Systems of Equations Unit
(Level IV Academic Math)
Draft
NSSAL
C. David Pilmer
©2008
(Last Updated: October 2013)
Use our online
math videos.
YouTube:
nsccalpmath
This resource is the intellectual property of the Adult Education Division of the Nova Scotia
Department of Labour and Advanced Education.
The following are permitted to use and reproduce this resource for classroom purposes.
 Nova Scotia instructors delivering the Nova Scotia Adult Learning Program
 Canadian public school teachers delivering public school curriculum
 Canadian nonprofit tuition-free adult basic education programs
The following are not permitted to use or reproduce this resource without the written
authorization of the Adult Education Division of the Nova Scotia Department of Labour and
Advanced Education.
 Upgrading programs at post-secondary institutions
 Core programs at post-secondary institutions
 Public or private schools outside of Canada
 Basic adult education programs outside of Canada
Individuals, not including teachers or instructors, are permitted to use this resource for their own
learning. They are not permitted to make multiple copies of the resource for distribution. Nor
are they permitted to use this resource under the direction of a teacher or instructor at a learning
institution.
Acknowledgments
The Adult Education Division would like to thank the following university professors for
reviewing this resource to ensure all mathematical concepts were presented correctly and in a
manner that supported our learners.
Dr. Genevieve Boulet (Mount Saint Vincent University)
Dr. Robert Dawson (Saint Mary’s University)
The Adult Education Division would also like to thank the following NSCC instructors for
piloting this resource and offering suggestions during its development.
Charles Bailey (IT Campus)
Elliott Churchill (Waterfront Campus)
Barbara Gillis (Burridge Campus)
Barbara Leck (Pictou Campus)
Suzette Lowe (Lunenburg Campus)
Floyd Porter (Strait Area Campus)
Brian Rhodenizer (Kingstec Campus)
Joan Ross (Annapolis Valley Campus)
Jeff Vroom (Truro Campus)
Table of Contents
Introduction to Level IV Academic Math (for Learners) ………………………….....
Introduction to Systems of Equations Unit (for Learners) …………………………...
Tracking Your Progress……………………………………………………………….
Negotiated Completion Date………………………………………………………….
Introduction to Level IV Academic Math (for Instructors) …………………………..
The Big Picture ……………………………………………………………………….
Course Timelines ……………………………………………………………………..
Customized Online Videos …………………………………………………………...
ii
iii
iii
iii
iv
vi
vii
vii
2 by 2 System of Equations
Interpretation Involving Two Linear Functions………………………………………
Solving Systems of Equations Graphically…………………………………………...
Solving Systems by Substitution……………………………………………………..
Solving Systems Using Elimination………………………………………………….
Inconsistent and Equivalent Equations……………………………………………….
Putting It Together Part 1 ……………………………………………………………
1
6
12
20
31
34
Planes in 3-Space
Introduction to Three Dimensions…………………………………………………….
The Phone Plan………………………………………………………………………..
Graphing Planes……………………………………………………………………….
The Plane Truth……………………………………………………………………….
From Graphs to Equations…………………………………………………………….
3D Graphs in the Real World…………………………………………………………
Putting It Together Part 2 ……………………………………………………………
38
42
47
52
56
62
71
3 by 3 System of Equations
Systems of Equations with Three Variables…………………………………………..
Checking Answers Using a TI-83 or TI-84…………………………………………...
Making Predictions……………………………………………………………………
Word Problems………………………………………………………………………..
77
84
87
89
Appendix
Grid for “The Phone Plan” Activity…………………………………………………..
3D Coordinate Graph Paper…………………………………………………………..
94
95
Answers……………………………………………………………………………….
97
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Introduction to Level IV Academic Math (for Learners)
Welcome to Level IV Academic Math. In this 200 hour, double credit course, you will learn
about set notation, systems of equations, graphing in 3D, sinusoidal functions, quadratic
functions, quadratic equations, exponential functions, logarithms, rational expressions, radicals,
and inferential statistics. There is an extensive amount of math content in this course, yet
learners, who are appropriately placed in the course and who work diligently, will be able to
complete this course within one school year. The developers of this math curriculum and its
resources, and the NSCC math instructors have the following expectations for all of their Level
IV Academic Math learners.
Expectations:

Learners are expected to attend most, if not all, scheduled classes. Most learners need the
200 hours of scheduled class time to complete the program. Failure to attend will likely
mean that you will not complete the course within the school year.

Learners are expected to be self-motivated, have a strong work ethic and use class time
wisely. Your instructor will be there to support you throughout the course but ultimately
the motivation to complete the work will come from you.

Learners are expected to be able to read and comprehend written explanations provided
in their math resources. With our flexible system of instruction (FSI), learners in the
same classroom are often working on different materials. It is unreasonable to expect an
instructor in this environment to be able to introduce every new topic to every learner.
For this reason, the Level IV Academic Math faculty working group spent hundreds of
hours creating resources that have complete explanations and numerous worked
examples. It is your responsibility to read these materials and understand many of the
concepts independently of your instructor.

When learners encounter difficulties with a particular mathematical concept, they are
expected to look for solutions or assistance beyond simply asking their instructor. In the
FSI environment, an instructor can be tied up with multiple learner issues and not be able
to address your concern in a timely manner. For this reason, you may have to remedy the
situation yourself. This may mean asking another learner for assistance, viewing one of
our Level IV Academic Math videos found on YouTube (Channel: nsccalpmath),
accessing other online videos (e.g. Khan Academy, Patrick Just Math Tutorials), or
referring to other textbooks found in your instructor's classroom.

Learners are expected to complete all the discovery activities in their math resources.
These activities require learners to think, which requires both effort and time on the part
of learner. There are only seven such activities in the whole course (three in the final
unit), and most include an accompanying YouTube video that provides the necessary
introduction and explanation. Use these videos.

