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MTH-4101 C1
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E
inequalities II
MTH-4101-2
quations
and
MTH-4101-2
EQUATIONS
AND
INEQUALITIES II
This course was produced in collaboration with the Service de l'éducation
des adultes de la Commission scolaire catholique de Sherbrooke and the
Department of the Secretary of State of Canada.
Author: Monique Pagé
Content revision: Jean-Paul Groleau
Daniel Gélineau
Mireille Moisan-Sanscartier
Consultant in adult education: Serge Vallières
Coordinator for the DGFD: Jean-Paul Groleau
Coordinator for the DFGA: Ronald Côté
Photocomposition and layout: Multitexte Plus
Translation: Consultation en éducation Zegray
Linguistic revision: Kay Flanagan and Leslie Macdonald
Translation of updated sections: Claudia de Fulviis
Reprint: 2006
© Société de formation à distance des commissions scolaires du Québec
All rights for translation and adaptation, in whole or in part, reserved for all countries.
Any reproduction, by mechanical or electronic means, including micro-reproduction, is
forbidden without the written permission of a duly authorized representative of the
Société de formation à distance des commissions scolaires du Québec (SOFAD).
Legal Deposit – 2006
Bibliothèque et Archives nationales du Québec
Bibliothèque et Archives Canada
ISBN 978-2-89493-279-7
Answer Key
MTH-4101-2
Equations and Inequalities II
TABLE OF CONTENTS
Introduction to the Program Flowchart ................................................. 0.4
The Program Flowchart .......................................................................... 0.5
How to Use this Guide ............................................................................ 0.6
General Introduction ............................................................................... 0.9
Intermediate and Terminal Objectives of this Module ......................... 0.10
Diagnostic Test on the Prerequisites ..................................................... 0.13
Answer Key for the Diagnostic Test on the Prerequisites .................... 0.21
Analysis of the Diagnostic Test Results ................................................. 0.27
Information for Distance Education Students ....................................... 0.29
UNITS
1.
2.
3.
4.
5.
6.
7.
Graphing a System of Equations ............................................................ 1.1
Solving a System of Equations by Comparison ..................................... 2.1
Solving a System of Equations by Substitution .................................... 3.1
Solving a System of Equations by Elimination Through Addition....... 4.1
Solving a System of Equations: Four Possible Methods ....................... 5.1
Solving Everyday Problems .................................................................... 6.1
Graphing a System of Inequalities ......................................................... 7.1
Final Summary........................................................................................ 8.1
Answer Key for the Final Review ........................................................... 8.11
Terminal Objectives ................................................................................ 8.18
Self-Evaluation Test................................................................................ 8.21
Answer Key for the Self-Evaluation Test .............................................. 8.35
Analysis of the Self-Evaluation Test Results ........................................ 8.49
Final Evaluation...................................................................................... 8.51
Answer Key for the Exercises ................................................................. 8.53
Glossary ................................................................................................... 8.159
List of Symbols ........................................................................................ 8.163
Bibliography ............................................................................................ 8.164
Review Activities ..................................................................................... 9.1
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MTH-4101-2
Equations and Inequalities II
INTRODUCTION TO THE PROGRAM FLOWCHART
Welcome to the World of Mathematics!
This mathematics program has been developed for the adult students of the
Adult Education Services of school boards and distance education. The learning
activities have been designed for individualized learning. If you encounter
difficulties, do not hesitate to consult your teacher or to telephone the resource
person assigned to you. The following flowchart shows where this module fits
into the overall program. It allows you to see how far you have progressed and
how much you still have to do to achieve your vocational goal. There are several
possible paths you can take, depending on your chosen goal.
The first path consists of modules MTH-3003-2 (MTH-314) and MTH-4104-2
(MTH-416), and leads to a Diploma of Vocational Studies (DVS).
The second path consists of modules MTH-4109-1 (MTH-426), MTH-4111-2
(MTH-436) and MTH-5104-1 (MTH-514), and leads to a Secondary School
Diploma (SSD), which allows you to enroll in certain Gegep-level programs that
do not call for a knowledge of advanced mathematics.
The third path consists of modules MTH-5109-1 (MTH-526) and MTH-5111-2
(MTH-536), and leads to Cegep programs that call for a solid knowledge of
mathematics in addition to other abiliies.
If this is your first contact with this mathematics program, consult the flowchart
on the next page and then read the section “How to Use This Guide.” Otherwise,
go directly to the section entitled “General Introduction.” Enjoy your work!
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MTH-4101-2
Equations and Inequalities II
THE PROGRAM FLOWCHART
CEGEP
MTH-5112-1
MTH-5111-2
MTH-536
MTH-5104-1
MTH-5103-1
Introduction to Vectors
MTH-5109-1
Geometry IV
MTH-5108-1
Trigonometric Functions and Equations
MTH-5107-1
Exponential and Logarithmic Functions
and Equations
Optimization II
MTH-5106-1
Real Functions and Equations
Probability II
MTH-5105-1
Conics
MTH-5102-1
Statistics III
MTH-5101-1
MTH-436
MTH-426
MTH-4110-1
MTH-216
MTH-116
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The Four Operations on
Algebraic Fractions
Sets, Relations and Functions
MTH-4108-1
Quadratic Functions
MTH-4107-1
Straight Lines II
MTH-4106-1
Factoring and Algebraic Functions
MTH-4105-1
Exponents and Radicals
MTH-4103-1
MTH-4102-1
MTH-4101-2
Complement and Synthesis I
MTH-4109-1
MTH-4104-2
MTH-314
Optimization I
MTH-4111-2
Trades
DVS
MTH-416
Complement and Synthesis II
MTH-5110-1
MTH-526
MTH-514
Logic
Statistics II
Trigonometry I
Geometry III
Equations and Inequalities II
MTH-3003-2
Straight Lines I
MTH-3002-2
Geometry II
MTH-3001-2
The Four Operations on Polynomials
MAT-2008-2
Statistics and Probabilities I
MTH-2007-2
Geometry I
MTH-2006-2
Equations and Inequalities I
MTH-1007-2
Decimals and Percent
MTH-1006-2
The Four Operations on Fractions
MTH-1005-2
The Four Operations on Integers
0.5
You ar e h er e
25 hours
= 1 credit
50 hours
= 2 credits
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MTH-4101-2
Equations and Inequalities II
HOW TO USE THIS GUIDE
Hi! My name is Monica and I have been
asked to tell you about this math module.
