, home health aide jobs are expected to increase from Use the point–slope formula, the Pythagorean theorem,
913,500 in 2014 to 1,261,900 in 2024. Assuming that the and Fig. 12.]
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This famous sea-captain was the grandson of John Hawkins of
Tavistock, who was a merchant in the service of Henry VIII. John was born
at Plymouth in the year 1520, and drank in the love of the salt seas from his
earliest years. His father, William Hawkins, was known to be one of the
most experienced sea-captains in the west of England: he had fitted out a
"tall and goodly ship," the Paul of Plymouth, and made in her three voyages
to Brazil and to Guinea. He treated the savage people so well that they
became very friendly, and in 1531 he brought one of their chiefs to England,
leaving a Plymouth man behind as hostage. This chief was presented to King
Henry and became the lion of society. On his way home to Brazil he died of
sea-sickness; but Hakluyt tells us that the savages, being fully persuaded of
the honest dealing of William Hawkins with their king, believed his report
and restored the hostage, without harm to any of his company.
William Hawkins married Joan Trelawny and had two sons, John and
William, both of whom made their way as seamen and merchants.
John made some voyages to the Canary Islands when quite a youth, and
with his quick eye for gain soon learnt that negroes might be cheaply gotten
in Guinea and profitably sold in Hispaniola. John Hawkins was not the first
to make and sell slaves, but he was the first Englishman to take part in this
cruel and inhuman barter. The Spaniards and Portuguese had used slaves,
both Moors and negroes, and Hawkins no doubt had seen plenty of cases of
slave-holding along the west coast of Africa, where savage warfare was
carried on between native tribes, and such of the conquered as were not
eaten were retained as slaves. He may have thought therefore that he was
only carrying them to a less barbarous captivity; and we should remember
that slavery was defended even by some religious people until quite recent
times: but we must deplore the fact that this daring sea-dog, who certainly
was not without religious feelings, found in this traffic a source of gain.
No doubt John, on his return to England, discussed the matter openly

with men of influence, for in October 1562, being now more than forty years
old, he led an expedition of a hundred men in three ships, the Solomon of
120 tons, the Swallow of 100 tons, and the Jones of 40 tons burden, and
sailed direct for the coast of Sierra Leone, in West Africa, just north of
Guinea. Hakluyt draws a veil over the exact methods by which John
Hawkins got possessed of 300 fine negroes, besides other merchandise; but
he probably took sides in some local quarrel and carried off his share of the
prisoners. These poor wretches were carried across the Atlantic in the stuffy
holds of small ships, and landed at San Domingo, one of the largest of the
Spanish islands in the West Indies.
John made due apologies for entering the Spanish port: they could see he
was really in want of food and water. The Spaniards too were polite, and as
they peeped into his hold they saw the very thing they wanted—negroes. A
bargain was quickly made, and John Hawkins took off in return for his
captives quite a goodly store of pearls, hides, sugar, and other innocent
materials.
Hawkins himself arrived safely in England with his three ships, but his
partner, Thomas Hampton, who took what was left over in two Spanish ships
to Cadiz, did not fare so well. For when it became known at Cadiz that
English merchants had been trading with Spain's colonies, Philip II.
confiscated the cargo, and Hampton narrowly escaped the prisons of the
Inquisition. Queen Elizabeth was warned by her ambassador at Madrid that
further voyages of this nature might lead to war.
For it seems that Philip had been an admirer of the Maiden Queen, and
had been rebuffed as a suitor; whereby his love had changed to hate, and he
lost no opportunity of showing his resentment.
But Queen Bess had her father's spirit in her, and answered the Spanish
threat by permitting one of her largest ships, the Jesus of Lübeck, to be
chartered for a new voyage. The Earls of Leicester and Pembroke joined in
raising money for the expedition—this time it was a Court affair; there sailed
a hundred and seventy men in five vessels, and they were to meet another
Queen's ship, the Minion, before they got out of the Channel.
Again Hawkins raided the West African coast, "going every day on shore
to take the inhabitants, with burning and spoiling of their towns."
It is strange how men engaged in such ruthless work could yet believe
that they were specially preserved by Providence. For on New Year's Day
1565 they were well-nigh surprised by natives as they were seeking water.
But a pious seaman wrote thus in his journal: "God, who worketh all things
for the best, would not have it so, and by Him we escaped without danger—
His name be praised for it!" Then they set sail for the West Indies with a
goodly cargo of miserable slaves; but for eighteen days they were becalmed
—"as idle as a painted ship, upon a painted ocean." "And this happened to us
very ill, being but reasonably watered for so great a company of negroes and
ourselves. This pinched us all: and, that which was worst, put us in such fear
that many never thought to have reached the Indies without great dearth of
negroes and of themselves; but the Almighty God, which never suffereth His
elect to perish, sent us the ordinary breeze."
Let us hope that they felt some pity for the poor negroes too, who must
have suffered agonies of thirst on that hideous journey.
But King Philip had ordered his Christian subjects to have no dealings
with heretics; and for some time they could sell no negroes in Dominica.
But some heathen Indians presented cakes of maize, hens, and potatoes,
which the English crews bought for beads, pewter whistles, knives, and other
trifles. "These potatoes be the most delicate roots that may be eaten, and do
far exceed our parsnips and carrots."
We need to remind ourselves occasionally of some of the luxuries which