Learners will likely have to do one to two hours per week of math work outside of
regularly scheduled classes. This allows one to more quickly transition through the
course and forces learners to review concepts discussed and/or worked on during regular
class time.
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Introduction to Systems of Equations Unit (for Learners)
This unit is designed specifically for ALP Level IV Academic Math. Prior to starting this unit,
learners should:
 be able to solve a variety of linear equations,
 be able to interpret a variety of graphs, and
 have a solid understanding of linear functions.
Tracking Your Progress
This page allows you to keep track of your progress through this material.
2 by 2 System of Equations
Interpretation Involving Two Linear Functions….
Solving Systems of Equations Graphically………
Solving Systems by Substitution…………………
Solving Systems Using Elimination……………..
Inconsistent and Equivalent Equations…………..
Putting It Together Part 1 ………………………..
Date Started
Date Completed
Date Started
Date Completed
Date Started
Date Completed
1
6
12
20
31
34
Planes in 3-Space
Introduction to Three Dimensions………………..
The Phone Plan…………………………………..
Graphing Planes………………………………….
The Plane Truth…………………………………..
From Graphs to Equations………………………..
3D Graphs in the Real World…………………….
Putting It Together Part 2 ………………………..
38
42
47
52
56
62
71
3 by 3 System of Equations
Systems of Equations with Three Variables……..
Checking Answers Using a TI-83 or TI-84………
Making Predictions………………………………
Word Problems…………………………………..
77
84
87
89
Negotiated Completion Date
After working for a few days on this unit, sit down with your instructor and negotiate a
completion date for this unit.
Start Date:
_________________
Completion Date:
_________________
Instructor Signature: __________________________
Student Signature:
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Introduction to Level IV Academic Math (for Instructors)
In a 2006 survey, NSCC/ALP instructors identified a need to revisit the Level IV Academic
Math curriculum. Through the faculty working group (FWG), it was learned that instructors had
concerns with the excessive amount of time required to complete the existing curriculum, the
over reliance on the prescribed textbook, the lack of relevant real world applications, and the
actual content, which appeared more pre-calculus in nature. From the FWG and Department's
perspective, we had well-founded concerns that the existing curriculum did not align with grade
11 and 12 public school academic math curriculum elsewhere in this country. The existing
course covered many concepts (e.g. absolute value equations and inequalities, conic sections,
rational equations, irrational equations) traditionally found in public school pre-calculus
programs, did not incorporate the use of technology, lacked any multi-step, multi-concept
application questions, and failed to incorporate any discovery learning: critical components in
any public school curriculum. Obviously, changes needed to be made. The FWG worked
diligently over two years to ensure that our revised curriculum and its accompanying resources
met the standards expected from learners in similar programs elsewhere in our country, while
still meeting the needs of our learners.
Creating an academic math course in a learning environment which relies on a flexible system of
instruction (FSI) was a challenging task for the faculty working group. The biggest issue
centered on the selection or development of a student resource. We examined off-the-shelf postsecondary products from the major publishers but found that they lacked the andragogically
sound approaches needed for our learners and/or failed to align with our proposed curriculum.
Similar products developed for Canadian public high schools were problematic in that they were
designed for a traditional delivery model, which relied heavily on regular and ongoing direction
from a teacher. We experienced the same problem when attempting to select student resources
for Level IV Graduate Math. As with Graduate Math, the Academic Math faculty working
group chose to develop their own student resources.
Hundreds of hours were spent developing these resources. They were piloted by the faculty
working group members, and reviewed by those members, a math education professor from
MSVU, and a mathematics professor from SMU. The completed resources were deemed
andragogically sound, mathematically sound, and appropriate for our learners and our FSI
delivery model. We recognized that some learners would be intimidated by the amount of
reading required for this course and its accompanying resource. For this reason, we created 86
videos with titles that correspond to sections in the locally developed resources. We posted them
to YouTube under the channel name of nsccalpmath. All learners should be strongly encouraged
to use these videos. This is particularly important in relation to the seven discovery activities
found in this course. Learners often need a brief introduction before starting such activities;
however, instructors are often strapped for time to offer these introductions. The videos provide
a nice compromise.
All ALP math instructors have access to the Microsoft Word versions of the Academic Math
units on SharePoint. Instructors are welcome to alter the MS Word versions of these units to
meet the needs of their learners. When modifying a locally developed math resource, instructors
should refer to the curriculum document to ensure that all of the required outcomes are still being
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met in their modified version of the resource. These units are the intellectual property of the
Department of Labour and Workforce Development (LWD), therefore modified versions should
still include the LWD copyright information.
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The Big Picture
The following flow chart shows the optional bridging unit and the eight required units in Level
IV Academic Math. These have been presented in a suggested order.
Bridging Unit (Recommended)
 Solving Equations and Linear Functions
Describing Relations Unit
 Relations, Functions, Domain, Range, Intercepts, Symmetry
Systems of Equations Unit
 2 by 2 Systems, Plane in 3-Space, 3 by 3 Systems
Trigonometry Unit
 Pythagorean Theorem, Trigonometric Ratios, Law of Sines,
Law of Cosines
Sinusoidal Functions Unit
 Periodic Functions, Sinusoidal Functions, Graphing Using
Transformations, Determining the Equation, Applications
Quadratic Functions Unit
 Graphing using Transformations, Determining the Equation,
Factoring, Solving Quadratic Equations, Vertex Formula,
Applications
Rational Expressions and Radicals Unit
 Operations with and Simplification of Radicals and Rational
Expressions
Exponential Functions and Logarithms Unit
 Graphing using Transformations, Determining the Equation,
Solving Exponential Equations, Laws of Logarithms, Solving
Logarithmic Equations, Applications
Inferential Statistics Unit
 Population, Sample, Standard Deviation, Normal Distribution,
Central Limit Theorem, Confidence Intervals
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Course Timelines
Academic Level IV Math is a two credit course within the Adult Learning Program. As a two
credit course, learners are expected to complete 200 hours of course material. Since most ALP
math classes meet for 6 hours each week, the course should be completed within 35 weeks. The
curriculum developers have worked diligently to ensure that the course can be completed within
this time span. Below you will find a chart containing the unit names and suggested completion
times. The hours listed are classroom hours. In an academic course, there is an expectation that
some work will be completed outside of regular class time.
Unit Name
Minimum
Completion Time
in Hours
0
6
18
18
20
36
12
20
20
Total: 150 hours
Bridging Unit (optional)
Describing Relations Unit
Systems of Equations Unit
Trigonometry Unit
Sinusoidal Functions Unit
Quadratic Functions Unit
Rational Expressions and Radicals Unit
Exponential Functions and Logarithms Unit
Inferential Statistics Unit
Maximum
Completion Time
in Hours
20
8
22
20
24
42
16
24
24
Total: 200 hours
As one can see, this course covers numerous topics and for this reason may seem daunting. You
can complete this course in a timely manner if you manage your time wisely, remain focused,
and seek assistance from your instructor when needed.
Customized Online Videos
There are 18 locally developed videos that have been created to match the various sections in this
Systems of Equations Unit. They have been posted on YouTube. When on the site, the easiest
way to find them is to search for nsccalpmath or Dave Pilmer. If you are unable to attend a class
or struggling at home with a new math concept, these videos will be of great assistance. Please
use them.
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Interpretation Involving Two Linear Functions
Often more than one line graph is drawn on the coordinate system. In this section, you will learn
to extract information from two similar real life situations graphed on the same coordinate
system. You will have to learn the significance of the point of intersection between the two
linear functions.
Cost in Dollars
Example:
Janice needs some gravel for her driveway. She can purchase it from two different companies.
The two companies charge different trucking fees and different rates for each tonne of gravel.
The following graphs show the relationship between the cost, c, in dollars and the number, n, of
tonnes of gravel ordered for the two companies.
110
105
(a) Determine the equation that describes the cost in terms
100
Company A
95
of the number of tonnes for Company A.
90
85
(b) Determine the equation that describes the cost in terms
80
75
of the number of tonnes for Company B.
Company B
70
65
(c) What are the trucking fees charged by the two
60
companies?
55
50
(d) How much is each company charging per tonne of the
45
40
gravel?
35
30
(e) How many tonnes of gravel would you need to order for
25
20
the cost to be the same from the two companies?
15
10
(f) When is it more economical to order from Company A?
5
0
(g) When is it more economical to order from Company B?
0
1
2
3
4
5
6
7
8
9
Tonnes of Gravel
Answers:
(a) Company A:
c-intercept = 10
slope 
rise 15

run 1
equation: c  15n  10
(b) Company B:
c-intercept = 25
slope 
rise 10

run 1
equation: c  10n  25
(c) Company A: Trucking Fee of $10
Company B: Trucking Fee of $25
(d) Company A: $15 per tonne
Company B: $10 per tonne
(e) Look for the point of intersection between the two straight line graphs. The coordinates of
this point are (3, 55). It means that if Janice orders 3 tonnes from either of these two
companies, she will have to pay $55 to have it delivered and dumped on her property.
(f) It is more economical to use Company A when Janice orders less than 3 tonnes of gravel.
(g) It is more economical to use Company B when Janice orders more than 3 tonnes of gravel.
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Questions:
1. Two different pumps are being used to empty two different storage tanks. The following
graphs show the relationship between the volume, v, of fluid remaining in the tanks and time,
t. The volume is measured in litres. The time is measured in minutes.
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120
110
100
Tank B
90
Volume in Litres
(a) Determine the equation that describes the volume
in terms of time for Tank A.
(b) Determine the equation that describes the volume
in terms of time for Tank B.
(c) How much fluid was initially in each of the
storage tanks?
(d) How fast is the fluid being removed from each of
the storage tanks?
(e) How much fluid remains in Tank B at 5 minutes?
(f) At what time do both containers have the same
amount of fluid?
(g) During what time interval does Tank A have less
fluid than Tank B?
80
70
60
Tank A
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
Time in Minutes
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2. Nancy is planning a surprise party for her parents who are celebrating their 40th wedding
anniversary. She’s decided to rent a banquet hall that also serves meals. She has two
banquet halls to choose from. The two halls charge different amounts for the hall and meals.
The following graphs show the relationship between the cost, c, of banquet expenses and the
number, n, of people who agree to attend and eat at the banquet.
2100
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2000
1900
1800
1700
1600
1500
1400
1300
Cost in Dollars
(a) Determine the equation that describes the cost in
terms of the number of people for Hall A.
(b) Determine the equation that describes the cost in
terms of the number of people for Hall B.
(c) What are the slopes and what do they represent in
these situations?
(d) What are the c-intercepts and what do they represent
in these situations?
(e) How many people would Nancy have to invite to
insure that the cost for either of the banquet halls
would be the same?
(f) If Nancy expects 30 people to attend, then which
hall should she select to minimize her expenses?
(g) When is it more economical to use Hall A?
(h) How much does it cost to use Hall A if 10 people
attend?
1200
Hall A
1100
1000
900
Hall B
800
700
600
500
400
300
200
100
0
0
10
20
30
40
50
60
Number of People
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3. Kamala’s son and daughter borrowed money from her. They are paying her back a little bit
each day. The following graph shows the relationship between the amount, a, of money each
owes her and time, t. The amount of money is measured in dollars. The time is measured in
weeks.
200
Amount Owed in Dollars
(a) Determine the equation that describes the amount of
money owed by Kamala’s son in terms of time.
(b) Determine the equation that describes the amount of
money owed by Kamala’s daughter in terms of time.
(c) What are the slopes and what do they represent in
these situations?
(d) What are the c-intercepts and what do they represent
in these situations?
(e) What are the t-intercepts and what do they represent
in these situations?
(f) Over what time period does Kamala’s son owe more
to Kamala than the daughter?
(g) Over what time period does Kamala’s daughter owe
more to Kamala than the son?
(h) When do the son and daughter owe the same amount
to their mother? How much do they both owe at that
time?
(i) When will Kamala’s daughter still owe her mother $150?
(j) How much will Kamala’s son owe on day 20?
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Kamala’s Daughter
180
170
160
150
140
130
120
110
100
90
Kamala’s Son
80
70
60
50
40
30
20
10
0
0
5
10
15
20
25
30
Time in Days
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4. Hamid needs to rent a car for the day. He has two rental agencies to choose from. In both
cases, the rental agencies charge a flat rate plus an amount based on the number of kilometres
he travels. The following graph shows the relationship between the rental car costs, c, and
the distance, d, travelled in the rental car. The cost is measured in dollars. The distance is
measured in kilometres.
100
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95
90
85
80
75
70
Rental Charges in Dollars
(a) Determine the equation that describes the cost of
rental car from Agency A in terms of the distance
travelled.
(b) Determine the equation that describes the cost of
rental car from Agency B in terms of the distance
travelled.
(c) If Hamid travels 100 km in Agency A’s rental car,
then how much will it cost?
(d) If Hamid’s rental charge is $300 for Agency B’s
car, then how many kilometres did he travel?
(e) What does Agency A charge for just picking up
the car (i.e. the flat rate)?
(f) What does Agency B charge per kilometre?
(g) How many kilometres would Hamid have to
travel in the rental car to insure that the charges
from either of the companies would be the same?
What would the charge be?
(h) If he only plans to travel 250 kilometres with the
rental car, then which agency should he go with?
65
Agency A
60
55
50
45
40
Agency B
35
30
25
20
15
10
5
0
0
50
100
150
200
250
300
Distrance Travelled in Kilometres
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Solving Systems of Equations Graphically
In the previous section, Interpretation Involving Two Linear Functions, you had to find the point
of intersection between two linear functions. Although you didn’t realize it, you were actually
solving a system of two equations in two variables when you found the point of intersection.
The two equations were the equations of the two linear functions. The two variables were the
two coordinates for the point of intersection.
There are many situations when we need to find the point of intersection that can also be called
the point that satisfies both equations simultaneously. This second part of statement is best
explained using an example.
Example 1:
Solve the following system of equations graphically. Check your answer.
5x  2 y  8
x y 3
Answer:
Change both equations to the slope, y-intercept form (i.e. y  mx  b ).
5x  2 y  8
x y3
2 y  5 x  8
 y  x  3
2 y  5x 8
y  x3