What’s your name?
Whether you are
registered at an
adult education
center or at
Formation à
distance, ...
Now, the module you have in your
hand is divided into three
sections. The first section is...
I’m Andy.
... you have probably taken a
placement test which tells you
exactly which module you
should start with.
... the entry activity, which
contains the test on the
prerequisites.
0.6
You’ll see that with this method, math is
a real breeze!
My results on the test
indicate that I should begin
with this module.
By carefully correcting this test using the
corresponding answer key, and recording your results on the analysis sheet ...
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MTH-4101-2
Equations and Inequalities II
... you can tell if you’re well enough
prepared to do all the activities in the
module.
And if I’m not, if I need a little
review before moving on, what
happens then?
In that case, before you start the
activities in the module, the results
analysis chart refers you to a review
activity near the end of the module.
I see!
In this way, I can be sure I
have all the prerequisites
for starting.
START
The starting line
shows where the
learning activities
begin.
Exactly! The second section
contains the learning activities. It’s
the main part of the module.
?
The little white question mark indicates the questions
for which answers are given in the text.
The target precedes the
objective to be met.
The memo pad signals a brief reminder of
concepts which you have already studied.
?
Look closely at the box to
the right. It explains the
symbols used to identify the
various activities.
The boldface question mark
indicates practice exercices
which allow you to try out what
you have just learned.
The calculator symbol reminds you that
you will need to use your calculator.
?
The sheaf of wheat indicates a review designed to
reinforce what you have just learned. A row of
sheaves near the end of the module indicates the
final review, which helps you to interrelate all the
learning activities in the module.
FINISH
Lastly, the finish line indicates
that it is time to go on to the self-evaluation
test to verify how well you have understood
the learning activities.
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MTH-4101-2
Equations and Inequalities II
There are also many fun things
in this module. For example,
when you see the drawing of a
sage, it introduces a “Did you
know that...”
It’s the same for the “math whiz”
pages, which are designed especially for those who love math.
For example. words in boldface italics appear in the
glossary at the end of the
module...
A “Did you know that...”?
Yes, for example, short tidbits
on the history of mathematics
and fun puzzles. They are interesting and relieve tension at
the same time.
Must I memorize what the sage says?
No, it’s not part of the learning activity. It’s just there to
give you a breather.
They are so stimulating that
even if you don’t have to do
them, you’ll still want to.
And the whole module has
been arranged to make
learning easier.
... statements in boxes are important
points to remember, like definitions, formulas and rules. I’m telling you, the format makes everything much easier.
The third section contains the final review, which interrelates the different
parts of the module.
Great!
There is also a self-evaluation
test and answer key. They tell
you if you’re ready for the final
evaluation.
Thanks, Monica, you’ve been a big
help.
I’m glad! Now,
I’ve got to run.
See you!
0.8
Later ...
This is great! I never thought that I would
like mathematics as much as this!
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MTH-4101-2
Equations and Inequalities II
GENERAL INTRODUCTION
MOVING TOWARDS A SOLUTION
Ideally detectives would like to find the guilty person in a case quickly, based on
the available information: pharmacists or cooks would like to find the exact
proportions of the ingredients for their new creation on their first attempt;
agronomists would like to know the ideal quantity of fertilizer for a given lot just
by looking at it. In short, all of us are looking for quick, easy and definitive
solutions! Unfortunately, instant solutions rarely exist.
As a result of the vast quantity of information which we must consider in order
to solve the numerous problems we encounter, we must interrelate this varied
information, arrive at possible solutions and make use of reliable tools to
evaluate these solutions.
A gardener must judge the quantity of fertilizer which is best for a garden, given
the information recorded about the garden in previous years and the results of
experiments carried out by agronomists. The gardener must know the method
used to dilute products to obtain an appropriate mixture and the exact time when
the fertilizer must be spread. To find the solution to growing the best possible
plants, he or she must consider all these factors.
To reach the objective of this module, you must be able to solve everyday
problems by translating the relationships which exist between the givens in a
problem using a system of equations or inequalities. You must first master
methods for solving these systems algebraically as well as for representing them
graphically. The latter is the concrete visualization of information. It allows you
to spot the required information and gives an overall view of the possible
solutions for a specific situation.
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MTH-4101-2
Equations and Inequalities II
INTERMEDIATE AND TERMINAL OBJECTIVES OF
THE MODULE
Module MTH-4101-2 consists of seven units and requires fifty hours of study
distributed as follows. Each unit covers either an intermediate or a terminal
objective. The terminal objectives appear in boldface.
Objectives
Number of Hours*
% (evaluation)
1
4
10%
2 to 5
10
40%
6
8
30%
7
6
20%
* Two hours are allotted for the final evaluation.
1. Graphing a System of Equations
To solve a system of two equations of the first degree in two variables
of the form Ax + By + C = 0 graphically. The coefficients A, B and C are
rational numbers ( ). The steps in the solution must be described.
2. Solving a System of Equations by Comparison
To solve a system of two equations of the first degree in two variables of the
form Ax + By + C = 0 algebraically by applying the method of solving by
comparison. The coefficients A, B and C are real numbers ( ). The steps in
the solution must be described.