our ancestors never knew till these old seadogs brought them home—
tobacco and potatoes! and later on, tea and coffee! It is difficult to imagine
what the want of such things would mean to us now.
But not all the Indians were so kind as these they first met; for on the
American mainland they fell in with a tribe whom the devilries of Spain had
turned to "ferocious bloodsuckers," and whom they only narrowly avoided.
Hawkins, according to his instructions from the Queen's Council, kept away
from the larger dependencies and islands, and tried to sell his cargo in out-
of-the-way places which Philip's orders might not have reached. At
Barbarotta he was refused permission to trade. But Hawkins sent in a
message: "I have with me one of Queen Elizabeth's own ships. I need
refreshment and without it I cannot depart; if you do not allow me to have
my way, I shall have to displease you."
Thereat he ran out a few of his guns to mark the form which his
displeasure might assume: the Spaniards improved in politeness. At Curacoa
they feasted on roast lamb to their heart's content: near Darien they again
had to use threats of violence in order to get licence to trade; but the price
offered by the Spaniards for the negroes so disgusted the equitable mind of
John Hawkins that he wrote the Governor a letter saying that they dealt too
rigorously with him, to go about to cut his throat in the price of his
commodities ... but seeing they had sent him this to his supper, he would in
the morning bring them as good a breakfast.
When that breakfast was served—and served hot—it proved to be

garnished with a handsome volley of ordnance, with ships' boats landing at
full speed a hundred armed Englishmen: the Spaniards fled.
"After that we made our traffic full quietly, and sold all our negroes."
Hawkins then sailed for Hispaniola, but being misled by his pilot he found
himself at Jamaica and then at Cuba, and so along the coast of Florida,
meeting many Indians whenever they landed who were of so fierce a
character that of five hundred Spaniards who had recently set foot in the
country only a very few returned; and a certain friar who essayed to preach
to them "was by them taken and his skin cruelly pulled over his ears and his
flesh eaten." "These Indians as they fight will clasp a tree in their arms and
yet shoot their arrows: this is their way of taking cover."
In coasting along Florida they found a Huguenot colony that had been
founded there at the advice of Admiral Coligny. They had been reduced by
fighting the Indians from two hundred to forty, and were glad to accept a
passage home in the Tiger. On the 28th of July the English ships started for
home, but, owing to contrary winds, their provisions fell so short they "were
in despair of ever coming home, had not God of His goodness better
provided for us than our deserving." On the 20th September they landed at
Padstow in Cornwall, having lost twenty persons in all the voyage, and with
great profit in gold, silver, pearls, "and other jewels great store." The Queen
was delighted with the bold way in which Hawkins had traded in defiance of
the Spanish king, and by patent she conferred on him a crest and coat of
arms.
The Spanish ambassador at once wrote off to his master, saying he had
met Hawkins in the Queen's palace, who gave him a full account of his
trading with full permission of the governors of towns (he did not say by
what means he had obtained such licence); "The vast profit made by the
voyage has excited other merchants to undertake similar expeditions.
Hawkins himself is going out again next May, and the thing needs
immediate attention." The result of this letter was that Hawkins was strictly
forbidden by Sir William Cecil from "repairing armed, for the purpose of
traffic, to places privileged by the King of Spain." So the ships went, but
Hawkins stayed at home; his ships returned next summer laden with gold
and silver. The crews did not publish any account of how they had obtained
their cargoes, and as the Queen had recently been assisting the Netherlands
in their struggle for liberty against Spain, she made no indiscreet inquiries,
and proceeded to lend the Jesus of Lübeck and the Minion for another
expedition. One of the volunteers was young Francis Drake, now twenty-two
years of age, whom Hawkins made captain of one of his six vessels.
As they left Plymouth they fell in with a Spanish galley en route for
Cadiz with a cargo of prisoners from the Netherlands. Hawkins fired upon
the Spanish flag, and in the confusion many of the captives escaped to the
Jesus, whence they were sent back to Holland.
The Spanish ambassador wrote strongly to the Queen, and the Queen
wrote strongly to Hawkins; but Hawkins had sailed away and was
encountering storms off Cape Finisterre, so that he had a mind to return for
repairs. But the weather moderating he went on to the Canaries and Cape
Verde. Here he landed 150 men in search of negroes, but eight of his men
died of lockjaw from being shot by poisoned arrows. "I myself," writes
Hawkins, "had one of the greatest wounds, yet, thanks be to God, escaped."
In Sierra Leone they joined a negro king in his war against his enemies,
attacked a strongly paled fort, and put the natives to flight. "We took 250
persons, men, women, and children, and our friend the king took 600
prisoners," which by agreement were to go to the English, but the wily negro
decamped with them in the night, and Hawkins had to be content with his
own few. They were at sea from February 3rd until March 27th, when they
sighted Dominica, but found it difficult to trade, until after a show of force
the Spaniards gave in and eagerly bought the slaves. At Vera Cruz the
inhabitants mistook our ships for the Spanish fleet. There is a rocky island at
the mouth of the harbour which Hawkins seized. The next morning the
Spanish fleet arrived in reality, but Hawkins would not admit them until they
had promised him security for his ships. Now there was no good anchorage
outside, and if the north wind blew "there had been present shipwreck of all
the fleet, in value of our money some £1,800,000." So he let them in under
conditions, for even Hawkins thought that he ought not to risk incurring his
Queen's indignation. On Thursday Hawkins had entered the port, on Friday
he saw the Spanish fleet, and on Monday at night the Spaniards entered the
port with salutes, after swearing by King and Crown that Hawkins might
barter and go in peace.
For two days both sides laboured, placing the English ships apart from
the Spanish, with mutual amity and kindness. But Hawkins began to notice
suspicious changes in guns and men, and sent to the Viceroy to ask what it
meant. The answer was a trumpet-blast and a sudden attack. Meanwhile a
Spaniard sitting at table with Hawkins had a dagger in his sleeve, but was
disarmed before he could use it. The Spaniards landed on the island and slew
all our men without mercy. The Jesus of Lübeck had five shots through her
mainmast, the Angel and Swallow were sunk, and the Jesus was so battered
that she served only to lie beside the Minion, and take all the battery from
the land guns.
ATTEMPT ON SIR JOHN HAWKINS' LIFE
As the Spanish and English fleets were anchored at Vera Cruz apparent amity and
goodwill existed between the two, but as Hawkins was sitting at dinner one day a Spaniard
sitting at table with him was discovered with a dagger up his sleeve, but fortunately was
disarmed before he could use it.
Hawkins cheered his soldiers and gunners, called his page to serve him a
cup of beer, whereat he stood up and drank to their good luck. He had no
sooner set down the silver cup than a demi-culverin shot struck it away.
"Fear nothing," shouted Hawkins, "for God, who hath preserved me from
this shot, will also deliver us from these traitors and villains."
Francis Drake was bidden to come in with the Judith, a barque of 50