2
2
2
5
y x4
2
Plot both graphs on the same coordinate system
by first plotting the y-intercepts and then rising
and running from that point based on the slope.
5
y x4
2
y-intercept = 4
4
3
y  x 3
2
1
y-intercept = -3
0
-2
rise  5
slope =

run
2
y
5
rise 1
slope =

run 1
-1
0
-1
1
2
3
4
x
(2, -1)
-2
-3
-4
The point of intersection is (2, -1). That means that the solution to this system of
equations is x = 2 and y = -1.
To check the answer, substitute the values of the variables into both of the original
questions and see if they satisfy the equations (i.e. see if both sides of the equation are
still equal to each other).
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Check:
5x  2 y  8
52  2 1  8
10  (2)  8
8  8 satisfies the first equation
x y3
2  (1)  3
3  3 satisfies the second equation
The answer x = 2 and y = -1 is correct.
Example 2:
Solve the following system of equations graphically. Check your answer.
x  3 y  18
2 x  y  1
Answer:
Change both equations to the slope, y-intercept form (i.e. y  mx  b ).
x  3 y  18
2 x  y  1
 3 y   x  18
y  2 x  1
 3 y  x 18


3 3 3
1
y  x6
3
y-intercept = 6
slope =
rise 1

run 3
y
y-intercept = -1
slope =
8
rise  2

run
1
7
(-3, 5)
6
5
The point of intersection is (-3, 5). That
means that the solution to this system of
equations is x = -3 and y = 5.
4
3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
x
-1
-2
-3
Check:
x  3 y  18
 3  35  18
 3  15  18
 18  18 satisfies the first equation
2 x  y  1
2 3  5  1
 6  5  1
 1  1 satisfies the second equation
The answer x = -3 and y = 5 is correct.
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Example 3:
After hurricane Juan, Tom needed to rent a chainsaw to clear up a few small trees that had been
uprooted on his property. Tools-R-Us charges a base fee of $18, plus $6 per hour. Another
company, Tool Depot, charges a base fee of $24, plus $4 per hour.
(a) Determine the equation of the linear function that describes the Tools-R-Us rental charge, C,
in terms of the number of hours, h, that the chainsaw is rented.
(b) Determine the equation of the linear function that describes the Tool Depot rental charge, C,
in terms of the number of hours, h, that the chainsaw is rented.
(c) Determine the point of intersection graphically.
(d) Explain what the point of intersection represents in this situation.
Answers:
(a) C  6h  18
(b) C  4h  24
C  6h  18
C  4h  24
y-intercept = 18
y-intercept = 24
slope =
rise 6

run 1
slope =
rise 4

run 1
The point of intersection is (3, 36).
(d) If Tom rents a chainsaw from either of
these companies for 3 hours, he has to pay
the same amount, $36.
Rental Cost
(c) Graph both linear functions on the same
coordinate system. The most challenging
thing with this question is choosing
appropriate scales on the two axes.
46
44
42
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
1
2
3
4
5
Time
Questions
1. Determine the slope and y-intercept of each line.
(a) 2 x  3 y  15
(b) 8x  6 y  12  0
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(c) 0.8x  0.2 y  1  0
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2. For each system of equations, determine whether the ordered pair is the solution.
(a) 3x  5 y  1 and 2 x  7 y  3 ; (-2, 1)
(b) 4 x  2 y  2 and 6 x  y  12 ; (3, -5)
(c) 5x  3 y  7 and  2 x  4 y  14 ; (-1, -4)
3. Solve each of the following systems graphically and verify your solution.
(a) 2 x  3 y  12 and 4 x  3 y  6
7
6
5
4
3
2
1
0
-7
-6
-5
-4
-3
-2
-1
-1
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
-2
-3
-4
-5
-6
-7
(b)
x  2 y  10 and 5x  4 y  8
7
6
5
4
3
2
1
0
-7
-6
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
-6
-7
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(c)
x  y  4 and 2 x  5 y  5
7
6
5
4
3
2
1
0
-7
-6
-5
-4
-3
-2
-1
-1
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
-2
-3
-4
-5
-6
-7
(d)
x  2 y  8 and 2 x  y  1
7
6
5
4
3
2
1
0
-7
-6
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
-6
-7
(e)
x  y  1  0 and 0.6 x  y  1  0
7
6
(Hint: Express 0.6 as a fraction.)
5
4
3
2
1
0
-7
-6
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
-6
-7
Scrambled Answer Key for Question 3
(-5, -1)
(3, 2)
(-6, 4)
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(-2, 3)
10
(4, -3)
(5, -4)
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4. Candice needs to have two small rooms in her house painted. She is going to hire a painter.
The first painter said that he would charge a flat fee of $150 to cover materials and travel,
plus $30 per hour. The second painter said that she would charge a flat fee of $165, plus $25
per hour.
(a) Determine the equation of the linear function that describes the cost, c, of using the first
painter in terms of the number of hours, h, he works.
(b) Determine the equation of the linear function that describes the cost, c, of using the
second painter in terms of the number of hours, h, she works.
(c) Determine the point of intersection graphically.
(d) Explain what the point of intersection represents in this situation.
260
255
250
245
240
235
230
225
220
215
Cost
210
205
200
195
190
185
180
175
170
165
160
155
150
0
1
2
3
4
5
6
7
8
Hours of Labour
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Solving Systems by Substitution
In the previous section you learned how to solve systems graphically. There are a few problems
with this method.
 It’s a time consuming process.
 Many linear functions are challenging to graph accurately. (e.g. y  0.165x  22.7 )
 If the coordinates of the point of intersection do not appear to be integers, then it is
almost impossible to determine the correct solution using only the graphs.
There are two algebraic methods of solving systems of equation; substitution and elimination. In
this section, we will focus on substitution. With this algebraic technique we substitute one
equation into the other equation such that we change the question from two equations with two
variables to one equation with one variable. This is best understood by looking at a few
examples.
Example 1:
Solve the following system using substitution.
x  3 y  2 and 5x  4 y  12
Answer:
x  3 y  2
x  3 y  2
Choose one of the equations and isolate one
of the variables by expressing that variable in
terms of the other.
In this example, we took the first equation
and isolated x. It was the easiest thing to do.
5 x  4 y  12
5 3 y  2  4 y  12
5 3 y  2   4 y  12
 15 y  10  4 y  12
 15 y  4 y  12  10
 11 y  22
 11 y
22

 11  11
y  2
x  3 y  2
x  3 2   2
x  6  2
x  2  6
x4
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Substitute the expression for x into the other
equation. By doing this, we are now down to
one equation with one variable, y.
Now solve for y.
Now solve for x by substituting the value of y
into either one of the original equations.
The solution for this system is x = 4 and
y = -2.
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Example 2:
Solve the following system using substitution.
5x  2 y  1 and 4 x  y  7
Answer:
4x  y  7
 y  4 x  7
y  4x  7
Choose one of the equations and isolate one
of the variables by expressing that variable in
terms of the other. In this example, we took
the second equation and isolated y. It was
the easiest thing to do.
5 x  2 y  1
5 x  24 x  7   1
5 x  24 x  7   1
5 x  8 x  14  1
5 x  8 x  1  14
13 x  13
x 1
5 x  2 y  1
51  2 y  1
5  2 y  1
2 y  1  5
2 y  6
y  3
Example 3:
Solve the system.
Answer:
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Substitute the expression for y into the other
equation. By doing this, we are now down to
one equation with one variable, x.
Now solve for x.
Now solve for y by substituting the value of x
into either one of the original equations.
The solution for this system is x = 1 and
y = -3.
2 x  3 y  18 and 3x  4 y  7
2 x  3 y  18
2 x  3 y  18
3
x   y9
2
3 x  4 y  7
 3

3  y  9   4 y  7
 2

9
 y  27  4 y  7
2
 9 
2  y   227   24 y   2 7 
 2 
 9 y  54  8 y  14
 17 y  14  54
 17 y  68
y4
13
3 x  4 y  7
3 x  44   7
3 x  16  7
3 x  7  16
3x  9
x3
The answer is x = 3
and y = 4.
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C. D. Pilmer
Example 4:
Porter Pools charges $30 for a service call, plus $45 per hour for labour. Bailey Pools and Spas
charges $50 for a service call, plus $40 per hour for labour. Find the length of a service call for
which both companies charge the same amount.
Answer:
Determine the equations of the linear functions that describe the cost, c, of the service call in
terms of the time, t, to complete the jobs.
Porter Pools: c  45t  30
Bailey Pools and Spas: c  40t  50
Solve by substitution.
c  45t  30
40t  50  45t  30
40t  45t  30  50
 5t  20
 5t  20

