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ECO 255

MATHEMATICS FOR ECONOMIST I

NATIONAL OPEN UNIVERSITY OF NIGERIA

SCHOOL OF ARTS AND SOCIAL SCIENCES

COURSE CODE: ECO 255

COURSE TITLE: MATHEMATICS FOR ECONOMISTS 1

1

ECO 255

MATHEMATICS FOR ECONOMIST I

NATIONAL OPEN UNIVERSITY OF NIGERIA

MATHEMATICS FOR ECONOMISTS 1

ECO 255 COURSE GUIDE

Course Developer:

ODISHIKA Vivian Anietem

Economics Unit,
School of Arts and Social Sciences,
National Open University of Nigeria
...


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ECO 255

MATHEMATICS FOR ECONOMIST I

NATIONAL OPEN UNIVERSITY OF NIGERIA
National Open University of Nigeria
Headquarters
14/16 Ahmadu Bello Way
Victoria Island
Lagos
Abuja Annex
245 Samuel Adesujo Ademulegun Street
Central Business District
Opposite Arewa Suites
Abuja
e-mail: centralinfo@nou
...
ng
URL: www
...
edu
...

For
National Open University of Nigeria Multimedia Technology in Teaching and Learning

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ECO 255

MATHEMATICS FOR ECONOMIST I

CONTENT
Introduction
Course Content
Course Aims
Course Objectives
Working through This Course
Course Materials
Study Units/m
Textbooks and References
Assignment File
Presentation Schedule
Assessment
Tutor-Marked Assignment (TMAs)
Final Examination and Grading
Course Marking Scheme
Course Overview
How to Get the Most from This Course
Tutors and Tutorials
Summary

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ECO 255

MATHEMATICS FOR ECONOMIST I

Introduction
Welcome to ECO: 255 MATHEMATICS FOR ECONOMIST 1
...
The course is made up of twelve study
units (subdivided into four modules), spread across fifteen lecture weeks
...
It
also suggests some general guidelines for the amount of time required of you on each
unit in order to achieve the course aims and objectives successfully
...
The topics covered includes the number system; inequalities; exponent and roots;
systems of equation; simultaneous equation; quadratic equation; set theory; logarithms;
calculus; optimization and linear programming
...

Course Aims
The general aim of this course is to give you an in-depth understanding of the
application of mathematics in economics
...

 Show the difference between the product rule and chain rule of partial derivative
as well as showing the difference between differentiation and integration
...
Although there are set out objectives for each unit, included at the beginning of
the unit- you should read them before you start working through the unit
...
You should
always look at the unit objectives after completing a unit
...
In this way, you can be sure you have
done what is required of you by the unit
...

 Understand the concept of inequalities, differentiate between the properties of
inequality, solve inequality problems and apply inequality to angle problems
...

 Have broad understanding of systems of equation, solve problems of linear
equation using substitution and the addition/subtraction method, and graph
systems of equations solution
...

 Apply the concept of set theory to economic problem, differentiate between
intersect and union, get acquainted with set properties and symbols, and solve
set theory problem using Venn diagram
...


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ECO 255

MATHEMATICS FOR ECONOMIST I

 Discuss the concept of differentiation and integration, understand the
difference between differentiation and integration, solve problem involving
higher order derivatives, master the rules of integration, solve definite and
indefinite integrals, use both the chain and the product rule to solve
differentiation problem, and apply definite integrals to economic problems
...

Each unit contains self-assessment exercises called Student Assessment Exercises (SAE)
...
At the end of the course there is a final examination
...

Course Material
The major component of the course, What you have to do and how you should allocate
your time to each unit in order to complete the course successfully on time are listed
follows:
1
...

3
...

5
...


Course guide
Study units
Textbooks
CDs
Assignment file
Presentation schedule

Study Units
There are 12 units in this course which should be studied carefully and diligently
...
Other areas
border on the Tutor-Marked Assessment (TMA) questions
...
You are advised to do so in order to understand and get acquainted with the
application of mathematics to economic problem
...
They are meant to give you additional information if only you can
lay your hands on any of them
...
By doing so, the stated learning objectives of the
course would have been achieved
...
B and Tamblyn
...
Understanding Math- Introduction to Logarithms
(Kindle Edition), Solid Stae Press, Barkeley: CA
...
J and Howard F
...
Introduction to the Theory of Sets, (Dover Books
on Mathematics), Dover Publiations, In
...

Carter
...
Foundation of Mathematical Economics, The MIT Press,
Cambridge, Massachusetts
Chiang
...
C (1967)
...
C and Wainwright
...
Fundamental Methods of Mathematical
Economics
...
P, Samuelson
...
A and Solow
...
M
...
Linear Programming and
Economic Analysis
...

Ekanem
...
T (2004)
...
H
...
The Elements of Set Theory
...

Franklin
...
N
...
Methods of Mathematical Economics: Linear and Nonlinear
Programming, Fixed-Point Theorems (Classics in Applied Mathematics, 37)
...
W (2004)
...

Kamien
...
I and Schwartz
...
L
...
Dynamic Optimization, Second Edition: The
Calculus of Variations and Optimal Control in Economics and Management
...

Dowling, E
...
(2001)
...
Theory and Problems of Introduction
to Mathematical Economics
...
Third Edition
...
and Vereshchagin N
...
(2002)
...
American Mathematical
Society (July 9 2002)
...
This file presents
you with details of the work you must submit to your tutor for marking
...

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ECO 255

MATHEMATICS FOR ECONOMIST I

Additional information on assignments will be found in the assignment file and later in
this Course Guide in the section on assessment
...
The four course assignments will cover:
Assignment 1 - All TMAs’ question in Units 1 – 3 (Module 1)
Assignment 2 - All TMAs' question in Units 4 – 6 (Module 2)
Assignment 3 - All TMAs' question in Units 7 – 9 (Module 3)
Assignment 4 - All TMAs' question in Unit 10 – 12 (Module 4)
Presentation Schedule
The presentation schedule included in your course materials gives you the important
dates of the year for the completion of tutor-marking assignments and attending tutorials
...
You should
guide against falling behind in your work
...
First are the tutor-marked
assignments; second, the written examination
...
The assignments must be submitted to your tutor
for formal Assessment in accordance with the deadlines stated in the Presentation
Schedule and the Assignments File
...

At the end of the course, you will need to sit for a final written examination of two hours'
duration
...

Tutor-Marked Assignments (TMAs)
There are four tutor-marked assignments in this course
...
You are encouraged to work all the questions thoroughly
...

Assignment questions for the units in this course are contained in the Assignment File
...
However, it is desirable that you
demonstrate that you have read and researched more widely than the required minimum
...

When you have completed each assignment, send it, together with a TMA form, to your
tutor
...
If for any reason, you cannot complete your work on time,
contact your tutor before the assignment is due to discuss the possibility of an extension
...

Final Examination and Grading
The final examination will be of two hours' duration and have a value of 70% of the total
course grade
...
All areas of the course will be assessed
Revise the entire course material using the time between finishing the last unit in the
module and that of sitting for the final examination
...
The final examination covers information from all parts of the course
...

Assignment

Marks

Assignments (Best three assignments out of four that is
marked)

30%

Final Examination

70%

Total

100%

Course Overview
The Table presented below indicates the units, number of weeks and assignments to be
taken by you to successfully complete the course, Mathematics for Economist (ECO
255)
...
This is one of the great
advantages of distance learning; you can read and work through specially designed study
materials at your own pace and at a time and place that suit you best
...
In the same way that a
lecturer might set you some reading to do, the study units tell you when to read your
books or other material, and when to embark on discussion with your colleagues
...

Each of the study units follows a common format
...
Next is a set of learning objectives
...

You should use these objectives to guide your study
...
If you make a
habit of doing this you will significantly improve your chances of passing the course and
getting the best grade
...

This will usually be either from your textbooks or reading sections
...
You will be
directed when you need to embark on these and you will also be guided through what you
must do
...
First, it will enhance your
understanding of the material in the unit
...
In any event, most of the practical
problem solving skills you will develop during studying are applicable in normal working
situations, so it is important that you encounter them during your studies
...
Working through these tests will
help you to achieve the objectives of the unit and prepare you for the assignments and the
examination
...

The following is a practical strategy for working through the course
...
Remember that your tutor's job is to help you
...

1
...

2
...
Refer to the `Course overview' for more details
...
Important information, e
...
details of your tutorials, and the date of the
first day of the semester is available from study centre
...

Whatever method you choose to use, you should decide on and write in your own
dates for working breach unit
...
Once you have created your own study schedule, do everything you can to stick to
it
...
If you get into difficulties with your schedule, please let your tutor know
before it is too late for help
...
Turn to Unit 1 and read the introduction and the objectives for the unit
...
Assemble the study materials
...
You will also need both the study
unit you are working on and one of your textbooks on your desk at the same time
...
Work through the unit
...
As you work through the unit you will be instructed
to read sections from your textbooks or other articles
...

7
...

8
...
Keep in mind that you will
learn a lot by doing the assignments carefully
...

Submit all assignments no later than the due date
...
Review the objectives for each study unit to confirm that you have achieved them
...

10
...
Proceed unit by unit through the course and try to pace your
study so that you keep yourself on schedule
...
When you have submitted an assignment to your tutor for marking do not wait for
its return before starting on the next units
...
When the
assignment is returned, pay particular attention to your tutor's comments, both on
the tutor-marked assignment form and also written on the assignment
...

12
...
Check that you have achieved the unit objectives (listed at the
beginning of each unit) and the course objectives (listed in this Course Guide)
...

You will be notified of the dates, time and locations of these tutorials
...

Your tutor will mark and comment on your assignments, keep a close watch on your
progress and on any difficulties you might encounter, and provide assistance to you
during the course
...
They will be marked by your
tutor and returned to you as soon as possible
...
The following might be circumstances in which you would find help necessary
...

• You do not understand any part of the study units or the assigned readings
• You have difficulty with the self-assessment exercises
• You have a question or problem with an assignment, with your tutor's comments on an
assignment or with the grading of an assignment
...
This is the only chance to have face to
face contact with your tutor and to ask questions which are answered instantly
...
To gain the maximum benefit
from course tutorials, prepare a question list before attending them
...

Summary

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ECO 255

MATHEMATICS FOR ECONOMIST I

The course, Mathematics for Economist 1 (ECO 255), will expose you to basic concepts
in mathematics and economics
...

On successful completion of the course, you would have developed critical and practical
thinking skills with the material necessary for efficient and effective discussion and
problem solving skills on mathematical economic issues
...


TABLE OF CONTENTS
MODULE 1 ……………………………………………………………
...
78-100
SET THEORY, LOGARITHMS & PARTIAL DERIVATIVES
Unit 1

Set Theory
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ECO 255

Unit 2
Unit 3

MATHEMATICS FOR ECONOMIST I

Logarithms
Partial Derivative

MODULE 4……………………………………………………………………101-125
INTEGRAL CALCULUS, OPTIMIZATION AND LINEAR PROGRAMMING
Unit 1
Integral Calculus
Unit 2
Optimization
Unit 3
Linear Programming (LP)

MODULE 1 NUMBER SYSTEM, INEQUALITITES, EXPONENT AND ROOTS
Unit 1
Number System
Unit 2
Inequalities
Unit 3
Exponent and Roots
UNIT 1
NUMBER SYSTEM
1
...
0
Objectives
3
...
1
Introduction to Number System
3
...
3
Real Number
3
...
5
Imaginary Numbers
3
...
0
Conclusion
5
...
0
Tutor-Marked Assignment
7
...
0

INTRODUCTION
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ECO 255

MATHEMATICS FOR ECONOMIST I

A numeral system (or system of numeration) is a writing system for expressing numbers,
that is, a mathematical notation for representing numbers of a given set, using digits or
other symbols in a consistent manner
...

Ideally, a number or a numeral system will:
 Represent a useful set of numbers (e
...
all integers, or rational numbers)
 Give every number represented a unique representation (or at least a standard
representation)
 Reflect the algebraic and arithmetic structure of the numbers
...
However, when decimal representation is used for the rational or real numbers,
such numbers in general have an infinite number of representations, for example 2
...
310, 2
...
309999999
...
, all of which have the same
meaning except for some scientific and other contexts where greater precision is implied
by a larger number of figures shown
...
Two Indian mathematicians are credited with developing them
...

The simplest numeral system is the unary numeral system, in which every natural number
is represented by a corresponding number of symbols
...
Tally marks represent
one such system still in common use
...

The unary notation can be abbreviated by introducing different symbols for certain new
values
...
This is called sign-value
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ECO 255

MATHEMATICS FOR ECONOMIST I

notation
...


2
...

 Have broad understanding of binary, imaginary and complex numbers
...


3
...
1

INTRODUCTION TO NUMBER SYSTEM

Whenever we use numbers in our day to day activities, we apply certain conventions
...
We also accept that the symbols
are chosen from a set of ten symbols: 0,1,2,3,4,5,6,7,8 and 9
...
A number has a base or radix
...
A
base 10 number system has ten digits, and it is called the decimal number system
...
Here the 10 is a
subscript which indicates that the number is a base 10 number
...
It
shows us a few things about the number 3456
...


Table1
...
1
0
...
01
0
...
001
0
...
e
...

 As we move right through the columns the numbers decrease in weight
...
Starting from the left of the decimal point and moving
left, the numbers increase positively
...

 You can see in the third row that the number base (10) is raised to the power of the
column or index
...
10 is the
base, 3 is the index or column
...
7 + 0
...
009
Or as:
3 * 103 + 4 * 102 + 5 * 101 + 6 * 100 + 7 * 10-1 + 8 * 10-2 + 9 * 10-3
...
, and digit or coefficient
value of 7, 8, 9, 10, 11, 12, 13, 14 and 15
...