tons, and take in men from the sinking ships: at night the English in the
Minion and Judith sailed out and anchored under the island. The English
taken by the Spaniards received no mercy. "They took our men and hung
them up by the arms upon high posts until the blood burst out of their
fingers' ends."
The Judith under Drake sailed for England and reached Plymouth in
January 1569; the Minion, with 200 men, suffered hunger and had to eat rats
and mice and dogs. One hundred men elected to be landed and left behind to
the mercies of Indians and Spaniards. "When we were landed," said a
survivor, "Master Hawkins came unto us, where friendly embracing every
one of us, he was greatly grieved that he was forced to leave us behind him.
He counselled us to serve God and to love one another; and thus courteously
he gave us a sorrowful farewell and promised, if God sent him safe home, he
would do what he could that so many of us as lived should by some means
be brought into England—and so he did." Thus writes Job Hartop. So we see
that John Hawkins, the slave-dealer, sincerely tried after his fashion to serve
God as well as his Queen. His men loved him and spoke well of him when
he failed; a good test of a man's worth when men will speak well of you
though all your plans be broken and your credit gone. But alas! for the poor
hundred men left ashore on the Mexican coast! They wandered for fourteen
days through marshes and brambles, some poisoned by bad water, others
shot by Indians or plagued by mosquitoes, until they came to the Spanish
town of Panluco, where the Governor thrust them into a little hog-stye and
fed them on pigs' food. After three days of this they were manacled two and
two and driven over ninety leagues of road to the city of Mexico. One of
their officers used them very spitefully and would strike his javelin into neck
or shoulders, if from faintness any lagged behind, crying, "March on,
English dogs, Lutherans, enemies to God." After four months in gaol they
were sent out as servants to the Spanish colonists. For six years they fared
passing well, but in 1575 the Inquisition was introduced into Mexico, and
then their "sorrows began afresh." On the eve of Good Friday all were
dressed for an auto-da-fé and paraded through the streets. Some were then
burnt, others sent to the galleys, the more favoured ones got three hundred
lashes apiece. One who had escaped had spent twenty-three years in various
galleys, prisons, and farms.
Meanwhile Hawkins was taking his other hundred men back to England,
meeting violent storms, but "God again had mercy on them." Then food
became scarce and many died of starvation: the rest were so weak they could
hardly manage the sails. At last they sighted the coast of Spain and put in at
Vigo for supplies; here more died from eating excess of fresh meat after their
famine. At length, with the help of twelve English sailors they reached
Mount's Bay in Cornwall, in January 1569.
Here was a miserable ending of an ambitious expedition: no profits, no