5
5
t4
We only need to solve for the time, t, for this question. Both companies charge the same
amount if it takes 4 hours to complete the service call.
Example 5:
The difference of two numbers is 5. Twice the larger number plus three times the smaller
number is 45. Find the numbers.
Answer:
Let x represent the larger number.
Let y represent the smaller number.
Determine the two equations.
x  y  5 and 2 x  3 y  45
Solve by substitution.
x y5
2 x  3 y  45
x  y5
2 y  5  3 y  45
2 y  10  3 y  45
2 y  3 y  45  10
5 y  35
y7
x y5
x7 5
x 57
x  12
The larger number is 12. The smaller number is 7.
Note:
At the point of intersection, the x-coordinates for both graphs are equal, and the y-coordinates are
equal. The method of substitution uses this relationship to replace one variable with an
expression in terms of the other variable. This means that we go from two equations with two
unknown variables to one equation with one unknown variable.
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Questions:
1. Solve the following systems by substitution.
(a) 4 x  y  13 and 3x  5 y  31
(b) 2 x  6 y  14 and x  3 y  1
(c) 2 x  y  9 and 4 x  5 y  39
(d) 3x  4 y  7 and  x  3 y  11
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(e) y  2 x  5 and y  3x  15
(f) 3x  y  4 and 5x  2 y  25
(g) m  4n  4 and  2m  7n  6
(h) 4r  s  19 and 2r  3s  3
Scrambled Answer Key for Question 1
5, -2
-4, -2
2, 5
-1, 8
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-3, 5
16
4, 1
-6, -5
6, 3
4, 3
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2. Solve the following systems by substitution.
(a) x  3 y  5 and x  5 y  7
(b) 2 g  4h  10 and 6 g  3h  15
(c) 3 p  2q  14 and 2 p  4q  4
(d) 4 x  3 y  4 and 6 x  y  5
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3. Ajay is draining two different tanks using two different pumps. The first tank holds 85 litres
and is being drained by a pump that removes 7 litres per minute. The second tank holds 64
litres and is being drained by a pump that removes 4 litres per minute. Assuming that the
tanks were initially fully and that the pumps were turned on at the same time, at what time
will the tanks have the same number of litres of fluid?
4. Nita is going to organize a 50th wedding anniversary party for her parents. She wants to rent
a banquet hall and have a buffet. The local legion hall charges a flat rate of $185 for the hall,
plus $10 per person for the buffet. The voluntary fire department and its auxiliary charge a
flat rate of $215 for their hall and $8.50 per person for the buffet. Determine the number of
people who would have to attend in order for the two organizations to charge the same
amount.
5. The sum of two numbers is 12. If you double the first number and triple the second number,
then the new sum is 27. Find the numbers.
6. The difference of two numbers is 11. The larger number is five more than double the smaller
number. Find the numbers.
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7. The perimeter of a rectangular garden is 174 metres. The length of the garden is twelve more
than twice the width. Find the length and width.
8. Two gears have a total of 75 teeth. The smaller gear has 15 more teeth
than half the number of teeth on the larger gear. How many teeth does
each gear have?
9. If Jacob buys 4 hamburgers and 1 container of french fries, he pays $15. If he purchases 6
hamburgers and 3 containers of french fries, he pays $24. How much does hamburger cost?
How much does a container of french fries cost?
10. Your parents have retired in the Florida Keys. They have to purchase an air conditioner for
their condominium. Unit A costs $650 to purchase and $18 per month to operate. Unit B
costs $574 to purchase and $22 per month to operate. Find the number of months at which
the total cost of either unit is the same.
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Solving Systems Using Elimination
Up to this point we have learned how to solve systems of linear equations graphically and by
substitution. Although the method of substitution is vastly superior to the graphical method, it
can be problematic with some questions where neither of the equations have a coefficient that is
equal to 1 or -1. In many cases, it forces us to work with fractions.
Example: Solve the system. 3x  2 y  7 and 4 x  3 y  8
Answer:
3x  2 y  7
 2 y  3 x  7
 2 y  3x 7


2
2 2
3
7
y x
2
2
4x  3y  8
7
3
4 x  3 x    8
2
2
9
21
4x  x 
8
2
2
 9   21 
24 x   2 x   2   28
2   2 
8 x  9 x  21  16
 1x  5
x5
3x  2 y  7
35  2 y  7
15  2 y  7
 2 y  8
y4
The answer is x = 5
and y = 4.
Although we are capable of dealing with these types of questions, it would be nice if we could
find an easier method. Elimination is such a method. Elimination is an algebraic approach to
solving systems of linear equations in which two equations are added together in such a manner
that one of the unknown variables is eliminated. In many cases, to insure this elimination of a
variable, one or both of the equations must be altered before the two equations are added.
Example 1:
Solve the following system using elimination.
Answer:
3x  2 y  7
4x  3 y  8
4
 -3
12 x  8 y  28
 12 x  9 y  24
y4
3x  2 y  7
3 x  24   7
3 x  15
x5
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3x  2 y  7 and 4 x  3 y  8
Choose a variable to eliminate. We will eliminate
the x. Multiply each equation by a number that
gives the opposite coefficients for that variable in
both equations. If we multiply the first equation
by 4, the new coefficient of x will be 12. If we
multiply the second equation by -3, the new
coefficient of x will be -12. When we add the
two equations the 12x and -12x will cancel out
thus eliminating the variable x.
Now that we have solved for one variable, y, use
either of the two original equations to solve for
the other variable, x.
The answer is x = 5 and y = 4.
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Example 2:
Solve the following system using elimination.
Answer:
5 x  6 y  16
2 x  3 y  1
2
5 x  6 y  16
4x  6 y   2
9 x  18
x  2
5 x  6 y  16
5 2  6 y  16
 10  6 y  16
 6 y  6
y 1
5x  6 y  16 and 2 x  3 y  1
We will eliminate the y. We will leave the first
equation so that the coefficient of y will remain 6. If we multiply the second equation by 2, the
new coefficient of y will be 6. When we add the
two equations the -6y and 6y will cancel out thus
eliminating the variable y.
Now that we have solved for one variable, x, use
either of the two original equations to solve for
the other variable, y.
The answer is x = -2 and y = 1.
Example 3:
Solve the following system using elimination.
Answer:
2x  5 y  4
x  5 y  17
3 x  21
x7
x  5 y  17
7  5 y  17
 5 y  10
y  2
2 x  5 y  4 and x  5 y  17
On a few rare occasions, neither of the equations
has to be altered to eliminate a variable. This is
the case with this particular question. The 5y and
-5y will cancel out when the two equations are
added together.
Now that we have solved for one variable, x, use
either of the two original equations to solve for
the other variable, y.
The answer is x = 7 and y = -2.
Example 4:
0.5x  0.7 y  5  0 and
Solve the following system using elimination.
Answer:
0.5 x  0.7 y  5  0
0.5 x  0.7 y  5
2
1
x  y 1  0
3
5
Both equations have to be rearranged so that they
are in the form Ax  By  C .
2
1
x  y 1  0
3
5
2
1
x  y 1
3
5
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0.5 x  0.7 y  5
100.5 x   100.7 y   105
5 x  7 y  50
It would be far easier to eliminate a variable if the
coefficients were integers, rather than decimals
and fractions.
We can get rid of the decimals in the first
equation by multiplying everything on both sides
of the equation by 10.
2
1
x  y 1
3
5
2 
1 
15 x   15 y   151
3 
5 
10 x  3 y  15
We can get rid of the fractions in the second
equation by multiplying both sides of the
equation by 15; the least common multiple of
denominators 3 and 5.
5 x  7 y  50  -2  10 x  14 y  100
10 x  3 y  15
10 x  3 y  15
Now that the equations that are in far easier to
work with, solve for x and y using elimination.
 17 y  85
y5
5 x  7 y  50
5 x  75  15
5 x  35  50
5 x  15
x3
The answer is x = 3 and y = 5.
Example 5:
Tickets for a play cost $5 for adults and $3 for children. A total of 800 tickets were sold and the
total sales were $3600. How many adult tickets were sold?
Answer:
Let a represent the number of adult tickets sold
Let c represent the number of child ticket sold
Total Number Sold: 800 a  c  800
Total Sales: $3600
5a  3c  3600
Now solve for a and c using elimination.
a  c  800  -3  3a  3c  2400
5a  3c  3600
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5a  3c  3600
2a  1200
a  600
They sold 600 adult tickets.
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Example 6:
Janice has a total of $2000 to invest. She puts part of it in an investment that pays 6% per year.
The remaining money she puts in an investment that pays 8% per year. At the end of the one
year, Janice has earned $134 in interest. How much money did she put in each of the
investments?
Answer:
Let A represent the amount she invested at 6% per year (Investment Plan A)
Let B represent the amount she invested at 8% per year (Investment Plan B)
Total Amount Invested: $2000
A  B  2000
Interest Earned: $134
0.06 A  0.08B  134
A  B  2000
0.06 A  0.08B  134
 6
 6 A  6 B  12000
 100
6 A  8B  13400
2 B  1400
B  700
A  B  2000
A  700  2000
A  1300
Janice put $1300 in the investment that paid 6% interest per year. She put $700 in the
investment that paid 8% interest per year.
Example 7:
Andrew needs customized baseball hats with his company logo on them. He finds two local
companies that are able to produce the desired hats. Company A charges a $85 set-up fee, plus
$7.25 per hat. Company B charges a $120 set-up fee, plus $6 per hat. Under what circumstances
would Company A and Company B charge the same amount for the same number of hats?
Answer:
Find the equations of the two linear functions where c will represent the charges and n will
represent the number of hats that are produced.
Equation for Company A:
c  7.25n  85
Equation for Company B:
c  6n  120
Although this question is slightly easier to solve by substitution, we’ll use elimination. The
equations will have to be rearranged so that they are in the form Ax  By  C .
 7.25n  c  85
 6n  c  120
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 -1
7.25n  c  85
 6n  c  120
1.25n  35
n  28
The two companies would have
to manufacture 28 hats to keep
the costs the same.
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Questions
1. Solve each of the systems using elimination.
(a) 4 x  2 y  18 and 3x  5 y  20
(b) 6 x  5 y  19 and 5x  7 y  13
(c) 3g  2h  10 and 5g  9h  6
(d) 2m  7n  17 and 3m  2n  12
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(e) 5x  2 y  7 and 3x  4 y  27
(f) 6 x  7 y  14 and 3x  5 y  10
(g) 3 p  8q  39 and 7 p  4q  47
(h) 9 x  5 y  43 and 2 x  5 y  34
Scrambled Answer Key for Question 1
(0,-2)
(-4, 1)
(1, 6)
(3, -4)
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(5, -1)
25
(-5, -3)
(6, 4)
(7, -4)
(-2, 3)
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2. Solve the following systems using elimination.
(a) 2 y  3x  2 and 4 y  5x  8
(b) 3 p  2q  6  0 and 4 p  5q  31  0
(c) s  3r  15 and s  2r  5
(d) 3x  y  20 and 0.2 x  0.1y  1
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(e) 0.7 g  0.2h  2 and g  3h  16
(f) 0.07 x  0.06 y  200  0 and 4 x  5 y  50  0
(g)
1
1
x  y  1 and 2 x  3 y  22
4
2
(h)
1
1
p  q  4 and p  2q  2
3
2
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3. Marsha organized a fund raiser car wash for her son’s elementary school. They charged $7
to wash a car and $10 to wash a van. If they washed a total of 39 cars and vans, and earned
$321, how many cars were washed? How many vans were washed?
4. The sum of two numbers is 23. If the smaller number is doubled and then decreased by the
larger number, the answer is 4. Find the numbers.
5. An adult education instructor wants to purchase books from a second-hand bookstore for her
classroom. The softcover books cost $2. The hardcover books cost $5. If she buys 41 books
and pays $133, how many hardcover books did she purchase? How many softcover books
did she purchase?
6. Montez has to rent a truck to do some moving. Metro Truck Rental charges $60 per day plus
$0.18 per kilometre. U-Rent charges $75 for the day, plus $0.14 per kilometre. How far
would Montez have to travel with the truck in order that the charges from either rental
company would be the same?
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7. Neil has $1100 to invest. Some of the money is placed in an account that pays 7% interest
per year. The remaining money is placed in an account that pays 5% interest per year. If
after one year, he has accumulated a total of $70.60 in interest, how much was invested in
each of these accounts?
8. Nasrin runs a catering business. She is catering a wedding where guests must choose
between a chicken dinner and a beef dinner. The chicken dinners cost $15 per plate, and the
beef dinners cost $18 per plate. The 54 wedding guests ordered in advance, and the total cost
to prepare the dinner is $879. How many of each type of dinner will Nasrin be preparing?
9. Heather is ordering shirts for her daughter’s canoe club. The short-sleeved shirts cost $5
each. The long-sleeved shirts cost $7 each. She ended up ordering 42 shirts and spending
$236. How many of each type of shirt were ordered?
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Wrap-Up Statement:
In the last four sections you learned how to solve and check systems of equations questions
both in and out of context.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed the four sections titled “Interpretation
Involving Two Linear Functions”, “Solving Systems of Equations Graphically”, “Solving
Systems by Substitution”, and “Solving Systems Using Elimination.” Select your response
to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
I understand all of the concepts covered in the section,
“Interpretation Involving Two Linear Functions.”
(b) I do not need any further assistance from the instructor on the
material covered in this section.
(c) I do not need any more practice questions.
(d) I understand all of the concepts covered in the section,
“Solving Systems of Equations Graphically.”
(e) I do not need any further assistance from the instructor on the
material covered in this section.
(f) I can solve for two unknowns given two equations by graphing
the linear functions and finding the point of intersection. I do
not need any more practice questions.
(g) I understand all of the concepts covered in the section,
“Solving Systems by Substitution.”
(h) I do not need any further assistance from the instructor on the
material covered in this section.
(i) I can solve for two unknowns given two equations using
substitution. I do not need any more practice questions.
(j) I can solve word problems that require one to use substitution.
I do not need any more practice questions.
(k) I understand all of the concepts covered in the section,
“Solving Systems Using Elimination.”
(l) I do not need any further assistance from the instructor on the
material covered in this section.
(m) I can solve for two unknowns given two equations using
elimination. I do not need any more practice questions.
(n) I can solve word problems that require one to use elimination.
I do not need any more practice questions.
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Inconsistent and Equivalent Equations
Case 1: Inconsistent Equations
Ryan was asked to solve the following system of equations.
2 x  4 y  20 and 3x  6 y  6
He completed the following work but was unable to solve for x or y.
2 x  4 y  20  3
6 x  12 y  60
3x  6 y  6  2  6 x  12 y  12
0  48 ???
Questions Regarding Case 1
1. Did he make any mistakes in his calculations?
______
2. To check Ryan’s work, change both linear functions to their slope y-intercept forms
( y  mx  b) , and graph them on the same coordinate system.
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4
3
2
1
0
-7
-6
-5
-4
-3
-2
-1
-1
0
1
2
3
4
5
6
7
-2
-3
-4
-5
-6
-7
3. How are the two lines oriented with respect to each other?
4. Do the lines ever intersect?
______
5. Is there no solution, one solution or an infinite number of solutions to this system of
equations? Explain.
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Case 2: Equivalent Equations
Michelle was asked to solve the following system of equations.
0.2 x  0.1y  0.3 and  10 x  5 y  15
She completed the following work but was unable to solve for x or y.
0.2 x  0.1y  0.3
 50
 10 x  5 y  15
10 x  5 y  15
 10 x  5 y  15
0  0 ???
Questions Regarding Case 2
1. Did she make any mistakes in her calculations? ______
2. To check Michelle’s work, change both linear functions to their slope y-intercept forms
( y  mx  b) , and graph them on the same coordinate system.
7
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5
4
3
2
1
0
-7
-6
-5
-4
-3
-2
-1
-1
0
1
2
3
4
5
6
7
-2
-3
-4
-5
-6
-7
3. How are the two lines oriented with respect to each other?
4. Do the lines ever intersect?
______
5. Is there no solution, one solution or an infinite number of solutions to this system of
equations? Explain.
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Conclusions:

If two lines are parallel, they will never intersect. If you attempted to complete
substitution or elimination with these two equations, then you would end up with a false
statement like 0 = 48, as you did in Case 1. If a false statement occurs then you are
dealing with inconsistent equations and there is no solution for that system of equations.

If two lines are actually the same line, then they intersect at an infinite number of points.
If you attempted to complete substitution or elimination with these two equations, then
you would end up with the true statement 0 = 0, as you did in Case 2. If this true
statement occurs then you are dealing with equivalent equations and there are an infinite
number of solutions for that system of equations.
Questions:
1. Solve the following systems of equations.
(a)  9 x  6 y  15 and 6 x  4 y  10
(b) 4 x  3 y  5 and 2 x  5 y  9
(c) 10 x  4 y  2 and  25x  10 y  17
(d)  4 x  10 y  8 and 0.2 x  0.5 y  0.4
(e) 0.2 x  0.5 y  6 and 6 x  15 y  30
(f) 3x  4 y  17 and 2 x  y  4
2. For equation 4 x  5 y  7 , write an equation that would make the system inconsistent, and
another equation that would make the system equivalent.
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Putting It Together Part 1
In the last four sections we learned how to solve systems of equations (two equations with two
variables) graphically, by substitution, and using elimination. In this section, we will review
these three techniques with a focus on word problems.
Questions
1. Solve the following system using all three techniques.
7 x  4 y  24 and x  2 y  6
(a) Graphically
7
6
5
4
3
2
1
0
-7
-6
-5
-4
-3
-2
-1
-1
0
1
2
3
4
5
6
7
-2
-3
-4
-5
-6
-7
(b) Substitution
(c) Elimination
For the remaining questions, solve using an algebraic technique (i.e. substitution or elimination).
2. The sum of two numbers is 19. If you double the smaller number and then decrease it by the
larger number, the result is 5. Find the numbers.
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3. The two numbers differ by 8. If you double the larger number and then increase this by triple
the smaller number, the result is 81. Find the numbers.
4. Kiana is buying canned peas and canned corn for the local food bank. The peas cost $0.89
per can. The corn costs $1.19 per can. She bought 75 cans and paid $80.25. How many of
each type did she purchase?
5. If Thomas buys 6 hamburgers and 2 chili dogs, he pays $21. If he buys 4 hamburgers and 8
chili dogs, he pays $29. How much does a hamburger cost? How much does a chili dog
cost?
6. Two very large containers, that initially have some water in them, are being filled using
different pumps. The first container initially has 14 litres of water in it and is being filled at a
rate of 1.5 litres per minute. The second container initially has 5 litres of water in it and is
being filled at a rate of 2.1 litres per minute. At what time will the two containers have the
same amount of water?
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7. The perimeter of a rectangle is 68 cm. The length of the rectangle is one centimeter more
than double the width. Find the length and width of the rectangle.
8. Monique has $850 to invest. Some of the money is placed in an account that pays 7%
interest per year. The remaining money is placed in an account that pays 6% interest per
year. If after one year, she has accumulated a total of $57.10 in interest, how much was
invested in each of these accounts?
9. Solve the following systems algebraically.
(a) 2 x  3 y  4 and 0.8x  1.2 y  3
(b) 9 x  6 y  15 and
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Wrap-Up Statement:
In the last two sections you learned about equivalent and inconsistent equations and reviewed
the concepts addressed in the previous four sections.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed the two sections titled “Inconsistent and
Equivalent Equations”, and “Putting It Together.” Select your response to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
(g)
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I understand all of the concepts covered in the section,
“Inconsistent and Equivalent Equations.”
I understand that inconsistent equations are formed when two
linear functions are parallel. There is no solution to this system
of equations because the lines never intersect.
I understand that equivalent equations are formed when two
equations actually represent the same linear function. There
are an infinite number of solutions because the two equations
intersect each other along their entire length.
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions on the material
covered in this section.
I understand all of the concepts covered in the review section,
“Putting It Together.”
After completing the review, I feel confident about the material
I learned over the last few weeks.
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Introduction to Three Dimensions
Up to this point, all of the graphing that you have been doing has been limited to two
dimensions. This type of graphing works nicely when you are working with a flat piece of
paper. You had two axes where the vertical axis was typically labeled the y-axis and the
horizontal axis was typically labeled the x-axis. These two axes divided the region into four
parts called quadrants.
y-axis
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3
2nd quadrant 2 1st quadrant
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
x-axis
-1
-2
3rd quadrant -3
4th quadrant
-4
-5
Points in two dimensions were described using ordered pairs. The first coordinate of the ordered
pair typically represented the x-value of the point and the second coordinate typically represented
the y-value of the point. For example, the ordered pair (-4, 3) would be found in the second
quadrant because the x-coordinate is negative and the y-coordinate is positive.
y
10
9
When points are joined together, they form lines of many
different shapes. In this case, we have three lines
graphed on our coordinate system. The straight line is
formed by a linear function. The parabola (i.e. U-shaped
curve) is formed by a quadratic function. The wavy
curve is formed by a sinusoidal function.
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3
2
1
-4
-3
-2
-1
0
-1 0
-2
1
2
3
4
x
-3
-4
z
In this unit, you will learn to graph in three dimensions. When we
graph in three dimensions, we have three axes. That means we work
with three variables, typically x, y, and z, rather than two variables.
y
x
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