0.10
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MTH-4101-2
Equations and Inequalities II
3. Solving a System of Equations by Substitution
To solve a system of two equations of the first degree in two variables of the
form Ax + By + C = 0 algebraically by applying the method of solving by
substitution. The coefficients A, B and C are real numbers ( ). The steps in
the solution must be described.
4. Solving a System of Equations by Elimination Through Addition
To solve a system of two equations of the first degree in two variables of the
form Ax + By + C = 0 algebraically by applying the method of solving by
elimination through addition. The coefficients A, B and C are real numbers
( ). The steps in the solution must be described.
5. Solving a System of Equations: Four Possible Methods
To solve a system of two equations of the first degree in two variables
of the form Ax + By + C = 0 by applying one of these methods:
• the graphic method
• the algebraic method by comparison
• the algebraic method by substitution
• the algebraic method by elimination through addition.
The coefficients A, B and C are real numbers ( ). The steps in the
solution must be described.
6. Solving Everyday Problems
To solve word problems which must be expressed as a system of two
equations of the first degree in two variables in order to be solved.
The situations presented are borrowed from everyday life. The
numbers used are rational numbers ( ). The five steps in the solution
must be described.
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MTH-4101-2
Equations and Inequalities II
7. Graphing a System of Equations
To solve a system of two inequalities of the first degree in two
variables of the form Ax + By + C ≤ 0 or of the form
Ax + By + C ≥ 0 by applying graphic methods of solution. The
coefficients A, B and C are real numbers ( ).
0.12
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MTH-4101-2
Equations and Inequalities II
DIAGNOSTIC TEST ON THE PREREQUISITES
Instructions
1° Answer as many questions as you can.
2° You may use a calculator.
3° Write your answers on the test paper.
4° Do not waste any time. If you cannot answer a question, go on
to the next one immediately.
5° When you have answered as many questions as you can, correct
your answers using the answer key which follows the diagnostic
test.
6° To be considered correct, answers must be identical to those in
the key. For example, if you are asked to describe the steps
involved in solving a problem, your answer must contain all the
steps.
7° Transcribe your results onto the chart which follows the answer
key. It gives an analysis of the diagnostic test results.
8° Do only the review activities listed for each of your incorrect
answers.
9° If all your answers are correct, you have the prerequisites to
begin working on this module.
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MTH-4101-2
Equations and Inequalities II
1. Calculate the numerical value of the following algebraic expressions. The
steps in the solution of the problem and the answer are required. Round your
answers to the nearest hundredth, if necessary.
a) 8[3(3 + 2 × 23) – 7] – (4 – 2 × 7)
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b) 23 + 7 × 2 – 7 + 3 – 8 × 2 – 13 + 4
4 ×3+ 2
7(2 × 3 – 5)
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MTH-4101-2
Equations and Inequalities II
c) [0.5(2.4 – 4.2)] ÷ (7.2 + 3.7 ÷ 4.2)
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2. a) Using the given tables of values, plot the points corresponding to the
ordered pairs described in the Cartesian plane, and graph the lines
through these points.
y
1
1
x
x
0
2
4
5
y
2
4
6
7
x
–3
–3
–3
–3
y
–1
0
4
5
x
–1
0
4
6
y
–2
–2
–2
–2
➀
➁
➂
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MTH-4101-2
Equations and Inequalities II
b) Which of the preceding tables of values contains ordered pairs which
satisfy the equation x = y – 2 ?
Answer: ...................................
3. For each of the equations below, complete a table of values and graph the
corresponding line.
➀ 3x + y = 6
➁ y=5
➂ x+3=9
y
x
➀
y
1
x
➁
1
x
y
x
➂
y
4. Given the algebraic expression 4x + 48, identify
3
a) the variable: ..........................................
b) the coefficient: .......................................
c) the inverse of the constant term: .........
d) the inverse of the coefficient: ................
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MTH-4101-2
Equations and Inequalities II
5. Solve the following equations. A complete solution is required. Round your
answers to the nearest hundredth, if necessary.
a) 3x – 7 = 2x + 8
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b) 7x + 4 – 2x = 4x – 3
3
8
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c) 4x – 2 = 6x + 2
7
3
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MTH-4101-2
Equations and Inequalities II
d) 0.5x – 2.4 = 0.2x – 3
0.5
0.7
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6. Using the distributive property, isolate the variable y in the following
equations.
a) 3(2y – 4) + 10y – 6 = 2(9y + 3)
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b)
8(2 – 3y) (3 – 4 y)2
=
3
4
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MTH-4101-2
Equations and Inequalities II
7. Yolanda earns a gross weekly salary of $404.00. She works 8 hours per day,
5 days a week. If 24% of this amount is withheld each week for various
deductions, calculate her annual net salary (52 weeks).
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MTH-4101-2
Equations and Inequalities II
8. Louise and Paul prepare their budget. Louise sets aside 1 of her weekly net
4
1
of his own pay. If Louise's
pay for her savings and Paul does as well with
5
and Paul's annual net salaries amount to $l9 825.00 and $20 930.00 respectively, calculate how much money Paul sets aside weekly for his current
expenses.