gold, no silver for the rich merchants and courtiers who had subscribed for
the fitting out of the ships; no jewels for the lady who graced the throne.
Sadly John Hawkins wrote to Sir William Cecil: "All our business hath had
infelicity, misfortune, and an unhappy end: if I should write of all our
calamities, I am sure a volume as great as the Bible will scarcely suffice."
Thus our hero, ruined but not broken, bided his time for revenge. As the
years wore on England and Spain grew more embittered. Private warfare had
existed for some time, and Philip had wished to declare open war in 1568;
but the Duke of Alva cautioned him against making more enemies, while
they still found it hard to subdue the Low Countries. So, for a while, the
King contented himself with underhand efforts to stir up rebellion in Ireland
and England.
In the year 1578 John Hawkins was summoned by the Queen from
Devon and appointed Comptroller of the Navy. His business was to see to
the building of new ships, the repairing of old ones, and the victualling and
manning of all about to take the sea. Hawkins is said to have invented "false
netting" for ships to fight in, chain-pumps and other devices. Acting with
Drake he founded the "Chest" at Chatham, a fund made up by voluntary
subscriptions from seamen on behalf of their poorer brethren. In fact he
entered upon his work with the same zeal which he had shown in the West
Indies. Lucky was it for him that he had a mistress like Elizabeth; for under
the craven James he would certainly have been handed over to the
Inquisition, or put to death by Spanish order, like Raleigh. In 1572 Hawkins
and George Winter were commissioned to do their utmost to clear the British
seas of pirates and freebooters, for of late the coasts of Norfolk and the East
had been much troubled by sea-robbers. But through all his multifarious
duties the old sea-rover was ever most bent on paying off old scores against
King Philip. So many of his friends, beside himself, had lost their all or
endured sharp punishment in Spanish dungeons, that he grimly chuckled
when he heard of Drake having "singed King Philip's beard"; and when the
news came that the invasion of England was only put off, and Pope Sixtus V.
had spurred his Spanish Majesty to quick action by the oft-quoted taunt,
"The Queen of England's distaff is worth more than Philip's sword," then
John Hawkins rubbed his hands gleefully, and lost no time in getting all the
Queen's ships taut and in order, well victualled and well manned. But
Hawkins did not mince matters when he saw anything amiss; any hesitation
or signs of parsimony met with his blunt disapproval. He writes in February
1588 to urge that peace could only be won by resolute fighting: "We might
have peace, but not with God. Rather than serve Baal, let us die a thousand
deaths. Let us have open war with these Jesuits, and every man will
contribute, fight, devise or do, for the liberty of our country."
Hawkins also wrote to ask for the use of six large and six small ships for
four months, with 1800 mariners and soldiers, which he would employ in
another raid upon the Spanish coast, so as to hinder Philip's grand Armada.
"I promise I will distress anything that goeth through the seas: and in
addition to the injury done to Spain, I shall acquire booty enough to pay four
times over the cost of the expedition."
But Burghley, like his mistress, kept a tight hand over slender resources,
and he rejected Hawkins' offer. Macaulay says that even Burghley's jests
were only neatly expressed reasons for keeping money carefully. Lord
Howard bitterly complained to Walsingham that "her Majesty was keeping
her ships to protect Chatham Church withal, when they should be serving
their turn abroad"; and again, when Drake was being prevented from getting
his Plymouth squadron in order for sea-service, he writes: "I pray God her
Majesty do not repent her slack dealing.... I fear ere long her Majesty will be
sorry she hath believed some so much as she hath done." Lord Burghley's
task was to defeat the Armada with an almost empty exchequer. We find
calculations of his as to whether it will not be cheaper to feed the sailors of
the fleet on fish three days a week and bacon once, instead of the usual
ration of four pennyworth of beef each day. And naturally these attempts to
cut down expenses were misconstrued into parsimony. But with all her rigid
economy, Elizabeth could show a brave front when the crisis came; as in the
camp at Tilbury, when she addressed the little army that was expecting every
hour to be called to meet the fierce onset of the invaders: "I have placed my
chiefest strength and safeguard in the loyal hearts and goodwill of my
subjects; and therefore am I come amongst you, as ye see, at this time,
resolved in the midst and heat of the battle, to live or die amongst you all; to
lay down for my God, for my kingdom, and for my people, my honour and
my blood even in the dust. I know I have the body but of a weak and feeble
woman; but I have the heart of a king, and of a King of England too, and
think it foul scorn that Parma, or Spain, or any Prince of Europe, should dare
to invade the borders of my realm; to which, rather than any dishonour shall
grow by me, I myself will take up arms, I myself will be your General,
Judge, and Rewarder of every one of your virtues in the field."
The Great Armada had left the Tagus on the 20th of May 1588. It
consisted of one hundred and thirty-two ships under the command of the
Duke of Medina Sidonia. Besides 8766 sailors, there were on board 2088
galley slaves, 21,855 officers and soldiers ready for action as soon as they
should land; 300 monks and friars were pacing the decks, sent to take
spiritual charge in partibus infidelium.
Against this force Queen Elizabeth had only thirty-four of her own ships,
but all the seaports from Bristol to Hull sent small armed vessels, while
noblemen and merchants contributed to swell the total, which came to nearly
two hundred in all.
John Hawkins was there as Rear-Admiral under Howard, making with