3
...
Thus, addition, subtraction and multiplication as well as division of real
numbers are feasible in the field of real numbers
...
142) and e
(2
...

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ECO 255

MATHEMATICS FOR ECONOMIST I

The properties of number systems are as follows:
1
...

2
...
Associativity: (A + B) + C = A + (B + C) and (A × B) × C = A × (B × C)
4
...
Identity: A + 0 = A, A × 1 = A
6
...

SELF ASSESSMENT EXERCISE
State the properties of number system
...
3

REAL NUMBERS

A real number is a value that represents a quantity along a continuous line
...
41421356… the square root of two, an irrational
algebraic number) and π or e (3
...

Real numbers can be thought of as points on an infinitely long line called the number line
or real line, where the points corresponding to integers are equally spaced
...
632, where each consecutive digit is measured in units one tenth the size of the
previous one
...


Figure1
...
Real numbers are used to measure continuous
quantities
...
823122147… The ellipsis (three dots) indicates that there would still
be more digits to come
...

 Real numbers can be ordered (obeys the addition and multiplicative rule)
...
First, an order can be lattice-complete (a partial
ordered set in which all subsets have supremum (joint) and an infimum (meet))
...
This simply means that this ordered property of real number is not true if
imaginary numbers are brought into the picture
...
This means that √3 comes before √11 on the number
line and that they both come before √3 + √11
...
Think about the rational numbers 3 and 5, we know that we
can order 3 and 5 as follows
...

One main reason for using real numbers is that the reals contain all limits
...
This means that the reals are complete
...
In other words, a sequence is a Cauchy sequence if
its elements xn eventually come and remain arbitrarily close to each other
...
In other words, a sequence has limit x if its elements eventually come and
remain arbitrarily close to x
...
4

BINARY NUMBERS

A binary number is a number expressed in the binary numeral system, or base-2 numeral
system, which represents numeric values using two different symbols: typically 0 (zero)
and 1 (one)
...


DECIMAL COUNTING
Decimal counting uses the ten symbols 0 through 9
...
When the available symbols for the low-order digit are exhausted, the nexthigher-order digit (located one position to the left) is incremented and counting in the
low-order digit starts over at 0
...
007, 008, 009, (rightmost digit starts over, and next digit is
incremented)

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ECO 255

MATHEMATICS FOR ECONOMIST I

010, 011, 012,
...
097, 098, 099, (rightmost two digits start over, and
next digit is incremented) 100, 101, 102,
...

Table1
...
Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes
an increment of the next digit to the left:
0000,
0001, (rightmost digit starts over, and next digit is incremented)
0010, 0011, (rightmost two digits start over, and next digit is incremented)
0100, 0101, 0110, 0111, (rightmost three digits start over, and the next digit is
incremented)
1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111
...
To determine
the decimal representation of a binary number simply take the sum of the products of the

23

ECO 255

MATHEMATICS FOR ECONOMIST I

binary digits and the powers of 2 which they represent
...


SELF ASSESSMENT EXERCISE
Convert 1002312to decimal form
...
5

IMAGINARY NUMBERS

An imaginary number is a number that can be written as a real number multiplied by the
imaginary unit i, which is defined by its property i2 = −1
...
For example, 5i is an imaginary number, and its square is −25
...

An imaginary number bi can be added to a real number a to form a complex number of
the form a + bi, where a and b are called, respectively, the real part and the imaginary
part of the complex number
...
The name "imaginary number" was coined in the 17th
century as a derogatory term, as such numbers were regarded by some as fictitious or
useless
...

Put differently, the square root of a negative number is not a real number
...
An imaginary number is the square root of a negative number, e
...
√2, √-9, √-16, e
...
c
...
e
...

Thus, i2 = -1
...
i = -1, i4 = i2
...
, i =14
...


24

ECO 255

MATHEMATICS FOR ECONOMIST I

e
...
c
...
g
...


3
...
A complex
number is a number that can be expressed in the form a + bi, where a and b are real
numbers and i is the imaginary unit, which satisfies the equation i2 = −1
...
Complex
numbers extend the concept of the one-dimensional number line to the two-dimensional
complex plane by using the horizontal axis for the real part and the vertical axis for the
imaginary part
...
A complex number whose real part is zero is said to
be purely imaginary, whereas a complex number whose imaginary part is zero is a real
number
...


Figure 1
...

25

ECO 255

MATHEMATICS FOR ECONOMIST I

The figure above proves that a complex number can be visually represented as a pair of
numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the
complex plane
...
In the above figure, a and b represents the real
numbers, while i represents the imaginary number, thus their combination formed the
complex number
...
Complex numbers provide a solution to this problem
...
In this case the solutions are −1 + 3i and −1 − 3i, as can
be verified using the fact that i2 = −1:
((-2 + 4i) + 2)2
(4i)2 = (42)(i2)
16(-1) = -16
Also,
((-2 - 4i) + 2)2
(-4i)2 = (-42)(i2)
16(-1) = -16
In fact not only quadratic equations (an equation in which the highest power of an
unknown quantity is a square), but all polynomial equations (an expression consisting of
variables and coefficients that involves only the operations of addition, subtraction,
multiplication and non-negative integer exponents) with real or complex coefficients in a
single variable can be solved using complex numbers
...
If a= 0, Z becomes an imaginary number and if b = 0, Z becomes a real
number
...

For example, 3 – 2i and 3 + 2i are conjugate complex number
...
i
...
, the real part of the sum is equal to the sum of
real parts of the two numbers and the imaginary part of the sum is equal to the sum of the
imaginary parts of the two numbers
...


The difference (Z1 – Z2) between Z1 = a1 + b1i and Z2 = a2 + b2i is Z1 – Z2 = (a1 – a2) +
(b1 – b2)i
...

and
(3 + 6i) - (2 - 4i)
(3 - 2) + (6 + 4)i
1 + 10i
...
(Since i2 = -1)
...


Notice that the product of two conjugate complex numbers is:
27

ECO 255

MATHEMATICS FOR ECONOMIST I

Z1*Z2 = (a1 + b1i)(a2 – b2i)
a1a2 + a1b2i + b1ia2 + b1b2i2
(a1a2 – b1b2) + (a1b2+ b1a2)i
Remember, in our analysis of complex number, i2 = -1, that was why the sign in front of
b1b2 changed to negative
...

25 – 10i + 10i + 4
25 + 4 = 29
...

We can find p + q1 as follow:
p + qi = a1 + b1i
a2 + b2i
(a1 + b1i)(a2 – b2i)
(a2 + b2i)(a2 – b2i)
a1a2 + b1b2 + (b1b2 – a1b2)i
a22 + b22
a1a2 + b1b2+ b1a2 + a1b2
a2 2 + b2 2

a22 + b22

It can easily be shown that the commutative, associative and distributive laws also hold
for operations on complex numbers
...


4
...

28

ECO 255

MATHEMATICS FOR ECONOMIST I

 A number or numeral system will represent a useful set of numbers and will give
every numbers represented a unique representation which reflects the algebraic
and arithmetic structure of the numbers
...

 Imaginary numbers are numbers that can be written as a real number multiplied by
the imaginary unit i, which is defined by its property i2 = -1
...
It
can be expressed in the form a + bi, where a andb are the real numbers, while i is
the imaginary unit
...
0

SUMMARY

This unit focused on number system which is a way of expressing numbers in writing, or
the mathematical notation for representing numbers of a given set, using digits or other
symbols in a consistent manner
...
It was observed that binary numbers is
expressed in a base-2 system which represents numeric values using two symbols (0 and
1)
...
The square of an imaginary number bi is -b2
...

6
...

 What is the product of two conjugate complex number Z1*Z2, with Z1 = (17 +
11i), and Z2= (9 + 15i)

29

ECO 255

7
...
C and Wainwright
...
Fundamental Methods of Mathematical
Economics
...
T (2004)
...
0
Introduction
2
...
0
Main Content
3
...
2
Properties of Inequalities
3
...
4
Solving Inequalities using Inverse Operations
3
...
0
Conclusion
5
...
0
Tutor-Marked Assignment
7
...


INTRODUCTION

30

ECO 255

MATHEMATICS FOR ECONOMIST I

This unit seeks to expose the students to the concept of inequality
...
Solving inequality means finding all of its solutions
...
Inequality is a statement that holds between two values when
they are different
...
It does not say that one is greater than
the other, or even that they can be compared in size
...

The notation ab means that a is greater than b
...
These relations are known as strict inequalities
...

In contrast to strict inequalities, there are two types of inequality relations that are not
strict:
 The notation a ≤ b means that a is less than or equal to b (or, equivalently, not
greater than b, or at most b)
...

2
...


3
...
1

INTRODUCTION TO INEQUALITIES

This chapter is about inequalities
...


31

ECO 255

MATHEMATICS FOR ECONOMIST I

Where:

x < y means " x is less than y "
x≤y means " x is less than or equal to y "
x > y means " x is greater than y "
x≥y means " x is greater than or equal to y "
x≠y means " x is not equal to y "

Like an equation, an inequality can be true or false
...

1 + 3 < 6 - 2 is a false statement
...

1 + 3≠6 - 2 is a false statement
...

To determine whether an inequality is true or false for a given value of a variable, plug in
the value for the variable
...

Example 1
...


Thus, 5x + 3 ≤ 9 is true for x = 1
...
Does 3x - 2 > 2x + 1 hold for x = 3?
3(3) - 2 > 2(3) + 1?
Is

7 > 7?, the answer is no
...
Plug each of the values in the replacement set in for the
variable
...

Example 3: Find the solution set of x - 5 > 12 from the replacement set {10, 15, 20, 25}
...

15 - 5 > 12? False
...

32

ECO 255

MATHEMATICS FOR ECONOMIST I

25 - 5 > 12? True
...

Example 4: Find the solution set of -3x ≥ 6 from the replacement set {-4, -3, -2, -1, 0, 1}
...

-3(- 3)≥6? True
...

-3(- 1)≥6? False
...

-3(1)≥6? False
...

Example 5
...

02 ≠ 2(0)? False (they are, in fact, equal)
...

22 ≠ 2(2)?False
...

Thus, the solution set is {1, 3}
...
2 + 6 > 2 × 4
2
...
2

PROPERTIES OF INEQIALITIES

There are formal definitions of the inequality relations >, <,≥,≤ in terms of the familiar
notion of equality
...
Recall that zero is not a positive number, so this cannot
hold if a = b
...
The following are the
properties of inequalities
...


TRICHOTOMY AND THE TRANSITIVITY PROPERTIES

TRICHOTOMY PROPERTY
For any two real numbers a and b, exactly one of the following is true: ab
...

If a>b and b>c, then a>c
...

If a≥b and b≥c, then a≥ c
...


ADDITION AND SUBTRACTION

ADDITION PROPERTIES OF INEQUALITY
If aIf a>b, then a + c>b + c
SUBTRACTION PROPERTIES OF INEQUALITY
If aIf a>b, then a - c>b - c
These properties also apply to ≤ and ≥:
If a ≤ b, then a + c≤b + c
If a≥b, then a + c≥b + c
If a≤b, then a–c≤b - c
If a≥b, then a–c≥b–c

3
...
Similarly, -2 < 0 and 2 > 0
...
This is also true when both numbers are nonzero: 4 > 2 and -4 < - 2 ; 6 < 7 and -6 > - 7 ; -2 < 5 and 2 > - 5
...
For instance, 4 > 2, so 4(- 3) < 2(- 3): -12 < - 6
...
This leads to the multiplication and division properties of inequalities for
negative numbers
...


4
...

If the numbers on both sides have the same sign, the inequality sign flips
...
Similarly,-1/3>-2/3, but -3 <-3/2
...

If a > 0 and b >0 , or a < 0 and b < 0 , and a > b , then 1/a<1/b
...


35

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MATHEMATICS FOR ECONOMIST I

SELF ASSESSMENT EXERCISE
What property is demonstrated by the following statement?
1
...
Name the property or operation used in the inequality 2x + x < 5 – x + x
...
3

SOLVING INEQUALITY PROBLEM

Solving linear inequalities is very similar to solving linear equations, except for one small
but important detail: you flip the inequality sign whenever you multiply or divide the
inequality by a negative
...

Solution: When we substitute 8 for x, the inequality becomes 8-2 > 5
...
On the other hand, substituting -2 for x yields the false
statement (-2)-2 > 5
...
Inequalities usually
have many solutions
...

The basic strategy for inequalities and equations is the same: isolate x on one side, and
put the "others" on the other side
...

We accomplish this by subtracting 5 on both sides (Rule 1) to obtain:
(2x + 5) – 5 < 7 – 5
2x< 2
...

All real numbers less than 1 solve the inequality
...

36

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MATHEMATICS FOR ECONOMIST I

Example 2
...

Solution: Let's start by moving the ``5'' to the right side by subtracting 5 on both sides
...

How do we get rid of the negative sign in front of x? Just multiply by (-1) on both sides,
changing "≤" to "≥" along the way:
(-x)*(-1) ≥ 1*(-1),
x ≥ -1
...

Example 3: Solve the inequality 2(x – 1) > 3(2x + 3)
...

(2x – 2) – 2x> (6x + 9) – 2x,
-2 > 4x + 9
...

-2 - 9 > (4x + 9) – 9
-11 > 4x
...

-11 > 4x
4
-11/4 >x
...


3
...

Follow these steps to reverse the order of operations acting on the variable:
1
...

2
...
When multiplying or dividing by a negative number, flip the
inequality sign
...

3
...