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MTH-4101-2
Equations and Inequalities II
ANSWER KEY FOR THE DIAGNOSTIC TEST
ON THE PREREQUISITES
1. a) 8[3(3 + 2 × 23) – 7] – (4 – 2 × 7)
8[3(3 + 46) – 7] – (4 – 14)
8[3(49) – 7] – (– 10)
8[147 – 7] + 10
8[140] + 10
1120 + 10
1130
b) 23 + 7 × 2 – 7 + 3 – 8 × 2 – 13 + 4
4×3+2
7(2 × 3 – 5)
23 + 14 – 7 + 3 – 16 – 13 + 4
12 + 2
7(6 – 5)
37 – 7 + 3 – 3 + 4
14
7(1)
30 + 3 – 7
7
14
33 – 14
14 14
19 or 1 5
14
14
c) [0.5(2.4 – 4.2)] ÷ (7.2 + 3.7 ÷ 4.2)
[0.5(– 1.8)] ÷ (7.2 + 0.88)
– 0.9 ÷ 8.08
– 0.11
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MTH-4101-2
Equations and Inequalities II
y
2. a)
➀
•
➁
•
•
•
•
•
1
•
•
1
• •
x
•
•➂
b) ➀
3. ➀ y = – 3x + 6
➁ y=5
➂ x=6
y
➀
x
0
1
2
y
6
3
0
➁
➂
•
•••
➀
x
0
1
•
1
2
➁
1
y
5
5
5
x
6
6
6
y
0
1
2
•
•
•
•
x
➂
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MTH-4101-2
Equations and Inequalities II
b) 4
3
4. a) x
5. a)
c) – 48
3x – 7 = 2x + 8
3x – 2x = 8 + 7
x = 15
b)
7x + 4 – 2x = 4x – 3
3
8
21x + 96 – 16x = 96x – 72
24
24
24
24
24
21x + 96 – 16x = 96x – 72
21x – 16x – 96x = – 72 – 96
– 91x = – 168
x = – 168
– 91
x = 168 = 24 or 1 11 or 1.85
91
13
13
c)
4x – 2 = 6x + 2
3
7
7(4x – 2) = 3(6x + 2)
28x – 14 = 18x + 6
28x – 18x = 6 + 14
10x = 20
x = 20
10
x= 2
d)
0.5x – 2.4 = 0.2x – 3
0.7
0.5
0.5(0.5x – 2.4) = 0.7(0.2x – 3)
0.25x – 1.2 = 0.14x – 2.1
0.25x – 0.14x = – 2.1 + 1.2
0.11x = – 0.9
x = – 0.9
0.11
x = – 8.18
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d) 3
4
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MTH-4101-2
Equations and Inequalities II
6. a) 3(2y – 4) + 10y – 6 = 2(9y + 3)
6y – 12 + 10y – 6 = 18y + 6
6y + 10y – 18y = 6 + 6 + 12
– 2y = 24
y = 24
–2
y = – 12
b)
8(2 – 3y)
(3 – 4y)2
=
4
3
16 – 24y
6 – 8y
=
4
3
3(16 – 24y) = 4(6 – 8y)
48 – 72y = 24 – 32y
– 72y + 32y = 24 – 48
– 40y = – 24
y = – 24
– 40
y = 3 or 0.6
5
7. • We want to determine Yolanda’s annual net salary.
• State the problem in mathematical language.
[$404 – ($404 × 24%)] × 52
• Estimate the result.
$400.00 – $400 × 25
100
× 50 = [$400 – $100] × 50 = [$300] × 50 =
$15 000
• Solve the problem.
$404.00 – $404 × 24
100
× 52 = [$404 – $96.96] × 52 =
[$307.04] × 52 = $15 966.08
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Equations and Inequalities II
• The result is close to the estimate.
• Yolanda’s net annual salary is $15 966.08.
8. • We want to determine how much money Paul sets aside weekly for his
current expenses.
• State the problem in mathematical language.
$20 930 – $20 930 × 1 ÷ 52
5
• Estimate the result.
$20 000 – $20 000 × 1 ÷ 50 = ($20 000 – $4 000) ÷ 50 =
5
$16 000 ÷ 50 = $320
• Solve the problem.
$20 930 – $20 930 × 1 ÷ 52 = ($20 930 – $4 186) ÷ 52 =
5
$16 744 ÷ 52 = $322
• The result is close to the estimate.
• Paul sets aside $322 weekly for his current expenses.
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MTH-4101-2
Equations and Inequalities II
ANALYSIS OF THE DIAGNOSTIC
TEST RESULTS
Question
1. a)
b)
c)
2. a)
b)
3.
4. a)
b)
c)
d)
5. a)
b)
c)
d)
6. a)
b)
7.
8.
Answer
Correct
Incorrect
Section
Review
Page
Before Going
to Unit(s)
9.2
9.2
9.2
9.3
9.3
9.3
9.4
9.4
9.4
9.4
9.4
9.4
9.4
9.4
9.4
9.4
9.1
9.1
9.19
9.19
9.19
9.25
9.25
9.25
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.4
9.4
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
1 and 6
1 and 6
• If all your answers are correct, you may begin working on this module.
• For each incorrect answer, find the related section listed in the Review
column. Do the review activities for that section before beginning the units
listed in the right-hand column under the heading Before Going to Unit(s).
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MTH-4101-2
Equations and Inequalities II
INFORMATION
FOR
EDUCATION STUDENTS
DISTANCE
You now have the learning material for MTH-4101-2 together with the homework assignments. Enclosed with this material is a letter of introduction from
your tutor indicating the various ways in which you can communicate with him
or her (e.g. by letter, telephone) as well as the times when he or she is available.
Your tutor will correct your work and help you with your studies. Do not hesitate
to make use of his or her services if you have any questions.
DEVELOPING EFFECTIVE STUDY HABITS
Distance education is a process which offers considerable flexibility, but which
also requires active involvement on your part. It demands regular study and
sustained effort. Efficient study habits will simplify your task. To ensure
effective and continuous progress in your studies, it is strongly recommended
that you:
• draw up a study timetable that takes your working habits into account and
is compatible with your leisure time and other activities;
• develop a habit of regular and concentrated study.
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The following guidelines concerning the theory, examples, exercises and assignments are designed to help you succeed in this mathematics course.