Drake and Frobisher his headquarters at Plymouth. "For the love of God," he
writes to Walsingham, on the 19th of June, "let her Majesty care not now for
charges," and in the same vein he wrote also to the Queen.
As he kept watch the Spanish fleet came slowly on, intending to surprise
Plymouth; but Hawkins and his vessels were already awaiting the foe
outside, so they anchored for the night off Looe. The next day was Sunday,
the 21st of July, and Medina Sidonia seems to have made up his mind to go
on to the Isle of Wight. All that day the little English ships were barking
round the unwieldy galleys of Spain. "We had some small fight with them
that Sunday afternoon," said Hawkins. By three o'clock the Spanish fleet
was in a pretty confusion, hasting to get away from their tormentors. On
Monday and Tuesday the fight continued, the details of which may be
reserved for a later chapter; but every day more reinforcements came to
Howard, as courtiers and merchants hurried down from London to serve in
pinnace or frigate. By Wednesday morning the English ships had spent
nearly all their ammunition, and were begging for powder and shot at every
village they passed. On Friday Lord Howard knighted Frobisher and
Hawkins for their valiant conduct; he then allowed the Armada to sail along
the Sussex coast and cross the Straits of Dover towards Calais. There
through Saturday and Sunday vast crowds of Flemings and Frenchmen
gathered to gaze at the two great fleets, which were waiting, the Spaniards
for the Prince of Parma to join them from Dunkirk, the English to carry out a
little device which Sir William Winter had suggested.
Six of the oldest vessels were filled with combustibles and guns loaded
to the mouth with old iron, and at midnight were conducted in the pitchy
darkness of a rising storm within bow-shot of the Armada.
A train was fired, and the fierce south-west wind bore the fire-ships into
the crescent of the Spaniards. The blaze, the explosions, the cannon-shot,
struck a panic into the Armada. "The fire of Antwerp!" they cried. "Cut
cable, up anchor!" In a few minutes they were all colliding together in their
hurry to get away from the flames, and all that night they sped away past
Dunkirk and Parma even to the mouth of the Scheldt.
"God hath given us so good a day in forcing the enemy so far to

leeward," wrote Drake, "as I hope in God, the Prince of Parma and the Duke
of Sidonia will not shake hands these few days.... I assure your honour, this
day's service hath much appalled the enemy."
Then on the Monday Hawkins in the Victory, Drake in the Revenge, and
Frobisher in the Triumph, led the English to the attack upon a fleet
disorganised and cowed, fearing alike the sands, the storm, and the foe.
Every ship in the Spanish fleet had received damage, some had been taken
and others sunk, while a score or so went on shore and were lost.
FIRE-SHIPS
The Spanish fleet lay safely moored in Calais Harbour, huge impregnable castles of
timber, but Howard's fire-ships caused them to scurry away before the wind like frightened
fowls.
"We pluck their feathers by little and little," wrote Lord Howard, and if
Burghley had only given them powder enough, the victory would have been
complete. The English had no more ammunition left, but still, as the
Spaniards forged ahead to the north, they grimly followed: "We set on a brag
countenance and gave them chase."
It was not until Friday the 2nd of August that Howard abandoned the
pursuit. He made for the Firth of Forth, took in victuals, powder, and shot,
and sailed southwards to be ready for Parma, should he cross from Dunkirk.
As they sailed, a storm burst upon them, scattering them so that they did not
assemble again in Margate Roads until the 9th of August. A note from Lord
Burghley suggests that as the danger is over, the ships shall be at once
discharged; but there was no money to pay the men who had saved England
in her hour of danger. Towards the end of August, Sir John wrote urgently to
Burghley for money to pay the seamen—£19,000 were already due to them
before the fight off Gravelines—and Lord Howard added a postscript:
"Hawkins cannot make a better return. God knows how the lieutenants and
corporals will be paid." Howard and Hawkins could not pay them off. The
men were kept hanging on, ill-fed, ill-clad, housed like hogs and dying as by
a pestilence. "'Tis a most pitiful sight to see how the men here at Margate,
having no place where they can be received, die in the streets. The best
lodging I can get is barns and such outhouses, and the relief is small that I
can provide for them here. It would grieve any man's heart to see men that
have served so valiantly die so miserably." So Howard writes to Burghley.
Burghley, at his wits' end, writes to Hawkins a melancholy letter: "Why