The answer should be an inequality; for example, x <5
...
Then, change the " = " sign in the answer to a " ≠ " sign
...

Example 1: 5x - 8 < 12
5x - 8 + 8 < 12 + 8
5x < 20
2x/5<20/5
x< 4
Example 2: 4 - 2x≤2x - 4
4 - 2x + 2x≤2x - 4 + 2x
4≤4x - 4
4 + 4≤4x - 4 + 4
8≤4x
8/4 ≤4x4
38

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MATHEMATICS FOR ECONOMIST I

2≤x
x≥2
Example 3:x -2 ≥ - 6
5
x -2× 5 ≥ - 6 × 5
5
The number being divided is negative, but the number we are dividing by is positive, so
the sign does not flip
...
(b) –(x + 2) < 4
...
5

APPLICATION OF INEQUALITIES TO ANGLES

Inequalities are useful in many situations
...
There are three types of angles: right angles, acute angles, and obtuse
angles
...
Acute angles have a measure
of less than 90 degrees
...

Thus, we can write out inequalities classifying the three types of angles:
x = the measure of angle A in degrees
39

ECO 255

MATHEMATICS FOR ECONOMIST I

If x < 90, then A is an acute angle
...

If x > 180, then A is an obtuse angle
...

Is A acute if x = 15? If x = 65? If x = 90? If x = 135?
15 < 90? Yes
...

65 < 90? Yes
...

90 < 90? No
...

135 < 90? No
...

Example 2: If angle A measures 2x - 5 degrees, for which of the following values of x is
A obtuse? {25, 45, 65, 85}
2(25) - 5 > 90? No
...

2(65) - 5 > 90? Yes
...

Thus, A is obtuse for x = {65, 85}
...

SELF ASSESSMENT EXERCISE
If angle F measures 3(15x – 45) degrees, for which of the following values of x is G
obtuse? {3, 4, 5, 6}
...
0

MATHEMATICS FOR ECONOMIST I

CONCLUSION
 Solution of an inequality is a number which when substituted for a variable makes
the inequality a true statement
...

 x < y means " x is less than y ", x ≤ y means " x is less than or equal to y ", x > y
means " x is greater than y ", x ≥ y means " x is greater than or equal to y ", x ≠ y
means " x is not equal to y "
...
Also, if a
> b and b >c , then a > c
...
Also, if a ...

 In solving linear inequality problem is similar to that of linear equation with only
flipping the inequality sign whenever the inequality is multiplied or divided by a
negative value
...


5
...
In order to do
justice to this unit, properties of inequality such as the transitive, addition and subtraction,
multiplication and division as well as the reciprocal properties were reviewed
...
Several
solution examples and solution were given on how to solve inequality problem, and the
application of inequalities to angles was reviewed as well
...
0

MATHEMATICS FOR ECONOMIST I

TUTOR-MARKED ASSIGNMENT
 If angle F measures x(x – 3) + 90 degree, for which of the following values of x is
F right? {0, 1, 2, 3}
...


7
...
M (2001)
...
C and Wainwright
...
Fundamental Methods of Mathematical
Economics
...
T (2004)
...
0
Introduction
2
...
0
Main Content
3
...
2
Exponents
3
...
1 Forms of Exponents
3
...
3
...
3
...
3
...
4
The Negative Exponents
3
...
5
...
5
...
6
Simplifying and Approximating Roots
3
...
1 Simplifying Roots
42

ECO 255

4
...
0
6
...
0

3
...
2 Approximating Roots
Conclusion
Summary
Tutor-Marked Assignment
References/Further Readings

1
...
This unit provides an
introduction to the meaning of exponents and the calculations associated with them
...

2
...

 Have broad understanding on the effect of negative exponents and roots
...

 Simplify and approximate roots without problem
...
0

MAIN CONTENT

3
...

For example: In the expression a7, the exponent is 7, and a is the base
...
It indicates that 2 is to be multiplied by itself 4 times
...

The root of a number (say x) is another number, which when multiplied by itself a given
number of times, equal x
...

The second root is usually called the square root, while the third root is usually called the
cube root
...

3
...
An exponent indicates the
number of times we must multiply the base number
...

A number to the first power is that number one time, or simply that number: for example,
61 = 6 and 531 = 53
...
e
...

Here is a list of the powers of two:
20

=

1

21

=

2

22

=

2×2 = 4

23

=

2×2×2 = 8

4

=

2×2×2×2 = 16

25

=

2×2×2×2×2 = 32

2
And so on
...

Here is a list of the power of ten:
100

=

1

101

=

10

102

=

10×10 = 100

3

=

10×10×10 = 1, 000

104

=

10×10×10×10 = 10, 000

105

=

10×10×10×10×10 = 100, 000

10

And so on
...
This is the meaning of base ten--a "1" in each
place represents a number in which the base is 10 and the exponent is the number of
zeros after the 1
...
For
example, a 5 in the thousands place is equivalent to 5×1000, or 5×103
...
The
number 492 has a 4 in the hundreds place (4×102), a 9 in the ten places (9×101) and a 2 in
the one place (2×100)
...

Examples: Write out the following numbers as single-digit numbers multiplied by the
powers of ten
...

3
...
1 FORMS OF EXPONENTS
1
...
5 squared, denoted 5 2, is equal to
5×5, or 25
...
One way to remember the term "square" is that
there are two dimensions in a square (height and width) and the number being squared
appears twice in the calculation
...

A number that is the square of a whole number is called a perfect square
...
25 and 4 are also perfect squares
...

2
...
5 cubed, denoted 53which is
equal to 5×5×5, or 125
...
The term "cube" can be remembered
because there are three dimensions in a cube (height, width, and depth) and the number
being cubed appears three times in the calculation
...

Exponents can be greater than 2 or 3
...
We write
an expression such as "74" and say "seven to the fourth power
...
"
Since any number times zero is zero, zero to any (positive) power is always zero
...

SELF ASSESSMENT EXERCISE
Provide answers to the following: (a) 83, (b) 07, (c)79
...
3

MATHEMATICS FOR ECONOMIST I

EXPONENTS OF SPECIAL NUMBERS

3
...
1 EXPONENTS OF NEGATIVE NUMBERS
Since an exponent on a number indicates multiplication by that same number, an
exponent on a negative number is simply the negative number multiplied by itself a
certain number of times:
(- 4)3 = - 4× -4× - 4 = - 64
(- 4)3 = - 64 is negative because there are 3 negative signs--see Multiplying Negatives
...

Since an odd number of negative numbers multiplied together is always a negative
number and an even number of negative numbers multiplied together is always a positive
number, a negative number with an odd exponent will always be negative and a negative
number with an even exponent will always be positive
...

Example 1: (- 3)4= ?
1
...
34 = 81
...
The exponent (4) is even, so (- 3)4 = 81
...
Take the power of the positive opposite
...

2
...

3
...
2 EXPONENTS OF DECIMAL NUMBERS
When we square 0
...
46×0
...
46×
46
...

0
...
46×0
...
2116
...
Next, multiply that number by the exponent
...
Then, take the power of the base
number with the decimal point removed
...

Example 1: 1
...
There is 1 decimal place and the exponent is 4
...

46

ECO 255

MATHEMATICS FOR ECONOMIST I

2
...

3
...
1
...
0625
...
043= ?
1
...
2×3 = 6
...
43 = 64 = 000064
...
Insert the decimal point 6 places to the right
...
043 = 0
...

As we can see, decimals less than 1 with large exponents are generally very small
...
3
...
When we multiply fractions together, we multiply their numerators together and
we multiply their denominators together
...
Thus, (3/4)3 = (33)/(43)
...
To take the power of a mixed
number, convert the mixed number into an improper fraction and then proceed as above
...

SELF ASSESSMENT EXERCISE
Provide answers to the following: (a) (-18)3, (b) (-7)7, (c) 0
...

3
...
What if we
have a negative exponent, how do we treat it?
Taking a number to a negative exponent does not necessarily yield a negative answer
...
For example, 5 -4
= 1/54 = 1/625; 6-3 = 1/63 = 1/216; and (- 3)-2 = 1/(- 3)2 = 1/9
...
For example, (2/3) -4 = (3/2)4 = (34)/(24) = 81/16 and (- 5/6)-3 = (6/(- 5))3
= (63)/((- 5)3) = 216/(- 125) = - 216/125
...

10-1

=

1/101 = 1/10 = 0
...
01

10-3

=

1/103 = 1/1, 000 = 0
...
0001

10-5

=

1/105 = 1/100, 000 = 0
...

Now we can write out any terminating decimal as a sum of single- digit numbers
multiplied by the powers of ten
...
45 has a 2 in the tens place (2×101) , a 3
in the unit place (3×100) , a 4 in the tenths place (4×10 -1) and a 5 in the hundredths place
(5×10-2)
...
45 = 2×101 +3×100 +4×10-1 +5×10-2
...
81 = 5×102 +2×101 +3×100 +8×10-1 +1×10-2
3
...
904 = 4×101 +6×100 +9×10-1 +0×10-2 +4×10-3
SELF ASSESSMENT EXERCISE
Provide answers to the following: (a) 8-3and, (b) (2/5)-2
3
...
For example, the square root of a number is the number that,
when squared (multiplied by itself), is equal to the given number and the symbol “√” is
usually used to denote square root
...
The square root of 121, denoted √121, is 11, because
112 = 121
...
√81 = 9, because 92 = 81
...
The square root of a number is always positive
...
All fractions that have a
perfect square in both numerator and denominator have square roots that are rational
numbers
...
All other positive numbers have squares that are
48

ECO 255

MATHEMATICS FOR ECONOMIST I

non-terminating, non- repeating decimals, or irrational numbers
...
41421356 and √53/11 = 2
...

Usually when people say root of a number, what readily comes to mind is the square root
of that number, however, they are other roots like the cube root, forth roots, eight roots
etc
...
g
...

Also, polynomials can also be said to have roots
...

3
...
1 SQUARE ROOTS OF NEGATIVE NUMBERS
Since a positive number multiplied by itself (a positive number) is always positive, and a
negative number multiplied by itself (a negative number) is always positive, a number
squared is always positive
...

Taking a square root is almost the inverse operation of taking a square
...
However, squaring a negative number and then taking the square root of
the result is equivalent to taking the opposite of the negative number: √(-7)2 = √49 = 7
...
For example, √6 2 = | 6| =
6, and √(-7)2 = | - 7| = 7
...

When we take the square root of a positive number and then square the result, the number
does not change: (√121)2 = 112 = 121
...

3
...
2 CUBE ROOTS AND HIGHER ORDER ROOTS
A cube root is a number that, when cubed, is equal to the given number
...
For example, the cube root of 27 is 271/3 = 3
...

Roots can also extend to a higher order than cube roots
...
The 5th root
of a number is a number that, when taken to the fifth power, is equal to the given number,
and so on
...


49

ECO 255

MATHEMATICS FOR ECONOMIST I

An odd root of a negative number is a negative number
...
For example, (- 27)1/3 = - 3, but (- 81)1/4 does not exist
...
0412 as single-digit numbers multiplied by powers of ten
...

3
...
6
...
SIMPLIFYING ROOTS
Often, it becomes necessary to simplify a square root; that is, to remove all factors that
are perfect squares from inside the square root sign and place their square roots outside
the sign
...

To simplify a square root, follow these steps:
1
...

2
...
If the factor appears three times, cross out two of the factors
and write the factor outside the sign, and leave the third factor inside the sign
...
times, this counts as 2, 3, and 4 pairs,
respectively
...
Multiply the numbers outside the sign
...

4
...

To simplify the square root of a fraction, simplify the numerator and simplify the
denominator
...

√12 = √2x2x3
√2x2x3 = 2x√3
2√3 = 22 x 3 = 12
...

√600 = √2x2x2x3x5x5
√2x2x2x3x5x5 = 2x5x√2x3
2x5x√2x3 = 10√6
50

ECO 255

MATHEMATICS FOR ECONOMIST I

102x6 = 600
...

Similarly, to simplify a cube root, factor the number inside the "(3√)" sign
...

Example4: Find the cube root of 8
...
So the cube root of
8 is 2
...

3√216
3√6*6*6
Thus giving us an answer of 6
...
6
...
And one cannot simply divide by some given number every time to find
a square root
...

Here are the steps to approximate a square root:
1
...
Take its square root
...
Divide the original number by this result
...
Take the arithmetic mean of the result of I and the result of II by adding the two
numbers and dividing by 2 (this is also called "taking an average")
...
Divide the original number by the result of III
...
Take the arithmetic mean of the result of III and the result of IV
...
Repeat steps IV-VI using this new result, until the approximation is sufficiently
close
...

25 is close to 22
...
4
(5 + 4
...
7
22/4
...
68
(4
...
68)/2 = 4
...
69 = 4
...
69
...

71 is close to 64
...
9
(8 + 8
...
45
71/8
...
40
(8
...
40)/2 = 8
...
425 = 8
...
425 + 8
...
426
71/8
...
426
√71 = 8
...

√56 can be simplified: √56 = √2x2x2x7 = 2×√2x7 = 2× √14
Approximate √14:
14 is close to 16
...
5
(4 + 3
...
75
14/3
...
73
(3
...
73)/2 = 3
...
74 = 3
...
74
Thus, √56 = 2×√14 = 2×3
...
48

52

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MATHEMATICS FOR ECONOMIST I

SELF ASSESSMENT EXERCISE
Simplify √180, √189 and √150
...
0

CONCLUSION
 An exponent is a mathematical notation that implies the number of times a number
is to be multiplied by itself
...

 An exponent on a negative number is simply the negative number multiplied by
itself in a certain number of times
...