Theory
To make sure you thoroughly grasp the theoretical concepts:
1. Read the lesson carefully and underline the important points.
2. Memorize the definitions, formulas and procedures used to solve a given
problem, since this will make the lesson much easier to understand.
3. At the end of an assignment, make a note of any points that you do not
understand. Your tutor will then be able to give you pertinent explanations.
4. Try to continue studying even if you run into a particular problem. However,
if a major difficulty hinders your learning, ask for explanations before
sending in your assignment.
Contact your tutor, using the procedure
outlined in his or her letter of introduction.
Examples
The examples given throughout the course are an application of the theory you
are studying. They illustrate the steps involved in doing the exercises. Carefully
study the solutions given in the examples and redo them yourself before starting
the exercises.
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Exercises
The exercises in each unit are generally modelled on the examples provided.
Here are a few suggestions to help you complete these exercises.
1. Write up your solutions, using the examples in the unit as models. It is
important not to refer to the answer key found on the coloured pages at the
end of the module until you have completed the exercises.
2. Compare your solutions with those in the answer key only after having done
all the exercises. Careful! Examine the steps in your solution carefully even
if your answers are correct.
3. If you find a mistake in your answer or your solution, review the concepts that
you did not understand, as well as the pertinent examples. Then, redo the
exercise.
4. Make sure you have successfully completed all the exercises in a unit before
moving on to the next one.
Homework Assignments
Module MTH-4101-2 contains three assignments.
The first page of each
assignment indicates the units to which the questions refer. The assignments
are designed to evaluate how well you have understood the material studied.
They also provide a means of communicating with your tutor.
When you have understood the material and have successfully done the pertinent exercises, do the corresponding assignment immediately. Here are a few
suggestions.
1. Do a rough draft first and then, if necessary, revise your solutions before
submitting a clean copy of your answer.
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2. Copy out your final answers or solutions in the blank spaces of the document
to be sent to your tutor. It is preferable to use a pencil.
3. Include a clear and detailed solution with the answer if the problem involves
several steps.
4. Mail only one homework assignment at a time. After correcting the assignment, your tutor will return it to you.
In the section “Student’s Questions”, write any questions which you may wish to
have answered by your tutor. He or she will give you advice and guide you in your
studies, if necessary.
In this course
Assignment 1 is based on units 1 to 5.
Assignment 2 is based on units 6 and 7.
Assignment 3 is based on units 1 to 7.
CERTIFICATION
When you have completed all the work, and provided you have maintained an
average of at least 60%, you will be eligible to write the examination for this
course.
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MTH-4101-2
Equations and Inequalities II
START
UNIT 1
GRAPHING A SYSTEM OF EQUATIONS
1.1
SETTING THE CONTEXT
A Sticky Story
Mr Fixit has been running the family maple sugar business for the last few years.
One of his barrels, the same age as he is, is filled with 40 litres of maple sap.
Unfortunately, 2 litres of this precious liquid are escaping per hour through a
recently formed crack at the bottom of the barrel.
Mr Fixit quickly places another barrel beneath the first. The spill is now being
picked up by the second barrel, which is in good condition and already contains
10 litres of maple sap.
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Equations and Inequalities II
Fig. 1.1 Mr Fixit saves the day
Mr Fixit learned in a mathematics course that many everyday situations can be
translated into algebraic expressions. He begins to wonder how the levels of sap
in the two barrels will change and at which point the two barrels will contain the
same amount.
To find the answer, the situation can be translated into two equations of the first
degree, where
• x represents the time elapsed in hours since the start of the leak
• y represents the volume of sap in litres, in each barrel.
An equation is a sentence formed by two mathematical expressions containing one or more variables and related by
the = sign.
For example: 5x + 4 = 7 is an equation of the first degree.
1.2
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Let x be the number of hours since the start of the leak.
• The first equation deals with the barrel which is being emptied.
Let 2x be the volume of sap lost, since 2 L are lost per hour,
y the volume of sap when the barrels contain the same amount of sap,
40 L the volume of sap at the start.
Then, ➀ 2x + y = 40.
• The second equation deals with the second barrel, which is being filled.
Let 2x be the volume of sap saved,
y the volume of sap when the barrels contain the same amount of sap,
10 L the volume of sap at the start.
Then, ➁ 2x + 10 = y.
N.B. You will learn how to translate this type of situation into equations in
Unit 6. For the moment, concentrate on what Mr Fixit is leading up to.
The two variables x and y are related to each other, that is, the volume of sap y
in one barrel depends on the time elapsed x since the start of the leak.
You are therefore faced with two equations dealing with the same context; since
these two barrels cannot be separated, these two equations cannot be separated.
It is therefore the set of both these equations which represents the
situation at hand. This set is called a system of equations.
A system of equations of the first degree is a set of equations
which must be solved simultaneously.
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➀ 2x + y = 40
The equations
➁ 2x + 10 = y
represent a system of equations. You must find which values of x and y satisfy
these two equations simultaneously (at the same time).
To do so, equip yourself with a pencil, ruler and graph paper and solve this
problem graphically.
To reach the objective of this unit, you should be able to graph a system
of equations of the first degree in two variables, to specify the relative
positions of the two lines and state the solution(s) of the system.
The first step consists in graphing each of the equations in the system after
having completed a table of values.
To complete a table of values easily, it is recommended that you isolate one of the
two variables. Take Mr Fixit's equations and isolate y to obtain the form:
y = mx + b. You therefore have:
➀ 2x + y = 40
➁ 2x + 10 = y
y = – 2x + 40
y = 2x + 10
The original system has thereby been converted into an equivalent system of
equations:
➀ y = – 2x + 40
➁ y = 2x + 10
1.4
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Equations and Inequalities II
Then complete a table of values for each of these equations.