do you ask for money when you know the exchequer is so empty?"
Howard tells the Queen how the men sicken one day and die the next;
and as woman she pitied them, but could not find means to help her sailors.
"Alas! these things must be—after a famous victory!" Her minister may
have suggested some such reflection.
Meanwhile the great Armada, left to the judgment of God, was leaving
its wrecks on the coasts of Norway, Scotland, and Ireland. It was not until
October that fifty-three ships out of one hundred and thirty-two came back
wearily to a Spanish port.
Sir Francis Drake tells us that many Spanish seamen landed in Scotland
and Ireland, of whom a few remained to live amongst the peasantry, but the
most part were coupled in halters and sent from village to village till they
were shipped to England. "But her Majesty, disdaining to put them to death,
and scorning either to retain or entertain them, they were all sent back again
to their own country to bear witness to the worthy achievement of their
Invincible Navy."
Sir John Hawkins in 1589 proposed to Lord Burghley a scheme for

capturing Cadiz and sinking all the Spanish galleys he could find there. "It is
not honourable for her Majesty to seem to be in any fear of the King of
Spain." But Burghley did not approve of any more expenditure of money,
and the scheme was dropped.
But every year English merchantmen were held up by Spanish galleys

and had to fight for their existence: so in 1590 we hear of Sir John proposing
another attack on Spain; and when this too was rejected, he wrote to
Burghley, saying that he was now out of hope that he should be allowed to
perform "any royal thing." So out of heart was he now that he begged he
might be relieved of his duties as Treasurer of the Navy. "No man living hath
so careful, so miserable, so unfortunate, and so dangerous a life." This
request too was declined. But in May 1590 Sir John was sent, with Sir
Martin Frobisher as Vice-Admiral, in command of fourteen ships to try and
intercept a fleet of Portuguese carracks coming from India. They ransacked
nearly every port on the Spanish coast for five months, so that all valuables
were hastily removed inland, and all the Spanish galleys were hidden behind
rocky promontories. But Philip had ordered the trading fleets to be kept
back, and therefore no prizes were captured, and the adventurers returned
empty-handed.
Queen Elizabeth was mightily incensed, though it was through no fault

of the admirals that the expedition had been a financial failure.
Hawkins wrote the Queen a lengthy epistle explaining why they had
failed, and he finished with a Biblical allusion: "Paul might plant, and
Apollos might water, but it was God only who gave the increase." The
Queen, stamping her foot, exclaimed hotly, "This fool went out a soldier, and
is come home a divine!"
So the poor seaman returned to his hated desk as Treasurer, though he

wrote to Burghley that he was fain to serve her Majesty in any other calling.
"This endless and unsavoury occupation in calling for money is always
unpleasant." He was now over seventy years of age, and his son Richard had
for some years been distinguishing himself on the sea. But when that son had
to surrender to the Spaniards and was sent to a Spanish prison, the old sea-
dog thirsted to go abroad again and rescue his son, or at least take a great
revenge.
His old friend Sir Francis Drake was to go with him, for the Queen had
assented; but rumours of a fresh Armada kept them in England, for "all
men," says Camden, "buckled themselves to war," and mothers only
bewailed that their sons had been killed in France instead of being alive and
well to defend hearth and home in England. But all the Spaniards did was to
cross from Brittany with four galleys and land at Penzance. This town they
sacked and burnt, but as the inhabitants had fled inland, no lives were lost.
This was the last hostile landing made by the Spaniards on England's shore.
Next year, in August 1595, Drake and Hawkins left Plymouth with
twenty-seven ships and 2500 men, Drake sailing in the Defiance as Admiral,
Hawkins in the Garland as Vice-Admiral; Sir Thomas Baskerville was the
Commander by land. The plan was to sail to Nombre de Dios and march
across the isthmus to Panama, there to seize what treasure they could. But
just before they sailed letters came from the Queen informing them that they
were too late to intercept the West Indian fleet; it had already arrived in
Spain; but one treasure-ship had lost a mast and put back to Puerto Rico;
they were to seek for this and take it. So they sailed on the 28th of August,
and in four weeks reached the Grand Canary. Drake and Baskerville wished
to land, victual the fleet, and take the island; Hawkins, still smarting under
Elizabeth's lash, was for strictly obeying the Queen's instructions. However,
Baskerville promised he would take the place in four days, and Hawkins
consented to wait. But finding that a strong mole had been built, and that the
landing-place was defended by guns, and as a nasty sea was rising, they just
landed to get water on the western side of the island and made for Dominica.
After making some traffic in tobacco they went on to Guadaloupe, where