 To simplify the square root of a fraction, simplify the numerator and the
denominator
...
0

SUMMARY

This unit focused on Exponent and Roots
...
Sub-topics
such as the square, cubes and higher order exponents were reviewed
...

6
...

 Simplify √291
...


7
...
M (2001)
...
T (2004)
...
Mareh:Benin
City

53

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MATHEMATICS FOR ECONOMIST I

MODULE 2 EQUATIONS
Unit 1
Systems of Equation
Unit 2
Simultaneous Equation
Unit 3
Quadratic Equation
UNIT 1
SYSTEMS OF EQUATION
1
...
0
Objectives
3
...
1
Introduction to System of Equation and Graphing
3
...
0
5
...
0
7
...
3
Solving systems of Linear Equation by Addition/Subtraction
Conclusion
Summary
Tutor-Marked Assignment
References/Further Readings

2
...
It is also a statement which
shows that the values of two mathematical expressions are equal
...
Solving equations means finding the value (or set of values)or unknown
variables contained in the equation
...

We combine like terms to reduce the equation to:
7x – 2x = 8 + 2
5x = 10
x = 2
...
In this unit, we will begin to deal with systems of equations; that is, with a set
of two or more equations with the same variables
...

Systems of linear equations can have zero, one, or an infinite number of solutions,
depending on whether they are consistent or inconsistent, and whether they are dependent
or independent
...
Substitution
is useful when one variable in an equation of the system has a coefficient of 1 or a
coefficient that easily divides the equation
...
However, many systems of linear equations are
not quite so neat (not easy to calculate) and substitution can be difficult, thus an
alternative method for solving systems of linear equations (the Addition/Subtraction
method) is introduced
...
0
OBJECTIVES
At the end of this unit, you should be able to:
 Explain broadly, the equation system
 Solve problems of linear equation using substitution method
...

 Graph systems of equations solution
...
0

MAIN CONTENT
55

ECO 255

MATHEMATICS FOR ECONOMIST I

3
...
In general, we could find a limited number of solutions to a single equation
with one variable, while we could find an infinite number of solutions to a single
equation with two variables
...

A system of equations is a set of two or more equations with the same variables
...
In order to solve a system of equations, one must find all the
sets of values of the variables that constitute solutions of the system
...

(8, 3) is a solution of the both equations
...

(6, 4) is a solution of both equations
...

Thus, the solution set of the system is {(8, 3), (6, 4)}
...
For
instance, consider y = 8 and x = 3 in the first and second equation, the resulting outcome
will be:
8 + 2(3) = 8 + 4 = 14
...

Thus, the option gave us the solution for both equations
...

SOLVING SYSTEMS OF LINEAR EQUATIONS BY GRAPHING
When we graph a linear equation of two variables as a line, all the points on this line
correspond to ordered pairs that satisfy the equation
...

To solve a system of equations by graphing, graph all the equations in the system
...

Example: Solve the following system by graphing:
4x - 6y = 12
(1)
2x + 2y = 6 (2)
From equation (1),
4x - 6y = 12
6y = 4x - 12
y = (4x/6) – 12/6
y = 2x/3 - 2
56

ECO 255

MATHEMATICS FOR ECONOMIST I

From equation (2),

2x + 2y = 6
2y = -2x + 6
y = -x + 3
In the above solution, the slope for the first and second equation is 2/3 and -1
respectively, while the y-intercept is -2 and 3 respectively
...
Systems of Equation Graph
Since the two lines intersect at the point (0, 3), this point is a solution to the system
...

To check, plug (0, 3) with y = 0, and x = 3 in to the equationy = 2x/3 – 2 and y = -x + 3
...

y = 2x/3 – 2 = 2(3)/3 – 2 = 6/3 – 2 = 2 – 2=0
2
...

1
...

2
...

3
...

A system is consistent if it has one or more solutions
...

The following chart will help determine if an equation is consistent and if an equation is
dependent:
SELF ASSESSMENT EXERCISE
57

ECO 255

MATHEMATICS FOR ECONOMIST I

Which ordered pairs in the set {(2, 5),(2, -9),(5, 2),(- 5, -2),(- 5, 12)} satisfy the following
system of equations?
x+y=7
x2 + 3x = 10
...
2
SOLVING SYSTEMS OF LINEAR EQUATION BY SUBSTITUTION
Graphing is a useful tool for solving systems of equations, but it can sometimes be timeconsuming
...
Consider the
following example below
...
63
Now substitute this x -value into the "isolation equation" to find y:
y = 3(1
...
88 + 7 = 11
...

Example 2: Solve the following system, using substitution:
2x + 4y = 36
10y - 5x = 0
It is easier to work with the second equation, because there is no constant term:
5x = 10y
x = 2y
In the first equation, substitute for x its equivalent expression:
2(2y) + 4y = 36
4y + 4y = 36
8y = 36
y = 4
...
5) = 9
Thus, the solution to the system is (9, 4
...

Example 3: Solve the following system, using substitution:
2x - 4y = 12
58

ECO 255

MATHEMATICS FOR ECONOMIST I

3x = 21 + 6y
It is easiest to isolate x in the second equation, since the x term already stands alone:
x = 21 + 6y
3
x = 7 + 2y
In the first equation, substitute for x its equivalent expression:
2(7 + 2y) - 4y = 12
14 + 4y - 4y = 12
14 = 12
Since 14≠12, the system of equations has no solution
...
The two equations describe two parallel lines
...
We will isolate y in the second
equation:
2y = 5x + 34
y = 5x + 34
3
y = 5x + 17
2
In the first equation, substitute for y its equivalent expression:
10x = 4(5/2x + 17) - 68
10x = 10x + 68 - 68
10x = 10x
0=0
Since 0 = 0 for any value of x, the system of equations has infinite solutions
...
The system is dependent (and consistent)
...

SELF ASSESSMENT EXERCISE
Solve the following system of equations:
(a) 2x + 2y = 4
(b) 4y + 2x = 5
...

3
...
There is another method for solving systems of
equations: the addition/subtraction method
...
In order for the new equation
to have only one variable, the other variable must cancel out
...

We can produce equal and opposite coefficients simply by multiplying each equation by
an integer
...

Example 2: Add and subtract to create a new equation with only one variable:
4x - 2y
= 16
7x + 3y = 15
Here, we can multiply the first equation by 3 and the second equation by 2:
12x - 6y = 48
14x + 6y = 30
Adding these two equations yields 26x = 78
We can add and subtract equations by the addition property of equality, since the two
sides of one equation are equivalent, we can add one to one side of the second equation
and the other to the other side
...
Rearrange each equation so the variables are on one side (in the same order) and
the constant is on the other side
...
Multiply one or both equations by an integer so that one term has equal and
opposite coefficients in the two equations
...
Add the equations to produce a single equation with one variable
...
Solve for the variable
...
Substitute the variable back into one of the equations and solve for the other
variable
...
Check the solution; it should satisfy both equations
...

Check:
2(1/2) - 3(-2) = 7? Yes
...

Example 2: Solve the following system of equations:
4y - 5 = 20 - 3x
4x - 7y + 16 = 0
Rearrange each equation:
3x + 4y = 25
4x - 7y = -16
Multiply the first equation by 4 and the second equation by -3:
12x + 16y = 100
-12x + 21y = 48
Add the equations:
37y = 148
Solve for the variable:
y=4
Plug y = 4 into one of the equations and solve for x:
3x + 4(4) = 25
3x + 16 = 25
3x = 9
x=3
Thus, the solution to the system of equations is (3, 4)
...

4(3) - 7(4) = - 16? Yes
...

61

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MATHEMATICS FOR ECONOMIST I

Since 0≠50, this system of equations has no solutions
...
The equations describe two parallel lines
...

Every ordered pair (x, y) which satisfies 6x + 14y = 16 (or -9x - 21y = -24) is a solution
to the system
...

SELF ASSESSMENT EXERCISE
Solve the following system of equations using the addition/substitution method:
(a) 3x + 4y = 12
(b) 2x + 9y = 32
...

4
...

System of linear equations can have zero, one or an infinite number of solutions,
depending on whether they are consistent or inconsistent, and whether they are
dependent or independent
...

A quicker way to solving systems of linear equation is by substitution through the
isolation of one variable in one question, and substituting the resulting expression
for that variable in the other equation
...


5
...
Sub-topics such as the graphing of equations which emphasize that all the
points of intersection are the points which satisfy the equations was reviewed
...

62

ECO 255

MATHEMATICS FOR ECONOMIST I

The problem with this approach is the isolation of variables which often involves dealing
with exhausting results
...

6
...
0

TUTOR-MARKED ASSIGNMENT
 Solve the following using the addition or subtraction method:
8x + 9y = 24
7x +8y =21
 Solve by graphing:
y - 12x = -4
y - 3 = -2(x + 6)
REFERENCES/FURTHER READINGS

Chiang A
...
K (2005)
...
4th edition-McGraw-hill
Ekanem O
...
Essential Mathematics for Economics and Business
...
0
Introduction
2
...
0
Main Content
3
...
2
Substitution Method
3
...
0
Conclusion
5
...
0
Tutor-Marked Assignment
7
...
0

INTRODUCTION
63

ECO 255

MATHEMATICS FOR ECONOMIST I

The purpose of this unit is to look at the solution of elementary simultaneous linear
equations
...
A
simultaneous equation is two (or more) equations which contain more than one letter
term
...

2
...
0
MAIN CONTENT
3
...

A simultaneous equation is two (or more) equations which contain more than one letter
term
...
You can then substitute this value into the original
equations to find the value of the other letter term
...
Before we do that, let’s just have a
look at a relatively straightforward single equation
...

The above expression is a linear equation
...
The only terms we
have got are terms in x, terms in y and some numbers
...

We can rearrange it so that we obtain y on its own on the left hand side
...

Now let’s take 3 away from each side
...

Suppose we choose a value for x, say x = 1, then y will be equal to:
y = 4×1−3 = 1
Suppose we choose a deferent value for x, say x = 2
...

y = 4×0−3 = −3
For every value of x we can generate a value of y
...

5x – y = 6
2x + y = 8
For the first equation, the solution becomes:
5x – y = 6
y = 5x – 6
...

For the second equation, the solution is:
2x + y = 8
y = -2x + 8
...

For this set of equations, there is but a single combination of values for x and y that will
satisfy both
...
Plotted on a graph, this condition
becomes obvious:
y
y=5x-6 = (4,2)
8

x
y=-2x+8 = (4,2)

-6

Figure1
...
Each equation, separately, has an infinite number of ordered pair (x,y)
solutions
...

Usually, though, graphing is not a very efficient way to determine the simultaneous
solution set for two or more equations
...

3
...
Perhaps the easiest to
comprehend is the substitution method
...
In this case, we take the definition of y, which
is 24 - x and substitute this for the y term found in the other equation:
2y = 20 – x
...
5x
Now, substitute y= 10 – 0
...
5x) = -4
Now that we have an equation with just a single variable (x), we can solve it using
"normal" algebraic techniques:
2x – 20 + x = -4
Combining the like terms gives:
3x – 20 = -4
Adding 20 to both sides:
3x = 16
Dividing both side by 3
x = 5
...

Now that x is known, we can plug this value into any of the original equations and obtain
a value for y
...
3) into the equation we
just generated to define y in terms of x, being that it is already in a form to solve for y:
x = 5
...
5x
y = 10 – 0
...
3)
y = 10 – 2
...
3
Example: Solve the pair of simultaneous equations:
2x + 5y = 27
(1)
2x + 2y = 12
(2)
Solution:
In both equations, notice we have the same number of x terms
...
5 – 2
...
5 – 2
...

Now, we impute y = 5 into our equation (1) to get the actual value for x or we can impute
it in our equation (3) to make things faster
...

Impute y = 5 into equation (1)
...

Imputing y = 5 in equation (3)
...
5 – 2
...
5 – 12
...

Both options gave us the same value for x
...

3
...
Elimination is done by adding or subtracting the equations
(if needed, multiply each equation by a constant)
...
One such method is the so-called
addition/subtraction method, whereby equations are added or subtracted from one another
for the purpose of canceling variable terms
...

One of the most-used rules of algebra is that you may perform any arithmetic operation
you wish to an equation so long as you do it equally to both sides
...

An option we have, then, is to add the corresponding sides of the equations together to
form a new equation
...
Example of additive simultaneous problem is giving below
...

We can solve the above equation completely:
6x = 8
x = 1
...
33 into equation (1)
4(1
...
32 + 2y = 14
2y = 14 – 5
...
68
y = 4
...
33, y = 4
...

4x – 4y = 12
(1)
3x + 4y = 8
(2)
Substituting equation (2) from (1) gives:
x=4
(3)
We impute x = 4 into equation (1) in order to find y
...

We impute y = 1 into equation 1 to confirm the value for x
...

We can see that the value for our x corresponds to the initial value of 4, thus confirming
that our result is in line
...

Example: 7x + 2y = 47
(1)
5x – 4y = 1
(2)
Notice that the value of y in equation (1) and (2) is not the same, thus making it
impossible to eliminate them through addition or subtraction
...

Based on the question above, we multiply equation (1) with 4 which belongs to y in
equation (2), and we multiply equation (2) with 2 which belongs to y in equation (1) to
get a uniform model
...

Now that we have a value for x, we can substitute this into equation (2) in order to find y
...

SELF ASSESSMENT EXERCISE
Solve this pair of simultaneous equations using the substitution method:
3x + 7b = 27
5x + 2y = 16
...
0
CONCLUSION
 Simultaneous equation refers to a condition where two or more unknown variables
are related to each other through an equal number of equations
...

 Substitution method involves the transformation of one of the equations such that
one variable is defined in terms of the other
...