The vertical axis is the z-axis. The upper portion of
the z-axis is comprised of positive z-values. The
lower portion of the z-axis is comprised of negative
z-values.
The horizontal axis going from left to right is the yaxis. The right portion of the y-axis is comprised
of positive y-values. The left portion of the y-axis
is comprised of negative y-values.
The horizontal axis going in and out of the page is
the x-axis. The portion of the x-axis coming out of
the page is comprised of positive x-values. The
portion of the x-axis going back into the page is
comprised of negative x-values.
z
(+)
x
(-)
y
(+)
y
(-)
x
(+)
z
(-)
Points plotted in 3-space are not called ordered pairs, rather they are called ordered triples.
That is because each point is described by three coordinates, typically of the form (x, y, z).
The axes divide 3-space into eight regions called octants. The octants are numbered in a
specific order that is illustrated in the diagram below.
3rd
4th
2nd
1st
7th
8th
6th
5th
For any point in the first octant, all three coordinates of an ordered triple are positive. The four
points (5, 7, 1), (3, 5, 12), (2, 4, 1) and (6, 5, 1) are all found in the first octant because all of their
coordinates are positive numbers.
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For any point in the second octant, the x- and z-coordinates are positive and the y-coordinate is
negative. The four points (2, -3, 6), (5, -1, 2), (1, -12, 9) and (3, -4, 17) are all in the second
octant.
For any point in the sixth octant, the y- and z-coordinates are negative and the x-coordinate is
positive. The points (2, -8, -1), (8, -1, -2), (5, -6, -9) and (9, -2, -7) are all in the sixth octant.
Questions:
1. Google search Massey University Cartesian Coordinate System in 3-D. Once you are the
appropriate website, scroll to the bottom of the page and see how the point (2, 3, 3) is plotted
in 3-space by selecting each of the “Show on Grid” icons. Once you have done this select
the “More Exercises” option at the bottom of the page. You can see five other ordered triples
being plotted by selecting the five “Plot” icons.
2. Determine which octant each of these ordered triple is found in.
(a) (2, 5, 9)
_______
(b) (-3, -4, 1)
_______
(c) (3, 7, -3)
_______
(d) (5, -2, 6)
_______
(e) (-5, 2, -7)
_______
(f) (-4, -2, -9)
_______
(g) (-5, -8, -12)
_______
(h) (-1, 3, -2)
_______
(i) (14, 7, 2)
_______
(j) (3, -4, -8)
_______
(k) (4, 9, -2)
_______
(l) (2, -7, 3)
_______
(m) (-9, 2, 5)
_______
(n) (-6, 13, -1)
_______
(o) (-4, -7, 8)
_______
(p) (-8, -13, -2)
_______
(q) (3, 5, -18)
_______
(r) (-5, -7, 3)
_______
3. Complete the following chart by adding + and - signs. The first row has been completed for
you.
Point Found in the:
1st octant
2nd octant
3rd octant
4th octant
5th octant
6th octant
7th octant
8th octant
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+
y-coordinate
+
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Wrap-Up Statement:
In this section you learned how to identify which octant an ordered triple was located in.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 3. Select your response to
each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
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I understand all of the concepts covered in the section,
“Introduction to Three Dimensions.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can visualize and identify the various octants.
Given an ordered triple, I can determine which octant it is
located.
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The Phone Plan
As you previously learned, graphs in two dimensions come in many shapes and sizes and they
are formed by joining points (i.e. ordered pairs). The same is true about graphs in three
dimensions. The graphs come in many shapes and sizes and are formed by joining points (i.e.
ordered triples). Below you will find three 3D graphs. The one on the left has the equation
z  2 cos x  0.5 y 2 . The one in the center has the equation z  0.4 x 2  2 y 2  3 . The one on the
right has the equation z   x  y 2  4 .
In all three cases, these graphs were drawn using computer software. If you want to look at other
3D graphs, Google search 3D Graph in a Box. At this site, you can examine a variety of 3D
graphs and rotate the graphs around their axes using your mouse.
We are obviously not going to graph equations as complicated as the ones above. We are going
to work with equations of the form Ax  By  Cz  D . These equations form planes. To
understand how equations of this form create planes, we are going to complete an activity
concerned with phone plans.
The Phone Plane Activity
Jacob has a telephone plan that charges him a flat rate of $30 per month for local calls and then
charges him 5¢ per minute for long distance calls within Canada and 10¢ per minute for long
distance calls to the United States.
1. If Jacob made long distance calls within Canada totaling 40 minutes and long distance calls
to the United States totaling 20 minutes, what would his phone bill be for that month?
(a) $4
(b) $34
(c) $36
(d) $90
2. If Jacob made long distance calls within Canada totaling 60 minutes and long distance calls
to the United States totaling 50 minutes, what would his phone bill be for that month?
(a) $35
(b) $36
(c) $38
(d) $42
3. If Jacob made long distance calls within Canada totaling 42 minutes and long distance calls
to the United States totaling 78 minutes, what would his phone bill be for that month?
(a) $37.80
(b) $38.20
(c) $38.70
(d) $39.90
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4. Which one of these equations represents the relationship between Jacob’s telephone bill (B),
the length of time he makes long distance calls within Canada (C), and the length of time he
makes long distance calls to the United States (U)?
(a) B  0.10C  0.05U
(b) B  0.10C  0.05U  30
(c) B  0.05C  0.10U
(d) B  0.05C  0.10U  30
If we wanted to illustrate the relationship between these three
variables (B, C, and U), we would have to graph in three dimensions.
The vertical axis would represent Jacob’s monthly phone bill, B. We
will let the axis coming out of the page represent the time making
long distance calls within Canada. The other axis will represent the
time making long distance calls to the United States. Notice that
with this application (i.e. telephone plan), we are only going to work
in the first octant, where all three variables are positive. In the other
octants, we have negative values for at least one of the variables and
that doesn’t make sense when you consider the context we are
working in.
B
U
C
If we wanted to plot a point in this 3-space, the ordered triple would have the coordinates
C,U , B  . For example, the ordered triple (30, 20, 33.5) in this 3-space tells us that when Jacob
makes 30 minutes worth of long distance calls within Canada and 20 minutes worth of long
distance calls to the United States, his monthly bill will be $33.50.
5. In the context of this phone plan, what does the ordered (12, 46, 35.2) represent?
6. Based on the phone plan context, find the missing term in the ordered triple (52, 29, B).
7. Based on the phone plan context, find the missing term in the ordered triple (34, U, 34.6).
You might want to use the equation from question 4 to help you solve this question.
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8. Complete the following table. The ordered triples that you generate in this table will be used
future questions.
Minutes Worth of
Canadian Long
Distance Calls
0
200
400
600
0
0
0
200
400
600
200
400
600
200
400
600
Minutes Worth of
U.S. Long Distance
Calls
0
0
0
0
100
200
300
100
100
100
200
200
200
300
300
300
Jacob’s Phone Bill
Ordered Triple
(C, U, B)
30
(0, 0, 30)
Notice that all of these ordered triples are found in the first octant because all of the coordinates
are positive.
We are now going to plot these 16 points in
3-space. We are not going to do this on a
piece of paper. It’s almost impossible to
graph an ordered triple on a two
dimensional surface.
We will use a grid and cube-a-link blocks.
The grid can be found in the appendix. Lay
the grid on your desk. The U-axis will be
along one edge of the grid. The C-axis will
be along the other edge. You will have to
imagine the B-axis which will rise vertically
from where the other two axes intersect.
(See the diagram)
B
U
C
Cube-a-links will be joined together to represent the various bill amounts. For
example, if you wanted to represent a bill of $30, you would join three cubes
together. The height of each cube will represent $10.
$10
$10
$10
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Using the grid and the cube-a-link blocks, you can plot the 16 ordered triples that you previously
calculated.
For example, the ordered triple (0, 0, 30)
would be represented by the top of three
joined cube-a-links positioned where the Uaxis and C-axis intersect. (See diagram.)
B
For example, the ordered triple (200, 300,
70) would be represented by the top of seven
joined cube-a-links positioned on the grid
where U equals 300 and C equals 200. (See
diagram.)
300
U
200
C
Once you have the 16 points plotted, gently place a sheet of paper along the tops of the stacks of
cube-a-links. You will have to hold the paper in place to prevent it from sliding off. This
inclined piece of paper represents the plane that is formed from the ordered triples that you have
plotted. A plane is a flat two-dimensional surface in three-dimensional space.
9. Circle the diagram that best
represents the plane for this phone
plan situation. In each case we are
only viewing the plane in the first
octant.
B
(a)
(b)
U
C
U
C
B
(c)
(d)
U
C
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B
U
C
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Wrap-Up Statement:
In this section you learned how ordered triples generated from equations of the form
Ax  By  Cz  D can be plotted and joined to create flat surfaces called planes.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 9. Select your response to
each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
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I understand all of the concepts covered in the section, “The
Phone Plan.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I could figure out the ordered triples for the Phone Plan
Activity.
I understand how to use the cube-a-link blocks and grid to plot
ordered triples in 3-space.
By resting a sheet of paper on top of the cube-a-link blocks, I
could see the resulting plan from the Phone plan activity and
understood it orientation in 3-space.
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Graphing Planes
In the last section we examined a particular phone plan and its resulting graph. The technique
that we used to create the graph was not very practical. Plotting ordered triples using a grid and
cube-a-link blocks is too slow and still makes it difficult to visualize the resulting plane. We
need another more practical technique so that we can access a greater variety of real world
applications. These applications will be discussed in the section, 3D Graphs in the Real World.
When working in three
dimensions, space is divided into
eight regions called octants.
Trying to draw a plane within all
of the octants is extremely
challenging and is best left to
computers. To simplify the
process, we are going to limit
ourselves to only two octants;
the first octant and the fifth
octant. If you look at the
diagram, you will notice that a
grid pattern has been drawn
along two planes. One grid is on
the y-z plane where all x-values
are equal to zero. The other grid
is on the x-z plane where all the
y-values are equal to zero.
y-z plane
(x = 0)
z
x-z plane
(y = 0)
1st Octant
y
5th Octant
x
z
We are going to graph equations of
the form Ax  By  Cz  D in three
dimensions. We need to understand
where the plane we wish to graph
(shaded grey in the diagram)
intersects the x-z plane and the y-z
plane. The lines formed from these
intersections are called traces. The
x-z trace forms when the plane we
wish to graph intersects the x-z plane.
The x-z trace is a straight line where y
is equal to 0. The y-z trace forms
when the plane we wish to graph
intersects the y-z plane. The y-z trace
forms a straight line where x is equal
to 0.
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y-z plane
x-z plane
y-z trace
y
x-z trace
x
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Example 1:
Graph 2 x  y  z  1 .
Answer:
Rearrange the equation so that z is expressed in terms of x and y.
2x  y  z  1
 z  2 x  y  1
z  2x  y  1
The x-z trace occurs when y = 0.
z  2 x  (0)  1
z  2x  1
Graph this trace on the x-z plane by
plotting the z-intercept at -1 and then
rising 2 and running 1 from that zintercept.
The y-z trace occurs when x = 0.
z  2(0)  y  1
z  y 1
Graph this trace on the y-z trace by
plotting the z-intercept at -1 and then
rising 1 and running 1 from that zintercept.
Connect the two traces and shade in the resulting plane.
z
z
y-z trace
z=y-1
x-z trace
z = 2x - 1
(0, 5, 4)
(3, 0, 5)
y
y
(0, 0, -1)
x
x
(Completed Graph)
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Example 2:
Graph 2 x  8 y  2 z  6 .
Answer:
Rearrange the equation so that z is expressed in terms of x and y.
2x  8 y  2z  6
2 z  2 x  8 y  6
z  x  4 y  3
The x-z trace occurs when y = 0.
z   x  4(0)  3
z  x  3
Graph this trace on the x-z plane by
plotting the z-intercept at 3 and then rising
-1 and running 1 from that z-intercept.
The y-z trace occurs when x = 0.
z  (0)  4 y  3
z  4 y  3
Graph this trace on the y-z trace by
plotting the z-intercept at 3 and then rising
-4 and running 1 from that z-intercept.
Connect the two traces and shade in the resulting plane.
z
z
y-z trace
z = -4y + 3
x-z trace
z = -x + 3
(0, 0, 3)
y
y
x
x
(5, 0, -2)
(0, 2, -5)
(Completed Graph)
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Questions:
1. For each of the following equations:
 rearrange the equation so that z is expressed in terms of x and y,
 determine the equation of the y-z trace,
 determine the equation of the x-z trace, and
 graph the plane on the 3D coordinate graph paper (found in Appendix)
(a) 3x  y  2 z  2
(b) 5x  2 y  3z  12
(c) 3x  2 y  4 z  0
(d) 15x  y  5z  10
(e) 2 x  y  z  3
(f)  2 x  6 y  3z  3
(g) 2 x  3 y  4 z  8
(h) 12 x  2 y  3z  0
(i) x  y  z  4
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Give This a Try?
If you have the time you might want to try to use some free 3D graphing software. If your class
computer does not have the software called “Graphing Calculator 3D”, then you will have to
download this freeware off of the internet. Google search Graphing Calculator 3D free software
downloads or go directly to http://www.download.com/Graphing-Calculator-3D/3000-2053_410725117.html.
You can graph the equations for questions 1 (a) to (d) and print off the resulting graphs. You
will have to adjust the Axis Setup by selecting the small graph icon found in the lower left hand
corner. Change the Range settings to the following.
Range
From
0
0
-5
To
x
y
z
5
5
5
You have to enter the equation in the yellow box found in the upper left hand corner. The
equation must be rearranged such that z is expressed in terms of x and y.
Wrap-Up Statement:
In this section you learned how to graph equations of the form Ax  By  Cz  D using 3D
coordinate graph paper. This is accomplished by:
 determining the equation of the x-z trace and plotting it on the x-z plane,
 determining the equation of the y-z trace and plotting it on the y-z plane, and
 connecting the two traces so that the resulting triangle represents a portion of the
plane in 3-space.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 and 2. Select your response
to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
NSSAL
©2008
I understand all of the concepts covered in the section,
“Graphing Planes.”
I do not need any further assistance from the instructor on the
material covered in this section.
I understand why x = 0 on the y-z plane.
I understand why y = 0 on the x-z plane.
I understand how I can you the traces to graph planes on 3D
coordinate graph paper.
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The Plane Truth
In this section, you will
 determine if a point (i.e. an ordered triple) lies on a specific plane,
 learn how to find a missing coordinate for an ordered triple on a specific plane, and
 continue to graph equations of the form Ax  By  Cy  D on 3-D coordinate graph
paper.
Example 1:
Determine if the points (1, -2, -5) and (3, 2, 6) lie on the plane 2 x  5 y  z  3 .
Answer:
If a point lies on the plane, then the point must satisfy the equation of that plane. That
means that when the values of x, y and z are substituted into the equation, then both sides
of the equation must remain equal to each other.
2 x  5 y  z  3
2(1)  5(2)  (5)  3
2  (10)  5  3
 3  3
The point (1, - 2, - 5) lies
on the plane.
2 x  5 y  z  3
2(3)  5(2)  6  3
6  10  6  3
10  3
The point (3, 2, 6) does not
lie on the plane.
Example 2:
If the points (2, y, -3) and (4, -5, z) lie on the plane x  2 y  4 z  6 , determine the value of the
missing coordinates.
Answer:
Substitute the known coordinates into the equation and solve for the unknown coordinate.
x  2 y  4z  6
2  2 y  4(3)  6
2  2 y  12  6
 2 y  6  2  12
 2 y  8
 2y  8