they cleaned their ships and let the men land. The next day, seeing some
Spanish ships passing towards Puerto Rico, Drake concluded that the
treasure-ship was still there, and that this force had been sent to convoy it.
Captain Wignol in the Francis, having straggled behind out of Hawkins'
fleet, fell in with these Spanish ships under Don Pedro Tello, and was
captured. Tello put his men to torture, and drew from them the object and
proposed course of the English ships. When Hawkins heard of this from a
small vessel that had escaped the Spaniards, he suddenly fell sick: age, the
troubles of the voyage, this last disappointment—all were too much for him;
he struggled bravely against his malady, but every day he grew weaker. They
started in three days from Guadaloupe and reached the Virgin Islands, where
they took plenty of fish. Drake and Hawkins had some dispute here, some
difference of opinion, and this was the last straw to weigh down the balance
of death. For on the morning of the 12th of November the fleet passed
through the Strait; and at night, when it was off the eastern end of Puerto
Rico, Hawkins breathed his last.
Drake sailed in imprudently under the forts; one shot wounded his
mizzen-mast, another entered the steerage, where he was at supper, and
struck the stool from under him, killing two of his officers, but not hurting
him. The treasure had been removed from the galleon, and the empty galleon
had been sunk in the mouth of the channel. After a fierce fight Drake had to
give up the attempt to get the treasure and sail away.
He only survived his old friend by eleven weeks. Thus England lost two
of her grandest seamen in this expedition. Hawkins was seventy-five years
old—too old to be exposed to the burning sun and all the anxieties of
warfare. Then came bitter disappointment, and the feeling that he had done
nothing to rescue his son, and little to avenge his wrongs. For six weeks Sir
John Hawkins strove to make head against this "sea of troubles," but his
work had been done, his body was worn out, and he could endure no more.
In his adventurous life he had done many questionable things; but we ought
not to judge him by the moral standard of another age; in his day, the rights
of the slave had not yet been thought of. Hawkins had tried to do his duty to
God and man, as he conceived that duty, and to his unflagging labours and
zeal as Treasurer of the Navy, the success of England against the Armada
was largely due.
CHAPTER III
GEORGE CLIFFORD, EARL OF CUMBERLAND,

THE
CHAMPION OF THE TILT-YARD
When a nobleman neglects his private duties to his home estates and
spends his fortune in fitting out ships to seek gold and jewels, and to damage
the trade of a country he hates, different estimates will be formed of the
honesty and nobility of the motives by which he is prompted.
We shall see in the sketch of George Clifford's life how various motives,
good and indifferent, urged him to play the sea-king.
He was born in his father's castle at Brougham, Westmorland, in August

1558, being fourteenth baron Clifford of Westmorland. His family history
had been distinguished, for the Fair Rosamond of Henry II. was a Clifford,
and in the wars of York and Lancaster the Cliffords took a prominent part on
the Lancastrian side. When George was still a boy his father brought about
his betrothment to the Lady Margaret Russell, daughter of the second Earl of
Bedford. He was being educated at Battle Abbey when his father died, and
later he went to Peterhouse, Cambridge, to complete his studies under
Whitgift, afterwards Archbishop of Canterbury, who was then Master of
Trinity. He also proceeded to Oxford for a time, as was not unusual in those
days.
Before he was nineteen he was married to his betrothed in St. Mary

Overy's Church, Southwark: the young lady was scarce seventeen years old.
Clifford was a young man of expensive tastes. Amongst his other
qualifications for Court life he excelled all the nobles of his time in tilting,
and soon won the recognition of the Queen by his prowess in the
Westminster tilt-yard. So he was made a Knight of the Garter, and in the
twenty-eighth year of his age was appointed one of the forty Peers by whom
Mary Queen of Scots was tried at Fotheringay Castle, in Nottinghamshire.
Was it policy, or love of adventure, or just desire for gain? The Earl soon
began to form schemes for sending ships to plunder the Spaniards; he fitted
out at his own cost the Red Dragon of 260 tons, and the barque Clifford of
130. Raleigh sent a pinnace to join them, and they sailed in 1586 under the
command of Robert Withrington. They took a few merchantmen on their
way to Sierra Leone; the principle on which they acted was a mixture of
courtesy and bullying. Hakluyt says: "Our Admiral hailed their Admiral with
courteous words, willing him to strike his sails and come aboard, but he
refused; whereupon our Admiral lent him a piece of ordnance" (it sounds so
kind and friendly) "which they repaid double, so that we grew to some little
quarrel." The result was that the English boarded the hulks and helped
themselves. For they found the hulks were laden in Lisbon with Spanish
goods, and so thought them fair game. When they reached Guinea they went
ashore, and in their search for water and wood came suddenly upon a town
of negroes, who struck up the drum, raised a yell, and shot off arrows as
thick as hail.
The English returned the fire, having about thirty calivers, and retired to
their boats, "having reasonable store of fish"; "and amongst the rest we
hauled up a great foul monster, whose head and back were so hard that no
sword could enter it; but being thrust in under the belly in divers places,
much wounded, he bent a sword in his mouth as a man would do a girdle of
leather about his hand: he was in length about nine feet, and had nothing in
his belly but a certain quantity of small stones." Later on they found another
town of about two hundred houses and "walled about with mighty great trees
and stakes, so thick that a rat could hardly get in or out." They got in, for the
negroes had fled, and found the town finely and cleanly kept, "that it was an
admiration to us all, for that neither in the houses nor streets was so much
dust to be found as would fill an egg-shell." The English found little to take
except some mats and earthen pots; was it for this that they, on their
departure, set the town on fire? "It was burnt in a quarter of an hour, the
houses being covered with reeds and straw."
Thence they sailed across to America, having no little trouble with a