 When the coefficient of y or x in each equations are the same, and the signs of y
and x are opposite, then adding or subtracting each side of the equation will
eliminate y or x
...
0
SUMMARY
This unit focused on simultaneous equation, which is a condition where two or more
unknown variables are related to each other through an equal number of equations
...
The substitution method defines one variable in terms of the
other, while the elimination method (addition/subtraction) involves adding or subtracting
the equations from one another in order to form a unified equation
...
0
TUTOR-MARKED ASSIGNMENT
Solve the following equations using the substitution and the elimination method:
69

ECO 255

(a)
(b)

MATHEMATICS FOR ECONOMIST I

4x +5y = 1
-4x-5y = -1
5y - 2z = 5
-4y + 5z = 37

7
...
M (2001)
...
C and Wainwright
...
Fundamental Methods of Mathematical
Economics
...
O
...
Essential Mathematics for Economics and Business
...
0
Introduction
2
...
0
Main Content
3
...
2
Factoring Quadratic Equation
3
...
4
Graphing Quadratic Function
4
...
0
Summary
6
...
0 References/Further Readings
1
...
Polynomials of degree two Quadratic equations are equations
of the form y = ax2 + bx + c
...

The first sub-unit introduces us to the concept of quadratic equation, while the second
sub-unitfocuses on factoring quadratic equation
...
Not all equations ax2 + bx + c = 0 can be easily factored
...
This is the quadratic formula, and it is the focus of sub- unit
three
...

2
...

 Solve quadratic equation problems using the factorization and quadratic formula
...

 State the steps involved in solving quadratic equation
...

3
...
1
INTRODUCTION TO QUADRATIC EQUATION
The name quadratic comes from "quad", which means square; this is because the
variables get squared (like x2)
...
A quadratic equation is any equation having the form ax2 + bx + c = 0; where x
represents an unknown, and a, b, and c are constants with a not equal to 0
...
The parameters a, b, and c are called the quadratic
coefficient, the linear coefficient and the constant or free term respectively
...
The
quadratic equation only contains powers of x that are non-negative integers, and therefore
it is a polynomial equation, and in particular it is a second degree polynomial equation
since the greatest power is two
...

71

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MATHEMATICS FOR ECONOMIST I

STEPS INVOLVED IN SOLVING QUADRATIC EQUATION
Many quadratic equations with one unknown variable may be solved by using factoring
techniques in conjunction with the Zero Factor Property as described below:
1
...

2
...

3
...

4
...

The resulting solutions are solutions of the original quadratic equation
...

Stage 1:
x2 + 6x + 9 = 0
Stage 2:
(x + 3)(x + 3)
Stage 3:
(x + 3)(x + 3) = 0
x + 3 = 0 or x + 3 = 0
Stage 4:
x = -3
...
2
FACTORISING QUADRATIC EQUATION
We can often factor a quadratic equation into the product of two binomials (expression of
the sum or difference of two terms)
...

The zero product property states that, if the product of two quantities is equal to 0, then at
least one of the quantities must be equal to zero
...
Consequently, the two solutions to the equation are x = -d and x = -e
...

x - 7 = 0 or x + 2 = 0
x = 7 or x = -2
...

Example 3: Solve for x: 2x2 – 16x + 24 = 0
2x2 – 16x + 24 = 0
2(x2 – 8x + 12) = 0
72

ECO 255

MATHEMATICS FOR ECONOMIST I

2(x – 2)(x – 6) = 0
x – 2 = 0 or x – 6 = 0
x = 2 or x = 6
...
However, sometimes, it gets harder to solve some
quadratic equations; in such situations, solving the quadratic equations using the
quadratic formula is recommended
...
3
SOLVING USING THE QUADRATIC FORMULA
Trinomials (algebraic expression consisting of three terms) are not always easy to factor
...
Thus, we need a different way to solve
quadratic equations
...
75 = 0
...

Here, a = 1, since it’s a number assigned to the first element of the question “ i
...
x 2”
...
75
...

x = -8 + 1, or x = -8 - 1
2
2
x = -7 or x = -9
2
2
x = -3
...
5
...


73

ECO 255

MATHEMATICS FOR ECONOMIST I

Since the square root of 400 is 20, we continue with the solution
...
7
...


Since the square root of 0 is 0, we continue with the solution
...

Example 4: Solve for x: 2x2 – 4x + 7 = 0
...


74

ECO 255

MATHEMATICS FOR ECONOMIST I

Since we cannot take the square root of a negative number, there are no solutions
...

This expression has a special name: the discriminant
...

 If the discriminant is zero, that is; if b2 - 4ac = 0, then the quadratic equation has
one solution
...

Example: How many solutions does the quadratic equation 2x2 + 5x + 2 have?
Where a = 2, b = 5, and c = 2
...

Thus, the quadratic equation has 2 solutions
...

3
...

1
...
Thus, the y-intercept is (0, c)
...
Thus, the x-intercept(s) can be found
by factoring or by using the quadratic formula
...
If b2 - 4ac> 0, there are 2
solutions to ax2 + bx + c = 0 and consequently 2 x-intercepts
...
If b2 - 4ac< 0, there are no
solutions to ax2 + bx + c = 0, and consequently no x-intercepts
...

2
...
We do
not know the vertex or the axis of symmetry simply by looking at the equation
...
We do
75

ECO 255

MATHEMATICS FOR ECONOMIST I

this by completing the square: adding and subtracting a constant to create a perfect square
trinomial (an equation consisting of three terms) within our equation
...
In order to "create" a perfect
square trinomial within our equation, we must find d
...
Then
square d and multiply by a, and add and subtract ad2 to the equation (we must add and
subtract in order to maintain the original equation)
...
Factor ax2 +2adx + ad2 into a(x + d)2, and simplify -ad2 +
c
...
Compute d = b/2a
...
Substitute this value for b
in the above equation to get ax2 + 2adx + c
...
Add and subtract ad2 to the equation
...

3
...
This produces and equation of the form y =
a(x + d)2 - ad2 + c
...
Simplify ad2 + c
...

5
...
It should satisfy the
equation
...

a = 1, b = 8 and c = -14
...
d = 8/2(1) = 4
...
ad2 = 16
...

3
...

4
...

5
...
→ -30 = -30
...

a = 4 and b = 16, while c = 0
...
d = 16/2(4) = 2
...
ad2 = 16
...

3
...

4
...

5
...
→ -16 = -16
...

a = 2
...
d = -28/2(2) = -7
...
ad2 = 98
...

3
...

4
...

5
...
→ 2 = 2
...

GRAPHING QUADRATIC EQUATION
Let us plot the quadratic equation diagram using the model: x2 + 3x – 4 = 0
...

x = -4 or x = 1
...
How would our solution look in
the quadratic formula? Using a = 1, b = 3, and c = -4
...

x = -3 + 5, or x = -3 - 5
2
2
x = 2 or x = -8
2
2
x = -4 or x = 1
...

Suppose we have ax2 + bx + c = y, and you are told to plug zero in for y
...
So solving ax2 + bx + c = 0 for x
means, among other things, that you are trying to find x-intercepts
...
Graphing,
we get the curve below:
y

x
-4

1

Figure1
...

77

ECO 255

MATHEMATICS FOR ECONOMIST I

As you can see, the x-intercepts match the solutions, crossing the x-axis at
x = –4 and x = 1
...

SELF ASSESSMENT EXERCISE
Solve and graph 2x2 – 4x – 3 = 0
...
0






CONCLUSION
The word "quad" which means "square" is the origin of the name quadratic
equations because the variables are squared
...

Quadratic equation is any equation having the form ax2 + bx + c = 0
...

The zero product property states that if the product of two quantities is equal to
zero (0), then at least one of the quantities must be equal to zero
...


5
...
Quadratic equation implies any equation having
the form ax2 +bx + c = 0; where x represents an unknown, and a, b and c are constants
with a ≠ 0
...

6
...

 Find the y-intercept of:
y = 2x2 + 4x -6
y = 6x2 - 12x - 18
...
0
REFERENCES/FURTHER READINGS
Carter
...
Foundation of Mathematical Economics, The MIT Press, Cambridge,
Massachusetts

78

ECO 255

MATHEMATICS FOR ECONOMIST I

Ekanem
...
T (2004)
...
Mareh: Benin
City

MODULE 3:
SET THEORY, LOGARITHMS &PARTIAL DERIVATIVES
Unit 1
Set Theory
Unit 2
Logarithms
Unit 3
Partial Derivative

UNIT 1
SET THEORY
1
...
0
Objectives
3
...
1
Introduction to Set theory
3
...
2
...
0
5
...
0
7
...
2
...
3
Venn Diagram
Conclusion
Summary
Tutor-Marked Assignment
References/Further Readings

3
...
In order to discuss this topic, subtopic such as Venn diagram which was introduced in 1880 will be reviewed
...
Also, the properties of set operations and
symbols used in set operations will be discussed extensively in order to facilitate easy
understanding
...
0
OBJECTIVES
At the end of this unit, you will be able to:
 Explain the concept of set theory
 Differentiate between intersect and union
...

 State the properties of set operations
 Understand the concept of Venn diagram and its application to set theory
...
0

MAIN CONTENT

3
...
For example we could say a Chess set , this set is made up of chess pieces and a
chess board used for playing the game of chess
...
All these are collection of objects grouped together as a
common singular unit
...
g
...

Set theory is a branch of mathematics which deals with the formal properties of sets as
units (without regard to the nature of their individual constituents) and the expression of
80

ECO 255

MATHEMATICS FOR ECONOMIST I

other branches of mathematics in terms of sets
...
Set theory begins with a
fundamental binary relation between an object o and a set A
...

A derived binary relation between two set is the subset relation, which is also called
“set inclusion” If all the members of set A are also members of set B, then A is a
subset of B, and we denote it as A⊆B
...
From this definition, we can deduce that a set is a subset of itself
...
If A is called a proper subset of B, then this can only happen if and only if A is a
subset of B, but B is not a subset of A
...

 Union: The union of sets A and B is denoted as A∪B
...
The union of {1,2,3} and {2,3,4} is the set
{1,2,3,4}
...
The intersection of {1,2,3} and
{2,3,4} is the set {2,3}
...
The set difference {1,2,3}\{2,3,4}
is {1}, while conversely, the set difference {2,3,4}\{1,2,3} is {4}
...
In
this case, if the choice of U is clear from the context, the notation Ac is sometimes
used instead of U\A, particularly if U is a “universal set” as the study of Venn
diagram
...
For instance, for the
sets {1,2,3} and (2,3,4}, the symmetric difference set is {1,4}
...

 Cartesian Product: The Cartesian product of A and Bis denoted by A × B, which is
the set whose members are all possible ordered pairs (a,b) where a is a member of
A and b is a member of B
...

 Power Set: Power set of a set A is the set whose members are all possible subsets
of A
...

SELF ASSESSMENT EXERCISE
What is the intersection between A = {10,15,20,25) and B = {20,25,30,40}?
3
...
2
...


Table1
...
2
...
PROPERTIES OF SET OPERATIONS
Set operations have properties like those of the arithmetic operations
...
The same can also be said about multiplication and intersection of sets
...

Examples of properties in arithmetic are commutative (a+b = b+a or a×b = b×a) and
distributive {3× (2+4) = (3×2)+(3×4)} properties
...
Multiplication is
distributive over addition because “a × (b + c) = a × b + a × c” for all numbers
...
For example, in addition, we have “0 + a =
a”, and in multiplication, “1 × a = a”
...

Parentheses or brackets are used to clarify an expression involving set operations in the
same way that they are used to clarify arithmetic or algebraic expressions
...
By putting in parentheses, you can distinguish between (4 + 8) ×
9 = 108 and 4 + (8 × 9) = 76
...

By convention, complementation is done before the other operations
...
However, it is necessary to use parentheses or brackets in
the expression (A∪B)C to mean that the union must be performed before the
complementation
...
The events A, B, and C
are all taken from the same universal set U
...
The Identity Laws: In addition (+), 0 acts as an identity since “0 + a = a”, and in
multiplication (×), 1 acts as identity since “1 × a = a”
...

AUØ=A
U∩ Ø = A
2
...
For addition, – (– a) = a,
and for division, 1 / (1 / a) = a
...

3
...
In addition, 0 is idempotent since 0 + 0 = 0 and in
multiplication 1 is idempotent since 1 × 1 = 1
...

AUA=A
A∩A=A
4
...

AUB=BUA
A∩B=B∩A
5
...
In arithmetic,
addition and multiplication are associative since “(a + b) + c” is the same as “a +
(b + c)” and “(a × b) × c” is the same as “a × (b × c)”
...
Distributive Laws: In arithmetic, multiplication is distributive over addition, “a ×
(b + c) = a × b + a × c” for all numbers, but addition is not distributive over
multiplication since, for example, 3 + (2 × 5) is not the same as (3 + 2) × (3 + 5)
...

A U(B∩C) = (A UB)∩ (A U C)
A ∩(B UC) = (A ∩B)U (A ∩ C)
7
...

AUAC = U
A∩AC = Ø
UC = Ø
ØC = U
...
DeMorgan’s Laws: These laws show how complementation interacts with the
operations of union and intersection
...


3
...
Put differently, a Venn
diagram or set diagram shows all possible logical relations between a finite collection of
sets
...

A rectangle is used to represent the universe of all the elements we are looking at, and
circles are used to represent sets
...
Among your friends, you have the first five of them in your football
team; then you have another five counting from the back (from David) as members of
your handball team
...
The twelve (12) friends represent the
universal set

A

∩

B

Figure1
...

The combine area of set A and B is called the union (U) of A and B
...
Then the area in both A and
B, where the two sets overlap (∩) is called the intersection of A and B, which is denoted
by A∩B
...
e
...