➀
➁
x
y = – 2x + 40
x
0
– 2(0) + 40 = 40
0
2
– 2(2) + 40 = 36
2
4
– 2(4) + 40 = 32
4
8
– 2(8) + 40 = 24
6
y = 2x + 10
10
10
No doubt you obtained the ordered pair (10, 20) for equation ➀ and the ordered
pairs (0, 10), (2, 14), (4, 18), (6, 22) and (10, 30) for equation ➁.
Take the ordered pair (4, 32) from equation ➀. This ordered pair is a solutionpair of equation ➀, since if x is replaced by 4 and y by 32 in this equation, the
equation is satisfied or remains true. In effect, y = – 2x + 40 becomes
32 = – 2(4) + 40
32 = – 8 + 40
32 = 32.
An ordered pair is a solution-pair of an equation of the first
degree in two variables if, after substitution, the ordered pair
satisfies the equation or expresses a true statement.
Let us leave Mr Fixit's calculations for a moment. Before graphing these
equations, you need some practice in converting them to the form y = mx + b and
in preparing tables of values for equations in two variables.
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Equations and Inequalities II
Exercise 1.1
1. Transform each of these systems of equations to a system of the form:
➀ y = m1x + b1
➁ y = m2x + b2
a) ➀ 3x – 6y – 12 = 0
➁ 3x – 6y + 6 = 0
b) ➀ 8x – 2y + 3x = 5 + 12x – 4y
➁ 10x + y + 3 = 5 – 3y + 10x
c) ➀ 3x = 4y
➁ x=
1.6
2y + 1
3
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d) ➀ 3(x – 1) + 2(y + 3) = 8
➁ 4x – (y – 6) = 9
2. Complete the two tables of values for the following system of equations.
➀ 10x + y + 27 = 10y + x
➁ x+y=7
➀
➁
➀
x
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➁
y
x
1.7
y
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Answer Key
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MTH-4101-2
Equations and Inequalities II
Since you now know how to derive solution-pairs for an equation, move on to the
next step, that of graphing the equation.
Mr Fixit must now transfer the solution-pairs for each equation in his system to
the same Cartesian plane. The graphs of these equations are lines. The
procedure used is summarized below:
Mr Fixit has to:
1. Establish his system of equations.
➀ 2x + y = 40
➁ 2x + 10 = y
2. Transform these equations to the form y = mx + b.
➀ y = – 2x + 40
➁ y = 2x + 10
3. Complete the tables of values.
x
y
x
y
0
40
0
10
2
36
2
14
4
32
4
18
8
24
6
22
10
20
10
30
N.B. Each table of values lists a set of solution-pairs to be plotted in the Cartesian
plane. A minimum of 3 ordered pairs per equation is required to graph these lines
correctly.
1.8
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Volume of sap in a barrel in litres
4. Graph each of these equations.
y
60
50
40•
(0, 40)
(2, 36)
•
(4, 32)
•
30
+
– 2x
2
4
6
1
(10, 30)
•
• (8, 24)(10, 20)
•
(6,
22)
20
•
•(4, 18)
•(2, 14)
10 •
(0, 10)
0
➁
y
0=
8
2x
+y
=4
Scale used:
x-axis
0.5 cm ^
= 1h
y-axis
^ 5L
0.5 cm =
0
➀
x
10
Time elapsed in hours since the start of the leak
Fig. 1.2 Intersecting lines representing Mr Fixit's system of equations
?
Do these two lines meet at a point? .................................................................
?
What is this point called in mathematical language?
...........................................................................................................................
In effect, these two lines meet at a point called the point of intersection. This
point is common to the two lines and it is called the solution-pair of the system
of equations.
The graphic solution of a system of equations consists in
determining the coordinates of the point that is common to
the two lines of the system, if there is a point of intersection.
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We will now return to Mr Fixit, who is completing the last step in his
Volume of sap in litres
investigation.
y
60
50
10
lled
x+
g fi
n
i
e
el b
arr
2
y=
40
b
(7.5, 25)
30
•
20
bar
rel
10
0
2
4
6
8
y=
– 2x
bei
+4
ng
0
em
pti
ed
Scale used:
x-axis
0.5 cm ^
= 1h
y-axis
^ 5L
0.5 cm =
x
10
Time elapsed in hours
Fig. 1.3 Mr Fixit's graph
From the graph representing his system of equations, Mr Fixit can "read" the
coordinates of the point of intersection of the two lines. This pair of coordinates
(7.5, 25) is the solution-pair of the system since it satisfies both equation ➀
and equation ➁ simultaneously. It is also the only solution-pair of the system
given that it is the only place where the two lines intersect, that is, the only place
where the x and y variables take the same values in both equations simultaneously.
He can therefore state, proof in hand, that the two barrels will contain the same
amount of sap, namely 25 litres, after 7 1 hours have elapsed. This is what the
2
ordered pair (7.5, 25) indicates.
1.10
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Did you know that
graphic, graph or graphing derive from the same Greek
verb graphein, which means to write? Graphs are representations or illustrations of data in a plane, that is, a form
of display which provides information. They are used in fields as varied
as medicine, economics and weather forecasting.
For example, an electrocardiogram is a graphic record (graph) of the
electrical activity (electro) of the heart (cardio).
R
P
T
Q
Normal electrocardiogram
P wave, atrial depolarization;
QRS wave, ventricular depolarization;
T wave, ventricular repolarization.
Fig. 1.4
S
Electrocardiogram
The daily volume of stock market transactions in a given period, as
summarized in the graph in Figure 1.5, is another example of the use of
graphs.