2
2
y4
The coordinates of the point
are (2, 4, - 3).
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x  2 y  4z  6
4  2(5)  4 z  6
4  10  4 z  6
 4 z  6  4  10
 4 z  8
 4z  8

4 4
z2
The coordinates of the point
are (4, - 5, 2).
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Questions:
1. Determine if each of the following ordered triples lie on the plane 3x  2 y  z  8 . Also
state the octant the point is located in.
(a) (4, 1, 6)
(b) (2, 7, 5)
(c) (-5, 3, -17)
(d) (3, -1, -2)
2. Determine the missing coordinate for each of the ordered triples assuming that all of the
points are found on the plane 2 x  3 y  2 z  12 .
(a) (1, 0, z)
(b) (3, y, 9)
(c) (x, -2, 4)
(d) (0, y, 6)
3. For each of the following equations:
 rearrange the equation so that z is expressed in terms of x and y,
 determine the equation of the y-z trace,
 determine the equation of the x-z trace, and
 graph the plane on the 3D coordinate graph paper (found in Appendix)
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(a) 3x  2 y  z  4
(b) 2 x  y  3z  3
(c) 15x  y  5z  0
(d) 4 x  3 y  5z  5
(e) 2 x  3 y  5z  20
(f) x  4 y  2 z  2
(g) 10 z  30
(h) 3x  2 z  4
(i) 5 y  2 z  0
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Wrap-Up Statement:
In this section you learned how to:
 determine if an ordered triple lies on a particular plane, and
 determine the missing coordinate for an ordered triple lying on a particular plane.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 3. Select your response to
each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
NSSAL
©2008
I understand all of the concepts covered in the section, “The
Plane Truth.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
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From Graphs to Equations
We have learned how to take an equation of the form Ax  By  Cz  D and draw its graph on
3D coordinate graph paper. In this section we will be given the graph of a plane and determine
the equation of that plane.
z
Example 1:
Based on the graph that has been provided,
determine the equation of the plane. The resulting
equation must be in the form Ax  By  Cz  D
where A, B, C, and D are integers.
(2, 0, 5)
(0, 5, 3)
y
(0, 0, -3)
x
Answer:
 Determine the equation of the x-z trace.
z-intercept = -3
rise 4
 4
slope 
run 1
equation: z  4 x  3


z
Determine the equation of the y-z trace.
z-intercept = -3
rise 6
slope 

run 5
6
equation: z  y  3
5
Determine the equation of the plane.
6
z  4x  y  3
5
6
 4 x  y  z  3
5
6 
5 4 x   5 y   5 z   5 3
5 
 20 x  6 y  5 z  15
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y
y-z trace
x
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Example 2
Based on the graph that has been provided,
determine the equation of the plane. The resulting
equation must be in the form Ax  By  Cz  D
where A, B, C, and D are integers.
z
(0, 0, 2)
(5, 0, 4)
y
x
(0, 3, -5)
Answer:
 Determine the equation of the x-z trace.
z-intercept = 2
rise 2
slope 