Venn diagrams normally comprise overlapping circles
...

Example: Out of 40 students, 14 are taking English, and 29 are taking Mathematics
...
If 5 students are in both classes, how many students are in neither class?
b
...
What is the probability that a randomly chosen student from this group is taking
only the Mathematics class?
Solution: There are two classifications in this universal set of 40 students: English
students and Mathematics students
...

English = 14

Mathematics = 29

Since 5 students are taking both classes, we put 5 in the overlap, and the diagram
becomes:
English = 14

Mathematics = 29

5

Now, we have accounted for 5 of the 14 English students, leaving 9 students taking
English but not Mathematics
...

English = 14

Mathematics = 29

9

5

Now, we have also accounted for 5 of the 29 Mathematics students, leaving 24 students
taking Mathematics but not English
...


86

ECO 255

MATHEMATICS FOR ECONOMIST I

English = 14

Mathematics = 29

9

5

24

This tells us that the total of 9 + 5 + 24 = 38 students are in either English or Mathematics
or both
...

From the above Venn diagram, we can deduce the following:
 2 students are taking neither of the subjects
...

 There is a 24/40 (60%) probability that a randomly-chosen students in this group
is taking Mathematics but not English
...


4
...

A derived binary relation between two set is the subset relation, which is also
called set inclusion
...

Venn diagram shows the relationship between sets in picture form
...

The overlapping of set A and B is called the intersection of A and B
...


5
...
Set theory is a branch of mathematics which deals
with the formal properties of sets as units (without regard to the nature of their individual
constituents) and the expression of other branches of mathematics in terms of sets
...
1 introduced us to set theory, the next sub-unit, focused on the symbols and
properties of set operations, explaining the different properties such as the identity law,
the involution laws, idempotent laws, e
...
c
...

87

ECO 255

6
...
0

MATHEMATICS FOR ECONOMIST I

TUTOR-MARKED ASSIGNMENT
 Out of 100 students in a class, 65 are offering economics as a course while 55 are
offering mathematics
a
...

How many are offering either of the courses?
c
...
J and Howard F
...
Introduction to the Theory of Sets, (Dover Books on
Mathematics), Dover Publications, Inc
...
T (2004)
...
Mareh:Benin
City
Enderton
...
B (1997)
...
Academic Press: San Diego,
California
...
A and Vereshchagin
...
K (2002)
...
17), American Mathematical Society (July 9, 2002), United State of America

UNIT 2
LOGARITHMS
1
...
0
Objectives
3
...
1
Introduction to Logarithm
3
...
3
Properties of Logarithms
3
...
0
Conclusion
5
...
0
Tutor-Marked Assignment
88

ECO 255

7
...
0

MATHEMATICS FOR ECONOMIST I

INTRODUCTION

In its simplest form, a logarithm answers the question of how many of one number is to
be multiplied to get another number; i
...
, how many 2’s do we multiply together to get
8?, the answer to this question is 2 x 2 x 2 = 8, so we needed to multiply 2 three (3) times
in order to get 8; so in this case, the logarithm is 3, and we write it as log28 = 3
...
e
...

In order to do justice to this unit, the first sub-unit introduces us broadly to the meaning
of the logarithmic function
...
The second sub-unit presents the two special logarithmic functions (the
common logarithmic function and the natural logarithmic function)
...
Sub-unit three deals with the
properties of logarithms
...
They are also useful in
simplifying and solving equations containing logarithms or exponents
...

2
...

3
...
1
INTRODUCTION TO LOGARITHM
Logarithms are the opposite of exponents, just as subtraction is the opposite of addition
and division is the opposite if multiplication
...
In general, if a = x, then logax = y
...

The value of “a” is the base of the logarithms, just as “a” is the base in the exponential
expression “ax”
...
Whatever is inside the logarithm is called “the argument” of the log
...

For example, log264 = 6 because 26 = 2 x 2 x 2 x 2 x 2 x 2 = 64
...

To evaluate a logarithmic function, we have to determine what exponent the base must be
taken to in order to yield the number x
...
If this is the case, the use of logarithm table or a calculator is recommended
...
Since the cube root of 3 = 27, this solves the problem
...

y = log51/625 = -4
...

SIMPLIFYING THE LOGS
The log is normally equal to some number, which we can call y like some of the example
cited above
...

For example, given log216, in order to simplify it, we can re-write the model as:
log216 = y
...

4
2 = 2 x 2 x 2 x 2 = 16
...

Simplify log5125
...
This means that the given
log5125 is equal to the power y that, when put on 5, turns 5 into 125
...

Then, log5125 = 3
...

y
y
The Relationship says that, since log77 = y, then 7 = 7
...

That is: log77 = 1
SELF ASSESSMENT EXERCISE
Simplify log32187
...
2
TWO SPECIAL LOGARITHM FUNCTIONS
In both exponential functions and logarithms, any number can be the base
...
On our scientific calculators, special keys are included denoting the common
and the natural logs (the “log” and “ln” keys)
...
Recall that our number system is
base 10; there are ten digits from 0-9; and place value is determined by groups of ten
...
The common log
function is often written as f(x) = logx
...

Examples: Simplify log1000
...

Natural logarithms are different from common logarithms
...
Although this looks like a variable, it represents a fixed irrational
number approximately equal to 2
...
(It continues without a repeating
pattern in its digits
...

Example: Simplify ln100
...
6052, rounded to four decimal places
...
04 is equivalent to 5
...
2189 respectively
...

2) Simplify log5062500000
...
3
PROPERTIES OF LOGARITHMS
Logarithms have the following properties:
1
...

Property B: logaa = 1
...
Since a and logax are inverse:
Property C: logaax= x
...

p q
p+q
p q
p-q
3
...
Since loga(M ) = loga(M+M+M…M) = logaM + logaM + logaM + …+ logaM =
nlogaM
...
Logarithms have an additional property called property H, and a property H 1,
which is a specific case of property H
...

Property H1: logaM = logM/loga
...
Property H 1 is
especially useful when evaluating logarithm with a calculator: since most calculators only
evaluate logarithms with base 10, we can evaluate logaM by evaluating logM/loga
...

SELF ASSESSMENT EXERCISE
Using the property H1, solve for log72401 + log7343 – log749
...
4
SOLVING EXPONENTIAL AND LOGARITHM FUNCTION
To solve an equation containing a variable exponent, we need to isolate the exponential
quantity
...

For example: Solve for x:
x
5 = 20
x
log55 = log520
x = log520
...
8614
...

Example 2: Solve for x:
2x
5(5 ) = 80
Dividing both sides by 5 gives:
2x
5 = 16
2x
Log55 = log516
2x = log516
2x = log16/log5
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2x = 1
...
8614
...
We
then convert it to exponential form and evaluate
...

Consider the following examples
...

Using the logarithm property E: loga(pq) = logap + logaq
We can go ahead and solve this problem:
log33x + log3(x–2) = 2
log3(3x * (x-2)) = log332
3x * (x- 2)=32
3x2 -6x= 9
3x2 - 6x – 9 = 0
3(x2 – 2x – 3) = 0
3(x – 3)(x+ 1) = 0
x = 3 or x = -1
...

To check this answer, let us plug x = 3 into the equation:
log33(3) + log3(3 - 2) = 2
...

Log39/1 = log39 = log33 2 = 2
...

log(2x+1)(2x + 4)2 – log(2x+1)4 = 2
...

(2x+1)2 = (2x + 4)2/4
(2x+1)2 = 4x2 + 16x + 16)/4
4x2 + 4x + 1 = x2 + 4x + 4
3x2 – 3 = 0
...

x = 1 or x = -1
...
We
can confirm this by plugging 1 into the equation:
x = 1in :2log(2(1)+1)2(1) + 4 – log(2(1)+1)4 = 2
...

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2log36 – log34 = 2
...

log336 – log34 = 2
...

Thus 1 is the solution to the above equation
...
0
CONCLUSION
Logarithm answers the question of how many of one variable is to be multiplied together
to get another number
...
e
...

Simplifying the logs makes it easy for us to solve
...

The common logarithm function is any logarithm with base 10, while the base of a
natural logarithm function is denoted by “e”, which is mostly written as“lnx”
...

In order to solve an equation containing a logarithm, we need to consider the properties
of logarithm which compresses the expressions into one
...
0
SUMMARY
This unit focused on the logarithm function
...

It is also the inverse of an exponent
...


6
...
0
REFERENCES/FURTHER READINGS
Brian
...
I (2012)
...
Solid State Press: California
...
C and Wainwright
...
Fundamental Methods of Mathematical
Economics, 4th edition-McGraw-hill
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Ekanem O
...
Essential Mathematics for Economics and Business
...
0
Introduction
2
...
0
Main Content
3
...
2
Higher Order Partial Derivatives
3
...
4
The Product Rule of Partial Differentiation
4
...
0
Summary
95

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6
...
0

MATHEMATICS FOR ECONOMIST I

Tutor-Marked Assignment
References/Further Readings

1
...
The term partial derivation refers to the solution or the outcome of
differentiation
...
For example, z = 2xy; the partial
derivative of z with respect to x is 2y, while the partial derivative for z with
respect to y is 2x
...
1) of this unit, handles the introductory part
of the topic, while the second section (3
...
3 and 3
...

2
...

 Understand the difference between differentiation and derivation
...

 Use both the chain and the product rule to solve differentiation problem
...
0

MAIN CONTENT

3
...

For instance consider the differentiation of the single variable model y = 5x3
...

15x2 is the derivative obtained by differentiation
...

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MATHEMATICS FOR ECONOMIST I

The introductory analysis is the partial differentiation of a single variable, and it is
worthwhile moving to the analysis of more than one variable where there is more than
one variable with all other variables constrained to stay constant when a variable is
differentiated
...

For example, given the polynomial in variables x and y, that is: f(x,y) = ax2 + by2
The partial derivative with respect to x is written as:
ðf(x,y) /ðx = 2ax
...

The problem with functions of more than one variable is that there is more than one
variable
...
For instance, one variable could be changing faster than the other variables in
the function
...

Consider f(x,y) = 2x2y3 and lets determine the rate at which the function is changing at a
point, (a,b), if we hold y fixed and allow x to vary and if we hold x fixed and allow y to
vary
...

We can get the partial derivative of x:
ð(x,y)/ðx = 4xy3
...

Now, let’s do it the other way
...
We can do
this in a similar way
...

Note that these two partial derivatives are sometimes called the first order partial
derivatives
...

We will be looking at higher order derivatives later in this unit
...

Example 1:f(x,y) = x4 + 6√y - 10
Let’s first take the derivative with respect to x and remember that as we do so, the y will
be treated as constants
...

Notice that the second and the third term differentiate to zero in this case
...
It’s a constant and constants always
differentiate to zero
...

Remember that since we are differentiating with respect to x here, we are going to treat
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all our y’s as constants
...

Now, let’s take the derivative with respect to y
...
Here is the partial derivative with respect to y
...

Example 2: w = 2x2y – 20y2z4 + 40x – 7tan(4y)
With this function we’ve got three first order derivatives to compute
...
Since we are differentiating with respect to x we
will treat all y’s and all z’s as constants
...

This first term contains both x’s and y’s and so when we differentiate with respect to x the
y will be thought of as a multiplicative constant and so the first term will be differentiated
just as the third term will be differentiated
...

Applying the same principles, we can find the partial derivative with respect to y:
ðw/ðy = x2 – 40yz4 – 28sec2(4y)
...

Thus, we are have successfully differentiated the three variables in the equation
...

3
...
In this sub-unit, we will be
focusing on more than once variable higher order analysis
...
This means that for the case of a function of two variables there will be a total
of four possible second order derivatives
...
Note as well that
the order that we take the derivatives is given by the notation in front of each equation
...
g
...

In other words, in this case, we will differentiate first with respect to x and then with
respect to y
...
g
...
In these cases
we differentiate moving along the denominator from right to left
...

Example 1: Find all the second order derivatives for f(x,y) = cos(2x) – x2e5y + 3y2
First, we need to find the first order derivatives:
fx(x,y) = ð(x,y)/dx = -2sin(2x) – 2xe5y
...

Now, we are done with the first order derivative, we need to find the second order
derivatives:
fxx = -4cos(2x) – 2e5y
fxy = -10xe5y
fyx = -10xe5y
fyy = -25x2e5y + 6
...

We find the first order derivatives in order to find the higher order derivatives
...

fy(x,y) = ð(y,x)/dy = x2 + 5xcos(y)
...

Notice that in the two examples, fxy(x,y) = fyx(x,y)
...
Thus it does not matter if we take the partial derivative with respect to x first or with
respect to y first
...
3

THE CHAIN RULE OF PARTIAL DIFFERENTIATION
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The chain rule is a rule for differentiating compositions of functions
...
In order for us to
use the chain rule extensively, we need to first review the notation for the chain rule for
functions of one variable
...

However, we rarely use this formal approach when applying the chain rule to specific
problems
...
For example, it is sometimes easier
to think of the functions f and g as “layers” of a problem
...
Thus, the chain rule tells us to first differentiate the
t
outer layer, leaving the inner layer unchanged (the term f (g(x))), then differentiate the
t
inner layer (the term g (x))
...

Example 1: Differentiate y = (3x + 1)2
...
Differentiate the square first,
leaving (3x+1) unchanged
...
) Thus:
Ɖ(3x + 1)2
= 2(3x + 1)2-1Ɖ(3x + 1)
By differentiating further, we get:
= 2(3x + 1)*(3)
= 6(3x + 1)
...