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MTH-4101-2
Equations and Inequalities II
DAILY VOLUME OF
TRANSACTIONS: 5 492 000
13 000
12 000
11 000
10 000
9 000
8 000
7 000
6 000
5 000
4 000
3 000
2 000
FEBRUARY
MARCH
APRIL
Fig. 1.5 Transactions on the stock market
*Source: La Presse, April 16 1988
Another type of graph, a circle graph, is used to represent the portions of
a family budget in the form of "pie slices." It is in current use in written
as well as electronic media.
culture and leisure
C
hotel, restaurants
H
dressing
transport
food
F
D
T
He
health, hygiene
Ho
F: 28.4%
D: 10.0%
H: 20.7%
He: 11.8%
T: 10.3%
C: 8.8%
Ho: 10.0%
———
100%
housing
Fig. 1.6 A typical family budget
1.12
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Exercise 1.2
1. Given the system of equations ➀ 5x – 3y = 14 and ➁ 2x + y = 10.
a) Determine at least 3 solution-pairs for each equation by completing a
table of values for each.
➀
➁
➀
x
➁
y
x
y
b) In the Cartesian plane which follows, plot the points corresponding to
these pairs and graph the 2 lines representing this system of equations.
y
1
1
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Equations and Inequalities II
c) What are the coordinates of the point of intersection of the two lines?
.................................
d) Demonstrate that the ordered pair corresponding to the point of intersection is a solution-pair for each of the equations in the system.
➀ 5x – 3y = 14
➁ 2x + y = 10
To solve a system of equations of the first degree in two
variables graphically:
1. Transform each of the equations in the system to the form
y = mx + b.
2. Determine at least 3 solution-pairs for each equation and
complete a table of values.
3. Graph each equation in the same Cartesian plane.
4. Identify the point of intersection of the two lines, if there
is one.
5. Determine the solution of the system of equations.
N.B. It is recommended that you verify the solution of the system by substituting the values obtained for the x and y variables in the original equations.
Is there always only one solution to a system of equations? Interesting question!
However, the answer is not obvious. Look at the following examples.
1.14
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Example 1
Given the following system of equations:
➀ 2x – y = – 5
➁ 2x – y = 3.
1. Transform the equations in the system to the form y = mx + b.
➀ 2x – y = – 5
➁ 2x – y = 3
– y = – 2x – 5
– y = –2x + 3
y = 2x + 5
y = 2x – 3
2. Determine at least 3 solution-pairs for each equation and complete the
tables of values.
➀
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➁
x
y
x
y
–2
1
0
–3
0
5
2
1
1
7
5
7
1.15
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Equations and Inequalities II
3. Graph each equation in the same Cartesian plane.
y
•➀
•➁
•
•
•
1
1
x
•
Fig. 1.7 Parallel lines representing the system of equations
2x – y = – 5 and 2x – y = 3
4. Identify the point of intersection of the two lines, if there is one.
There is no point of intersection since the two lines are parallel.
5. Determine the solution of the system of equations.
This system has no solution since there is no point common to the two
lines.
Therefore, there are systems for which no common solution exists. These
systems are composed of distinct parallel lines.
Now take a look at another case where a unique solution cannot be found.
1.16
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Example 2
Given the system of equations:
y
=1
➀ x +
2 6
➁ x+
y
=2
3
1. Transform the equations in the system to the form y = mx + b.
y
➀ x +
=1
2 6
3x + y = 6
6
6
6
3x + y = 6
y
=2
3
3x + y = 6
3
3
3
3x + y = 6
➁ x+
y = – 3x + 6
y = – 3x + 6
As you can see, these two equations are identical.
2. Determine at least 3 solution-pairs for each equation and complete the
tables of values.
N.B. Only one table of values is needed here since the 2 equations in the
system are identical.
➀ and ➁
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x
y
0
6
1
3
2
0
1.17
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3. Graph each equation in the same Cartesian plane.
y
➀ and ➁
•
•
1
1
•
x
Fig. 1.8 Coinciding lines representing the system of equations
y
y
➀ x +
= 1 and ➁ x +
=2
2 6
3
4. Identify the point of intersection of the two lines, if there is one.
There are an infinite number of points of intersection since these two lines
coincide.
5. Determine the solution of the system of equations.
This system has an infinite number of solutions since the two lines are
superimposed.
Thus, systems exist for which there are an infinite number of solutions.
These systems are composed of coinciding lines.
Now for a summary!
1.18
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In solving a system of equations of the first degree in
two variables graphically, three situations can occur:
1. The two lines can intersect and, in this case, the solution
of the system is a unique solution-pair;
2. The two lines can be parallel and distinct and, in this case,
there is no solution;
3. The two lines can coincide and, in this case, there exist an
infinite number of solutions.
Intersecting
lines
Parallel and
distinct lines
y
Coinciding
lines
y
x
y
x
x
Fig. 1.9 Various types of lines that result from systems of equations
Now see whether you can identify these lines in the following practice exercises.
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Equations and Inequalities II
1.2
PRACTICE EXERCISES
1. Transform each of the following equations to the form y = mx + b.
a) x +
y
=6
2
c) x – 1 =
y
2
y–2
b) x – 2 –
= –2
2
3
d) 0.15x + 0.3y = 1.2
e) 3(x – 3) + 4 = 5(y – 2) + 3x
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2. Solve the following systems of equations of the first degree in two variables
graphically by following the procedure presented in this unit. Determine the
solution as well as the types of lines which represent each system.
a) ➀ 2x – 5y = 20
➁ 3x + 2y = 11
1. ➀
➁
2. Tables of values
3. Graph
y
➀
x
y
x
➁
x
y
4. ...............................................................................................................
5. • Solution-pair: ...................................................................................
• Type of lines: ...................................................................................
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b) ➀ 22x – 6y = 24
➁ 6y –15x = – 3
1. ➀
➁
2. Tables of values
3. Graph
y
➀
x
y
➁
x
x
y
4. ...............................................................................................................