disease which they had caught ashore. They came to Buenos Ayres, where
was great store of corn, cattle, and wine; but no gold or silver was to be had.
Here they fell into some contention as to their further course, but as food
was growing scarce they sailed northwards, till they came to Bahia, where
they met a hot welcome of balls and bullets, but took some prizes. Then
came a storm, and some of their prizes got loose and were lost. "Anon the
people of the country came down amaine upon us and beset us round, and
shot at us with their bows and arrows."
Of another fight which occurred shortly after, when the barque Clifford
was boarded by the enemy, the merchant Sarracoll says: "Giving a mighty
shout they came all aboard together, crying, 'Entrad! Entrad!' but our men
received them so hotly, with small shot and pikes, that they killed them like
dogs. And thus they continued aboard almost a quarter of an hour, thinking
to have devoured our men, pinnace and all ... but God, who is the giver of all
victories, so blessed our small company that the enemy having received a
mighty foil was glad to rid himself from their hands: whereas at their
entrance we esteemed them to be no less than betwixt two and three hundred
men in the galley, we could scarce perceive twenty men at their departure
stand on their legs; but the greater part of them was slain, their oars broken
and the galley hanging upon one side, as a sow that hath lost her left ear,
with the number of dead and dying that lay one upon another." While this
terrible havoc was a doing, others of the crew had gone ashore and fetched
sixteen young bullocks, "which was to our great comforts and refreshing."
After committing what havoc they could along the coast, with little profit
to themselves, the commander resolved to go home; which resolution was
"taken heavily of all the company—for very grief to see my Lord's hopes
thus deceived, and his great expenses cast away."
The Earl took part in the Armada fight on board the Bonadventure, and
the Queen, to mark her approbation, gave him a commission to go the same
year as General to the Court of Spain, lending him the Golden Lion; this ship
he victualled and furnished at his own expense. After taking one merchant
ship and weathering a storm, he was obliged to turn.
But the Queen was his good friend and lent him the Victory, and with
three smaller ships and 400 men he sailed from Plymouth in 1589. We do
not hear what his Countess thought of so much wandering into danger, but
duty, or profit, called her lord to the high seas. He made a few prizes, French
and German, for he was not too scrupulous about nationalities, and made for
the Azores, where he cut out four ships. At Flores he manned his boats and
obtained food and water from Don Antonio, a pretender to the throne of
Portugal. As they were rowing back to their ship, "the boat was pursued two
miles by a monstrous fish, whose fins many times appeared above water four
or five yards asunder, and his jaws gaping a yard and a half wide, not
without great danger of overturning the pinnace and devouring some of the
company"; they rowed at last away from the monster. They were now joined
by a ship of Raleigh's and two others.

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7.7 Normal Approximation to the Binomial Distribution 359

8 Markov Processes 369

9 The Theory of Games 404

10 The Mathematics of Finance 430

12 Difference Equations and Mathematical Models (Online*)

Appendix A Areas Under the Standard Normal Curve A-1

Appendix B Using the TI-84 Plus Graphing Calculator A-2

Appendix C Spreadsheet Fundamentals A-6

Appendix D Wolfram|Alpha A-10

Learning Objectives (Online*) A-11

Selected Answers SA-1

Index of Applications IA-1

fourteenth edition fourteenth edition

New to This Edition

• Difference Equation Chapter Moved Online. We moved former Chapter 11 online

Relevant and Varied Applications

Exercises to Meet All Student Needs

Check Your Understanding Problems

End-of-Chapter Study Aids

Technology and Supplements

Learning and Teaching Tools

• Instructional videos—238 example-based videos—are available as learning aids within

Student Solutions Manual

Instructor Answers / Instructor Solutions Manual (downloadable)

Larry J. Goldstein Martha J. Siegel

David I. Schneider Steven M. Hair

1.1 Coordinate Systems and Graphs

EXAMPLE 1 Plotting Points Plot the following points:

Now Try Exercise 1

(b) 8x - 4y = 4 Given equation

Table 1 Graphs of Linear Equations

Equation Description of Graph How to Draw Graph

An equation in general form having d ≠ 0 (that is, an equation in which y

SOLUTION Since y appears in the equation, solve for y.

The next example gives an application of linear equations.

(a) Draw the graph of this linear equation.

0 = 25,000 - 400x Set y = 0.

(b) After 5 years (or 60 months), the value of the van is

y = 25,000 - 400(60) = 25,000 - 24,000 = 1000.

15,000 = 25,000 - 400x Set y = 15,000.

The value of the van will be $15,000 after 25 months.

plot ax + by = c; plot y = ax + b; plot x = a

evaluate 2500 − 400x at x = 5

gives the result 500.

y 38. Which of the following equations is graphed in Fig. 9?

9. What are the coordinates of the point Q in Fig. 7? x

8 Markov Processes 369

9 The Theory of Games 404

10 The Mathematics of Finance 430

12 Difference Equations and Mathematical Models (Online*)

Thus, m = 4. Now Try Exercise 1

Figure 8 In other words, y2 = y1 + m, which is what we desired to show.

The salvage value is $200,000.

Now Try Exercise 21