The outer layer is the 10th power, and the inner layer is (2 – 4x + 5x4)
...
Then differentiate (2 – 4x + 5x4)
...

Example 3: Differentiate y = sin(5x)
The outer layer is the “sin function” and the inner layer is (5x)
...
Then differentiate (5x)
...

SELF ASSESSMENT EXERCISE
Using the chain rule, differentiate y = cos2(x3)
...
4
THE PRODUCT RULE OF PARTIAL DIFFERENTIATION
The difference between the chain and the product rule is that the chain rule deals with a
function of a function, that is, ð/ðx{f(g(x))}, while the product rule deals with two
separate functions multiplied together; that is, ð/ðx{f(x)*g(x)}
...
The rule follows from the limit definition of derivative and is
given by:
t
t
Ɖ{f(x)g(x)} = f(x)g (x) + f (x)g(x)
...

Example 1: Differentiate y = (2x3 + 5x – 1)*(4x + 2)
yt= (2x3 + 5x – 1)* Ɖ(4x + 2) + Ɖ(2x3 + 5x – 1)*(4x + 2)
= (2x3 + 5x – 1)* (4) + (6x2 + 5)*(4x + 2)
= 8x3 + 20x – 4 + 24x3 + 20x + 12x2 + 10
...

Example 2: Differentiate x-4(5 + 7x-3)
...

yt= Ɖ10x2 + sinx*Ɖ(cosx) + Ɖ(sinx)*cosx
= 20x +sinx*(-sinx) + (cosx)*cosx
= 20x + cos2x – sin2x
...

Thus, our answer changes to 20x + cos2x
...


4
...

 Partial derivative is the situation where there is more than one variable with all the
other variables constrained to stay constant with respect to the differentiated one
...


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MATHEMATICS FOR ECONOMIST I

 Mixed derivative involves taking the derivative of a variable while holding several
others constant
...

 The chain rule is a rule for differentiating composition of functions, while the
product rule is a rule for differentiating problems where one function is multiplied
by another
...

5
...
In order to
analyze the topic lucidly, sub-topics such as the higher order partial derivatives was
reviewed, with focus on the second order partial differentiation
...
Both rules like we found out,are
rules used for differentiating compositions of functions
...

6
...
0
REFERENCES/FURTHER READINGS
Carter
...
Foundation of Mathematical Economics, The MIT Press:Cambridge,
Massachusetts
Chiang A
...
K (2005)
...
4th edition-McGraw-hill
Ekanem, O
...
Essential Mathematics for Economics and Business
...
P
...
Partial Derivatives (Library of Mathematics)
...


MODULE 4 INTEGRAL CALCULUS, OPTIMIZATION AND LINEAR
PROGRAMMING (LP)
Unit 1
Integral Calculus
Unit 2
Optimization
Unit 3
Linear Programming (LP)

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UNIT 1:
INTEGRAL CALCULUS
1
...
0
Objectives
3
...
1
Introduction to Integral Calculus
3
...
3
Definite and Indefinite Integrals
3
...

4
...
0
Summary
6
...
0
References/Further Readings

1
...
Basically, integral is
the inverse of derivative
...
Integral is the function of which a given function is the derivative
...
To enable students understand this concept better, the main
content of this unit has been divided into four sections, with the first section focusing on
the introductory part of the topic, while the second and third sections will look at the
rules guarding integration and the definite and indefinite integrals
...

2
...

 Master the rules of integration
...

 Apply definite integrals to economic problems
...
0

MAIN CONTENT

3
...
The two main type of calculus are differential calculus and
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MATHEMATICS FOR ECONOMIST I

integral calculus
...

Broadly, an integral is a function of which a given function is the derivative which yield
that initial function when differentiated
...

Finding an integral is the reverse of finding a derivative
...
The
!
simple derivative of a function f with respect to a variable x is denoted either by f (x) or
df/dx
...
Definite integral is an integral
expressed as the difference between the values of the integral at specified upper and
lower limits of the independent variable
...

An indefinite integral is an integral expressed without limits, and so containing an
arbitrary constant
...

3
...

Using the above rule, the answer is 4x + c
...


Example 2: Evaluate ʃ12x3dx
= 12ʃx3dx = 12

= 12x4/4 + c =3x4+ c
...

Let us represent our u as x – 7, and u5 as (x – 7)5
...

= u6/6 + c
...

Integration by Substitution
This method of integration is useful when the function f(x) is difficult to solve or when
simpler methods have not been sufficient
...

!
This formula takes the form ʃf(g(x))g (x)dx
...

!
From the above equation, our f = cos, and our g = x2, while its derivative g (x) = 2x
...

So, let us complete our example
...

Now, we integrate:
ʃcos(u)du = sin(u) + c
...

Rule 4: Logarithm Rule
Given that y = lnx, then dy/dx = 1/x
...

In general,

ʃ1/xdz = lnz + c
...

We can still go further:
Since u = 2x + 3,
5lnu + c = 5ln(2x + 3) + c
...

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MATHEMATICS FOR ECONOMIST I

And ʃzdx = ʃaxdx = ax + c
...

du/dx = 5
...

dx = du/5

...

A simple integration by parts starts with:
d(uv) = udv + vdu, and integrates both sides:
ʃd(uv) = uv = ʃudv + ʃvdu
...

Example 6: Consider the integral ʃxcosxdx and let:
u = x, and dv = cosx dx, du = dx, and v = sinx;
Integration by parts gives:
ʃxcosxdx = xsinx - ʃsinxdx
= xsinx + cosx + c
...
(????)
Example 7: Integrate ʃx5cos(x4)dx
Note that:
ʃx5cos(x4)dx = ʃx4xcos(x4)dx
Let: u = x4, dv = xcos(x4), du = 4x3dx and v = (1/4) sin(x4)
...

= (1/4)x4 sin(x4) - ʃxsin(x4)dx
= (1/4)x4 sin(x4) – (1/4)(-cos(x4)) + c
= (1/4)x4 sin(x4) + (1/4)cos(x4) + c
...

ʃx2/3 dx, and Evaluate ʃ(x – 4)3dx
...
3
DEFINITE AND INDEFINITE INTEGRALS
In the last unit of module three, we talked about function of x (f(x))
...
Now, the question is “what function are we
going to differentiate to get the function f(x)
...

Definite integral is an integral of the form
with upper and lower limits
...
In
the formula above, a, b and x are complex numbers and the path of integration from a tob
is referred to as the contour
...
We will be applying the rule here
...

5
-4
Example 1: Evaluate ʃx + x dx
...

5
-4
6
-3
The integration of ʃx + x dx = (1/6)x – (1/3)x + c
...


ʃ4x10/x3 - 2x4/x3 + 15x2/x3dx
ʃ4x7 – 2x + 15/x)dx
Now, we will integrate the equation
...

Thus, our solution becomes:
(1/2)x8 – x2 + 15ln|x| + c
...

-5
20(1/4)x4 – 35(1/5)x + 7x + c
...

INDEFINITE INTEGRALS
An indefinite integral is an integral expressed without limits, and it also contains an
arbitrary constant
...
The first fundamental theorem of calculus allows definite
integrals to be computed in terms of indefinite integrals
...

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Since the derivative of a constant is zero, any constant may be added to an antiderivative
and will still correspond to the same integral
...

Example 1: What function did we differentiate to get the following?
F!(x) = 2x2 + 4x – 10?
To solve this problem, we follow these steps
...
This means that when we differentiate a
function, the answer became 2x2, so we need to find what we differentiated to get 2x2
...
If we assume the figure in front of the value to be 1, we would have x2
which is different from 2x2; therefore the figure is 2/3x3
...

For the second term, we have 4x after differentiating 4/2x2
...

Putting all this together gives the following equation: F(x) = 2/3x3 + 4/2x2 – 10x + c
...

If F(x) is any antiderivative of f(x), then the most general antiderivative of f(x) is called
an indefinite integral
...

This question is asking for the most general antiderivative; all we need to do is to undo
the differentiation
...

To solve this problem, we need to increase the exponent by one after differentiation
The indefinite integral forʃ2x2 + 4x – 10dx = (2/3)x3 + (4/2)x2 – 10x + c
...
With integrals, think of the integral
sign as an “open parenthesis” and the dx as a “close parenthesis”
...
Consider the following variations of the above
example
...

ʃ2x2 + 4x dx – 10 = 2/3x3 + 4/2x2 + c – 10
ʃ2x2 dx + 4x – 10 = 2/3x3 + c + 4x– 10
...
Each of the above
integrals end in a different place and so we get different answers because we integrate a
different number of terms each time
...
Likewise, in the third integral the “4x–
10” is outside the integral and so is left alone
...

Example 3: Integrate ʃ40x3 + 12x2 – 9x + 14dx
...

10x4 + 4x3 – 4
...

To confirm your answer, you just simply differentiate it, and it should give you the
original question
...
5x2 + 14x + c, and see if it will give us back
our integrand
...
5x2 + 14x + c as U
...

We can see that our answer gave back the integrand, and thus we can be satisfied
...
5x-4 –5x + 12dx
...

3
...

Consumer surplus is the deference between the price consumers are willing to pay for a
good or service and the actual price they paid
...
Producer’s
surplus on the other hand is the difference between the amount that a producer of a good
receives and the minimum amount that he or she would be willing to accept for the good
...
Consumers’ surplus and producers’ surplus are calculated using the
supply and demand curves
...

Let us consider some examples
...

First, we need to find the equilibrium price
...

Using Q = 5, and plugging it into either the supply or the demand function we find P =
25
...

Based on the above findings, the consumer’s surplus is calculated as:

Consumer Surplus (CS) = 12
...

The producer’s surplus is:

Producer Surplus (PS) = 25
...

4
...

Definite integral is an integral of the form
with upper and lower limits
Integration by substitution method is suitable or useful when the function f(x) is
hard or when simpler methods have not been sufficient
...

 An integral of the form
without upper or lower limit is called an
antiderivative
...

 Producer’s surplus on the other hand is the difference between the amount that a
producer of a good receives and the minimum amount that he or she would be
willing to accept for the good
...
0
SUMMARY
This unit focused on integral calculus which is the derivative that yields the function
when differentiated
...
1 exposed us to the concept of integration which is the
inverse of differentiation
...
2 reviewed the rules of integration which are the
constant rule, the power rule, the difference rule, integration by parts and substitution, the
logarithm rule, e
...
c
...
3 focused on definite and indefinite integral, and we
learnt that definite integral is an integral expressed as the difference between the values
of the integral at specified upper and lower limits of the independent variable, while
indefinite integral is an integral expressed without limits and also containing an arbitrary
constant
...

6
...

 Using the indefinite integral method, Integrate ʃ15x6 + 22x2 – 9x5 + 5dx
...
0
REFERENCES/FURTHER READINGS
Chiang A
...
(1967), Fundamental Methods of Mathematical Economics, Third Edition,
McGraw-Hill Inc
Dowling, E
...
(2001)
...
Theory and Problems of Introduction to
Mathematical Economics
...
Third Edition
...
P and Nicholas R
...
University of Toronto Press
...
(2003)
...
Second
Edition
UNIT 2
OPTIMIZATION
1
...
0
Objectives
3
...
1
Introduction to Optimization
3
...

3
...

4
...
0
6
...
0

MATHEMATICS FOR ECONOMIST I

Summary
Tutor-Marked Assignment
References/Further Readings

1
...
The
sub-unit one will introduce broadly the concept of optimization, while the sub-unit two
will consider the optimization problems
...

2
...

3
...
1
INTRODUCTION
Decision-makers (e
...
consumers, firms, governments) in standard economic theory are
assumed to be "rational"
...
We usually make
assumptions that guarantee that a decision-makers preference ordering is represented by a
payoff function (sometimes called utility function), so that we can present the decisionmakers problem as one of choosing an action, among those feasible, that maximizes the
value of this function or minimizes the cost
...

If the decision-maker is a classical consumer, for example, then a is a consumption
bundle, u is the consumer's utility function, and S is the set of bundles of goods the
consumer can afford
...

Even outside the classical theory, the actions chosen by decision-makers are often
modeled as solutions of maximization problems
...

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In the case of minimization, we can assume, for example, that firms choose input bundles
to minimize the cost of producing any given output; an analysis of the problem of
minimizing the cost of achieving a certain payoff greatly facilitates the study of a payoffmaximizing consumer
...

3
...

First, we want to look at the optimization of a single variable without constraint
...

In order to solve this problem, we differentiate the total utility equation, as the
differentiation of total utility gives the marginal utility
...

Since the marginal utility defines the slope of the total utility, and the slope of a function
is zero at its maximum or minimum point, we set MU = 0
...

MU = dTU/dx = 40x3 + 289x = 0
...
Here, we consider not only x1, but x2…xn
...
e
...

In order for us to find the marginal utility of a function similar to the one above, we need
to differentiate partially and hold other x constant
...

TU = 10x1 + 1
...

For us to solve this problem, we need to solve for the marginal utility of each x while
holding others constant
...
5x22 + 4x1x2 = 0
...

Now, let us determine the utility maximizing combination subject to income constraint
...

Where P1 represents the price for good x1, and Y represents the income
...

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Given Z = f(x1, x2, x3… xn) + λ(1 – p1x1 – p2x2 - p3x3… - pnxn), we proceed to differentiate
the function partially
...

We have successfully solved equation 1 to 5 simultaneously in order to determine the
consumption level of the commodities that would maximize the total utility (TU)
function
...

λ = MUx1/p1
Note, ðTU/ðx1 = MUx1, while P1 is the price for x1 good
...

For equation (2),
ðTU/ðx2 - λp2 = 0
ðTU/ðx2 = λp2
λ = ðTU/ðx2 /p2
...