5. • Solution-pair: ...................................................................................
• Type of lines: ...................................................................................
1.22
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c) ➀ x +
5
➁ x +
3
Equations and Inequalities II
y
=7
2
y
=7
4
1. ➀
➁
2. Tables of values
3. Graph
➀
x
y
y
➁
x
y
x
4. ...............................................................................................................
5. • Solution-pair: ...................................................................................
• Type of lines: ...................................................................................
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MTH-4101-2
Equations and Inequalities II
d) ➀ – 0.5x + y – 3 = 0
➁ x – 2y + 6 = 0
1. ➀
➁
2. Tables of values
3. Graph
➀
x
y
y
x
➁
x
y
4. ...............................................................................................................
5. • Solution-pair: ...................................................................................
• Type of lines: ...................................................................................
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Answer Key
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MTH-4101-2
Equations and Inequalities II
e) ➀ x + y – 2 = 0
2
➁ x +y–5 =0
2
1. ➀
➁
2. Tables of values
3. Graph
➀
x
y
y
➁
x
x
y
4. ...............................................................................................................
5. • Solution-pair: ...................................................................................
• Type of lines: ...................................................................................
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Answer Key
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3
MTH-4101-2
Equations and Inequalities II
f) ➀ 4x + y = 19
➁ y=5
1. ➀
➁
2. Tables of values
3. Graph
y
➀
x
y
x
➁
x
y
4. ...............................................................................................................
5. • Solution-pair: ...................................................................................
• Type of lines: ...................................................................................
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MTH-4101-2
Equations and Inequalities II
g) ➀ x = 4
➁ 2y = 6
1. ➀
➁
2. Tables of values
3. Graph
➀
x
y
y
➁
x
x
y
4. ...............................................................................................................
5. • Solution-pair: ...................................................................................
• Type of lines: ...................................................................................
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Answer Key
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3
MTH-4101-2
Equations and Inequalities II
3. In each case, identify among the 3 given systems of equations the system
which corresponds to the graph on the right. Give your answer by recording
the letter corresponding to your choice in the space provided at the bottom of
the list.
y
=3
a) A: ➀ 2x +
5
3
➁ x – 2y = 4
B: ➀ – 6x +
➁ 2x –
y
= – 29
6
y
=8
3
C: ➀ 7x + 2y = 31
2
➁ x – 5y = – 23
3
y
(6, 5)
•
(–9, 4)
•
1
x
1
•
(10, –2)
The graph in the figure above corresponds to the system of equations
described in ............................ .
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MTH-4101-2
Equations and Inequalities II
b) A: ➀ 0.5x + 2y = 19 B: ➀ x –
2
➁ 1x– 1y = 1
➁ x +
7
5
7
y
= 11
5
2
y
=6
2
C: ➀
x – y =0
10 7
➁ x – y = –3
y
(0, 12)
•
• (7, 10)
1
1
•
(3, 0)
x
The graph in the figure above corresponds to the system of equations
described in ............................ .
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Answer Key
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MTH-4101-2
1.3
Equations and Inequalities II
SUMMARY ACTIVITY
1. Complete the following sentences by writing the missing term(s) in the blank
spaces.
A system of equations of the first degree in two variables is a set of at
least .................... equations of the first degree in ................... variables for
which you wish to determine the common ....................... .
This system is represented graphically with as many ................... as there
are equations.
The solution of the system is the point ................ to these lines. There
are ........... possible types of systems.
A system of ................... lines has a unique solution. In this case, the solution
is given by a solution- ............................. representing the point which is
...................... to the two lines in the graph.
A system of .................. and ................ lines does not have a solution.
A system of ..................... and .................... lines has an infinite number of
common ............... . In this case the solution comprises the set of ...............
the points that are common to these ................. lines.
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MTH-4101-2
Equations and Inequalities II
2. Complete the flowchart of the steps involved in solving a system of equations
of the first degree in two variables graphically by describing each of the
operations performed in the lefthand column.
➀ x+y–5=0
1.
➁ 9x – 2y – 23 = 0
➁ y = 9x – 23
2
2
➀ y = –x + 5
➀
2.
➁
x
y
x
y
–3
8
0
– 23
2
0
5
2
4
1
–5
2
13
2
4
y
3.
(– 3, 8)•
• 4, 13
2
(0, 5) •
2
• (4, 1)
2
•
➁
•
x
2, – 5
2
0, – 23
2
➀
4.
These two lines intersect at point (3, 2).
5.
Solution: (3, 2)
Type of lines: intersecting
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Answer Key
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MTH-4101-2
1.4
Equations and Inequalities II
THE MATH WHIZ PAGE
The Hare and the Tortoise, a Well-known Fable
You no doubt have heard the famous fable of the hare and the tortoise?
Here is a modern version of it. The hare and tortoise are getting ready
for a big race. Sure of himself, the hare gives his friend a 2 000 metre
lead; he knows that he is much faster than the tortoise. The tortoise
only advances at a rate of 50 m/min while the hare moves at
350 m/min. The following table of values shows their respective
positions.
Hare:
Tortoise:
time in min
0
2
distance covered in m
0
700
time in min
0
2
2 000
2 100
distance covered in m
4
4
When will the hare overtake the tortoise? Complete the table of values
below and use the Cartesian plane which follows to determine the
time required!
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Answer Key
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Distance in metres
MTH-4101-2
Equations and Inequalities II
y
Scale used:
x-axis
^ 0.5 min
0.5 cm =
y-axis
0.5 cm ^
= 100 m
Time in minutes
x
Fig. 1.10 The hare and the tortoise
What are the coordinates of the point of intersection? ......................
How far from the start will the hare overtake the tortoise? ..............
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