For equation (3),

ðTU/ðx3 – λp3 = 0
ðTU/ðx3 = λp3
λ = ðTU/ðx3 /p3
λ = MUx3/p3
...


...

For equation (4)ðTU/ðxn – λpn = 0
ðTU/ðxn = λpn
λ = ðTU/ðxn/pn
λ = MUxn/pn
...

In summary, λ = ðTU/ðx1/p1 = ðTU/ðx2 /p2 = ðTU/ðx3 /p3 = … = ðTU/ðxn/pn
...

The above solution is known to be the necessary condition for consumer utility
maximization
...

Example: Maximize the consumer utility:
TU = 12x1 + 18x2 – 0
...
5x22
Subject to the income constraint (Y) = 60
...

Where px1 is the price of commodity x1, and px2 is the price for commodity x2
...

TU = 12x1 + 18x2 – 0
...
5x22+ λ(60 – 2x1 – 5x2)
TU = 12x1 + 18x2 – 0
...
5x22 + λ60 - λ2x1- λ5x2
ðTU/ðx1 = 12 – x1 - 2λ = 0
12 – x1 = 2λ
λ = 6 – 0
...
6 – 0
...


From equation (1) and (2), let us equate λ = λ at equilibrium; thus, our equation becomes:
6 – 0
...
6 – 0
...
6 + 0
...
5x1
2
...
2x2 = 0
...
8 + 0
...

60 – 2x1 – 5x2 = 0
60 – 2(4
...
4x2) - 5x2 = 0
60 – 9
...
8x2 - 5x2 = 0
50
...
8x2 = 0
50
...
8x2
...
7
...

From equation (4):
x1 = 4
...
4x2
x1 = 4
...
4(8
...
8 +3
...
8 +3
...
3
...
3 quantity of good x1 and 8
...

SELF ASSESSMENT EXERCISE
Maximize U = 10x12 + 15x24 – 21x12x2
Subject to: income = 100, px1 = 4, px2 = 3
...
4
SOLVING OPTIMIZATION USING MATRIX
Matrix can also be used to solve optimization problem
...
5x12 – 0
...

Solution:
For us to solve this equation, we will also introduce the Lagrange multiplier
...
5x12 – 0
...
5x12 – 0
...

ðTU/ðx1 = 12 – x1 - 2λ = 0
x1 + 2λ = 12
(1)
ðTU/ðx2 = 18 – x2 - 5λ = 0
x2 + 5λ = 18

(2)

ðTU/ðλ = 60 – 2x1 – 5x2 = 0
2x1 + 5x2 = 60

(3)

We can arrange the above equations into the matrix box using the Crammer’s rule
...

Take note that the right hand side values of the equation are depicted in the right hand
side of the diagram as well
...

Now, we can calculate the determinants of the matrix A to get |A| = -29, How did we get
this? follow the steps below
...
In order to evaluate the determinants of A, we first start with making up a checkboard array of + and – signs
...
We will start our analysis from the upper left with a sign +, and alternate signs
going in both directions
...
Choose any row or any column
...
2, 5 and 0
...
We choose 2 first, and we use it against (0:2) and (1:5)
...


x

0

2

1

2

5

0

2

5

2

x

1

0

1

x

0

5
0
0
1
,
,
5
...

6
...
If the entry
comes from a positive position in the checkerboard, add the product
...

2

x

5

2

x

1

0

1
5
5
0
7
...
The above gives the following: 2(0 – 2) – 5(0 – 5) + 0(1 – 0)
9
...
Thus |A| = -29
...

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MATHEMATICS FOR ECONOMIST I

Now that we have the bottom in the computation for all three unknown, let us expand the
top determinants in each case by replacing the first column by the resource column: i
...
,
12
18

=

2

1

5

60

x1

0

5

0

x1 is calculated by dividing determinants of x1{|x1|} by the equation determinant |A|
...

Now, let us choose the upper row of (12, 0 and 2)
...

x

1

5

5

12

0

0

18

x

-

5

60

0

2

+

x

18
60

12(0 – 25) – 0(0 – 300) + 2(90 – 60)
|x1| = -300 + 60 = -240
...

x1 = -240 ÷ -29
x1 = 8
...

This same approach was applied to get ourx2 and λ
...

Thus, x2 = -252 ÷ -29
x2 = 8
...

For λ, we have:
1

=

12

0

1

18

2

λ

0

5

60

Where |λ| = -54
...

λ = -54 ÷ -29
λ = 1
...

SELF ASSESSMENT EXERCISE
118

1
5

ECO 255

MATHEMATICS FOR ECONOMIST I

Using the Crammer’s rule and Matrix, Maximize U = 5x1 + 12x24 – 11x132x22
Subject to: income = 20, px1 = 3, px2 = 5
...
0







CONCLUSION
According to the cardinal theory, utility can be quantified in terms of the money a
consumer is willing to pay for it, thus MUx=Px
...
This
means that partial differentiation of the total utility function with respect to x1, x2,
x3, …, xn gives us the marginal utility for that variable
...

If we choose any row or column, we will get the same determinants of the matrix
|A|
...
0
SUMMARY
This unit focused on partial optimization which is the selection of a best element from
some set of available alternative
...
In order to analyze the topic, sub-units such as the
Lagrangian method for solving optimization and the application of matrix to solving
optimization problem were reviewed
...


6
...

 Solve the above question using the matrix approach
...
0
REFERENCES/FURTHER READINGS
Hands, D
...
Introductory Mathematical Economics, Second Edition, Oxford
Univ
...

Lasonde
...
Approximation, Optimization and Mathematical Economics
...
Kg (2001)

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MATHEMATICS FOR ECONOMIST I

UNIT 3
LINEAR PROGRAMMING (LP)
1
...
0
Objectives
3
...
1
Introduction to Linear Programming
3
...
2
...
2
...
3
Solving LP using Simplex Algorithm
4
...
0
Summary
120

ECO 255

6
...
0

MATHEMATICS FOR ECONOMIST I

Tutor-Marked Assignment
References/Further Readings

1
...
Linear
programming considers only linear relationship between two or more variables
...
Programming means planning or decision-making in a systematic way
...
Linear programming can also be referred to as
optimization of an outcome based on some set of constraints using a linear mathematical
model
...

The sub-unit one focuses on the introduction to LP, while sub-unit two andthreefocuses
on the assumptions, merits and demerits as well as the LP calculation using simplex
algorithm
...
0
OBJECTIVES
At the end of this unit, you should be able to:
 Understand the concept of linear programming
 State the assumptions, merits and demerits of linear programming
 Solve linear programming problem using simplex algorithm
...
0

MAIN CONTENT

3
...
It can also be referred to as the use of linear mathematical
relations to plan production activities
...
Put
differently, linear programming is a tool of analysis which yields the optimum solution
for the linear objective function subject to the constraints in the form of linear
inequalities
...
For instance, a company may want
to determine the quantity of good x and y to be produced in order to minimize cost
121

ECO 255

MATHEMATICS FOR ECONOMIST I

subject to price of those goods and the company’s budget
...
Before going
further, it is necessary to look at the terms of linear programming
...

Objective Function
Objective function, also called criterion function, describe the determinants of the
quantity to be maximized or to be minimized
...
If the linear programming
requires the minimization of cost, then this is the objective function of the firm
...
If the primal of the objective
function is to maximize output then its dual will be the minimization of cost
...

Technical Constraints
The maximization of the objective function is subject to certain limitations, which are
called constraints
...
Technical constraints are set by the state of
technology and the availability of factors of production
...

3
...

4
...

5
...
In other words, of all the
feasible solutions, the solution which maximizes or minimizes the objective function is
the optimum solution
...
Similarly, if the objective function is
to minimize cost by the choice of a process or combination of processes, then the process
or a combination of processes which actually minimizes the cost will represent the
optimum solution
...

SELF ASSESSMENT EXERCISE
List and explain the linear programming
...
2

ASSUMPTIONS, ADTANTAGES AND LIMITATIONS OF LP
122

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MATHEMATICS FOR ECONOMIST I

3
...
1 Assumptions of LP
The linear programming problems are solved on the basis of some assumptions which
follow from the nature of the problem
...
Linearity: The objective function to be optimized and the constraints involve only
linear relations
...
Non-negativity: The decision variable must be non-negative
...
Additive and Divisibility: Resources and activities must be additive and divisible
...
Alternatives: There should be alternative choice of action with a well defined
objective function to be maximized or minimized
...
Finiteness: Activities, resources, constraints should be finite and known
...
Certainty: Prices and various coefficients should be known with certainty
...
2
...

1
...

2
...

3
...
For example, in a factory, some
machines may be in great demand, while others may lie idle for some times
...

3
...
3 Limitations of Linear Programming
1
...
e
...
In real life situations, when constraints or objective
functions are not linear, this technique cannot be used
...
Factors such as uncertainty and time are not taken into consideration
...
Parameters in the model are assumed to be constant but in real life situations, they
are not constant
...
Linear programming deals with only single objectives, whereas in real life
situations, we may have multiple and conflicting objectives
...
In solving linear programming problems, there is no guarantee that we will get an
integer value
...

SELF ASSESSMENT EXERCISE 2
What are the assumptions, advantages and limitations of linear programming?
3
...

Let us look at how simplex algorithm method is applied to linear programming problem
...

In the table, Pj is the representation of the maximizing equation, as 200 and 240
correspond to x1 and x2 respectively, while no figure for s1 and s2 respectively, thus
represented by zero (0)
...
i
...
For the stage 1 in our table, the Pj value for
S1 and S2 are both 0, while the resources value are both 2400 each, thus the Zj value is
(0*2400) + (0*2400) = 0
...

Thus, we rewrite the model as:
30x1 + 15x2 + s1 + 0S2 = 2400
(1)
20x1 + 30x2 + 0s1 + S2 =2400
(2)
Equation (1) and (2) above fills the stage one of the figure below
...
Simplex Algorithm
Pj
0
0

Activity Resources
S1
2400
S2
2400
Zj
Pj - Zj

0
240

0
-

S1
X2

1200
80

Stage 1
200
240
X1
X2
30
15
20
[30]
0
0
200
240
Stage 2
[-19
...
67

124

0
S1
1
0

0
S2
0
1

0
0

0
0

1
0

0
...
03

ϴ
160
80

60
...
4

ECO 255

MATHEMATICS FOR ECONOMIST I

Zj
Pj - Zj
200
240

19,200
-

X1
X2

60
...
7

-160
...
8
Stage 3
1
0
0
1

Zj
21568
200
240
Pj - Zj
0
0
Source: Authors Computation
...


0
0

-7
...
2

0
...
03

-0
...
04

2
...
8

5
...
6

The bolded row and column are both our pivot row and column, while the pivot element
is acquired by observing the highest value in row Pj – Zj and the lowest values in column
ϴ
...

Since the pivot element belong to column X2 and S2 row, we replace S2 with X2 in stage
2, and the same principle applies to stage 3
...

20x1 + 30x2 + 0s1 + s2 =2400
30x2 = 2400 - 20x1 - 0s1 - s2
Dividing through by 30 gives:
x2 = 80 - 0
...
03s2
(3)
Equation (3) above is imputed in the X2 row in stage 2
...
67x1 - 0s1 - 0
...
05x1 – 0s1 – 0
...
95x1 + s1 – 0
...
95x1 + 0
...
Here, we will make x1 the subject of the formula in (4):
19
...
45s2 = 1200
x1 = 60
...
05s1 + 0
...

Imputing (6) into (3) gives:
x2 = 80 - 0
...
03s2
By rearranging the above equation gives:
0
...
03s2 = 80
Thus our model gives:
0
...
2 – 0
...
02s2) + x2 + 0
...
3 – 0
...
01s2 + x2 + 0
...
03s1 +0
...
7
Making X2 the subject of the formula:
x2 = 39
...
03s1 – 0
...

The equation is maximize when the values for X1,X2,S1 and S2 are either zero (0) or
negative
...
2) + 240 (39
...

This maximizes the value Z
...
e
...
2x 1 and
39
...

SELF ASSESSMENT EXERCISE
Using the simplex algorithm method,
Maximize Z = 2x1 + 4x2
Subject to: 15x1 + 25x2 ≤ 10
10x1 + 17x2 ≤ 15
4
...

 The objective function according to the linear programming assumption is that
the objective function is linear and that there is an additivity of resources and
activities
...

 Linear programming problems are solved based on some assumptions which
are; Linearity, Non-negativity, additive and divisibility, alternatives, finiteness
and certainty
...

 Linear programming is applicable to only problems where the constraints and
the objective function are linear
...


5
...
In order to construct the model, transformation of the
constraints using slack variable is required
...
The simplex algorithm proceeds by
performing successive pivot operations which each give an improved basic feasible
solution; the choice of pivot element at each step is largely determined by the
requirement that this pivot improve the solution
...
0

TUTOR-MARKED ASSIGNMENT
 max Z = 10x1 + 9x2
Subject to: x1 + 22x2 ≤ 50
3x1 + 5x2 ≤ 70
4x1 + 4x2 ≤ 32
 max Z = 5x1 + 7x2 + 2x3
Subject to: 15x1 + 2x2 + 2x3 ≤ 80
4x1 + 8x2 + 2x3 ≤ 90
3x1 + 2x2 + 5x3 ≤ 40

7
...
P, Samuelson
...
A and Solow
...
M (1987)
...
Dover Publications, Inc
...

Franklin, J
...
(2002)
...
Society for Industrial and Applied Mathematics
...

Kamien
...
I and Schwartz
...
L (1991)
...

Elsevier Science; 2nd Edition (October 25, 1991), North Holland

---

Title: Mathematics I
Description: Mathematics for Economists

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