run 5
2
equation: z  x  2
5
 Determine the equation of the y-z trace.
z-intercept = 2
rise  7
slope 

run
3
7
equation: z   y  2
3
 Determine the equation of the plane.
2
7
z  x y2
5
3
2
7
 x yz 2
5
3
 2 
7 
15  x   15 y   15 z   152 
 5 
3 
 6 x  35 y  15 z  30
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z
x-z trace
y
y-z trace
x
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C. D. Pilmer
Questions:
1. Determine the equation of each plane. The resulting equation must be in the form
Ax  By  Cz  D where A, B, C, and D are integers.
(a)
(b)
z
(0, 3, 5)
z
(0, 4, 5)
(2, 0, 5)
(5, 0, 4)
(0, 0, 1)
y
y
(0, 0, -1)
x
x
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(c)
(d)
z
z
(0, 5, 5)
(0, 0, 4)
(0, 0, 3)
(0, 5, 2)
y
y
x
x
(5, 0, -1)
(5, 0, -2)
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(e)
(f)
z
z
(2, 0, 5)
(0, 0, 0)
y
y
(0, 5, -2)
(0, 0, -2)
x
x
(0, 5, -4)
(5, 0, -3)
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2. Match each graph to its appropriate equation.
Graph A
Graph B
z
Graph C
z
z
y
y
y
x
x
x
Matches Equation _____
Matches Equation _____
Matches Equation _____
Equations:
Equation (i): z  mx  ny  p , m > 0, n > 0, p < 0
Equation (ii): z  mx  ny  p , m < 0, n < 0, p > 0
Equation (iii): z  mx  ny  p , m < 0, n > 0, p > 0
Wrap-Up Statement:
In this section you learned how to determine the equation of a plane from its graph. This is
accomplished by using our understanding of the graph’s x-z traces and y-z traces.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 and 2. Select your response
to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
NSSAL
©2008
I understand all of the concepts covered in the section, “From
Graphs to Equations.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
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3D Graphs in the Real World
With the exception of the phone plan activity, we have not looked at any other applications of 3D
graphs. In this section we will complete a variety of questions that deal with real world
applications of 3D graphs. In all cases, we will only be dealing with planes that exist in first
octant.
Example 1:
Kara is filling her swimming pool using two pumps. The volume, V, of
water in the pool is dependent upon the amount of time Pump A is on and
the amount of time Pump B is on. The length of time Pump A is
operating and length of time Pump B is operating are represented by the
variables A and B respectively. Don’t assume that both pumps are
necessarily operating for the same length of time. Pump A adds water at
a rate of 0.10 m3/min. Pump B adds water at a rate of 0.05 m3/min.
V
B
A
(a) Determine the equation of the plane that describes the volume of water in the pool in terms of
the times that each of the pumps is in operation.
(b) What is the V-intercept and what does it represent in this situation?
(c) Find the missing coordinate in the ordered triple (40, 60, V) and explain what the resulting
ordered triple represents in this situation.
(d) If Pump A operated for 50 minutes and the pool had 9 m3 of water in it, how long was Pump
B operating?
(e) What is the equation of the A-V trace?
(f) What is the equation of the B-V trace?
(g) Which one of these graphs best represents this situation?
(i)
(ii)
(iii)
V
V
V
B
A
B
B
A
A
(h) If the pool initially had 6 m3 of water in it before the pumps were turned on, what would the
equation of the plane be?
(i) If the A-V trace was V  0.08 A  10 and the B-V trace was V  0.04B  10 , then what would
be the equation of the plane that generated these traces? How would the situation have
changed?
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Answers:
(a) V  0.10 A  0.05B
(b) The V-intercept is equal to 0. That tells us that when neither of the pumps has been in
operation then there are 0 m3 of water in the pool.
(c) Substitute 40 in for A, substitute 60 in for B, and solve for V.
V  0.10 A  0.05 B
V  0.10(40)  0.05(60)
V  43
V 7
When Pump A and Pump B have been operating for 40 minutes and 60 minutes
respectively, the pool will have 7 m3 of water in it.
(d) We know A and V, but have to solve for B.
V  0.10 A  0.05 B
9  0.10(50)  0.05 B
9  5  0.05 B
4  0.05 B
B  80
Pump B was operating for 80 minutes.
(e) The A-V trace occurs when B = 0.
V  0.10 A  0.05B
V  0.10 A  0.05(0)
V  0.10 A
(f) The B-V trace occurs when A = 0.
V  0.10 A  0.05B
V  0.10(0)  0.05B
V  0.05B
(g) Graph (i) best represents the situation because its V-intercept is 0 and the slopes of both
traces are positive.
(h) V  0.10 A  0.05B  6
(i) New Equation: V  0.08 A  0.04B  10
New Situation: Pool initial has 10 m3 of water
Pump A is adding water at a rate of 0.08 m3/min.
Pump B is adding water at a rate of 0.04 m3/min.
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Questions:
1. Barb’s new phone plan charges $0.03 per minute for long distance calls
within Canada, $0.06 per minute for long distance calls to the United
States, and a flat fee of $25 for local calls. The monthly bill, B, is
dependent upon the time spent making long distance calls to Canada
and the United States. Let C represent the number of minutes spent
making long distance calls within Canada. Let U represent the number
of minutes spent making long distance calls to the United States.
B
U
C
(a) What is the equation of the plane that describes the monthly phone bill in terms of the
time spent making long distance calls within Canada and to the United States.
(b) Complete the ordered triple (37, 46, B) and explain what the ordered triple represents in
this situation?
(c) If Barb’s phone bill is $27.88 and she made long distance calls within Canada totaling 48
minutes, how many minutes did she spend making long distance calls to the United
States?
(d) What is the B-intercept and what does it represent in this situation?
(e) What is the equation of the U-B trace?
(f) What is the equation of the C-B trace?
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(g) Which one of these graphs best represents this situation?
(i)
(ii)
B
B
U
C
(iii)
B
U
C
U
C
(h) If the C-B trace was B  0.04C  22 and the U-B trace was B  0.07U  22 , then what
would be the equation of the plane that generated these traces? How would the situation
have changed?
2. Ryan was doing a project for his Level IV Science course and
discovered some interesting information. If someone does nothing all
day, essentially they are a couch potato, their body still requires 2400
calories a day just to survive. If an individual exercises by walking,
he/she will require an additional 120 calories for every hour of
walking. If an individual exercises by running, he/she will require an
additional 600 calories for every hour of running. The number of
calories, C, required each day is dependent on the time spent running
and the time spent walking. Let R represent the number of hours
spent running. Let W represent the number of hours spent walking.
C
W
R
(a) What is the equation of the plane that describes the required number of calories in terms
of the time spent running and time spent walking?
(b) What is the C-intercept and what does it represent in this situation?
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(c) Complete the ordered triple (2, W, 3960) and explain what it represents in this situation?
(d) If an individual ran for 3 hours and walked for 2.5 hours, how many calories would their
body require that day?
(e) What is the equation of the W-C trace?
(f) What is the equation of the R-C trace?
(g) Which one of these graphs best represents this situation?
(i)
(ii)
C
C
W
R
(iii)
C
W
R
W
R
(h) If your body only required 2000 calories per day to survive, how would the equation of
the plane and the graph of the plane change?
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(i) If the W-C trace was C  140W  2600 and the R-C trace was C  500R  2600 , then
what would be the equation of the plane that generated these traces? How would the
situation have changed?
3. Monique is using two different pumps to drain her swimming pool
that holds 80 m3 of water. The volume, V, of water remaining in the
pool is dependent upon the length of time Pump A is on and the
length of time Pump B is on. Pump A removes water at a rate of
0.06 m3/min and Pump B removes water at a rate of 0.09 m3/min.
Let A represent the length of time Pump A is operating. Let B
represent the length of time Pump B is operating.
V
B
A
(a) What is the equation of the plane that describes the volume of water remaining in the
pool in terms of length of time Pump A is operating and the length of time Pump B is
operating?
(b) Complete the ordered triple (240, B, 49.4) and explain what it means in terms of this
situation.
(c) What is the V-intercept and what does it represent in this situation?
(d) If Pump B operates for 160 minutes and the pool presently has 32 m3 of water, how long
has Pump A been operating?
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(e) What is the equation of the A-V trace?
(f) What is the equation of the B-V trace?
(g) Which one of these graphs best represents this situation? Assume that the same scale was
used on the A-axis and B-axis.
(i)
(ii)
(iii)
V
V
V
B
A
B
A
B
A
(h) If the A-V trace was V  0.08 A  75 and the B-V trace was V  0.12B  75 , then what
would be the equation of the plane that generated these traces? How would the situation
have changed?
4. Both Tyrus and his wife, Kimi, work at part-time jobs that pay
hourly wages. Tyrus makes $12.50 per hour and Kimi makes
$14.00 per hour. Their weekly earnings, E, before deductions
(taxes, employment insurance payments,…) are dependent upon the
number of ours each of them works. Let T represent the number of
hours Tyrus works in a week. Let K represent the number of hours
Kimi works in a week.
E
K
T
(a) Determine the equation of the plane that describes their weekly earnings in terms of the
number of hours each of them works.
(b) What is the equation of the K-E trace?
(c) What is the equation of the T-E trace?
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(d) What is the E-intercept and what does it represent in this situation?
(e) If Kimi works 30 hour and their earnings before deductions are $845, then how many
hours did Tyrus work?
(f) Complete the ordered triple (38, K, 937) and explain what the ordered triple represents in
this situation.
(g) Which one of these graphs best represents this situation? Assume that the same scale was
used on the T-axis and K-axis.
(i)
(ii)
(iii)
E
E
E
K
T
K
T
K
T
(h) If the T-E trace was E  13T and the K-E trace was E  14.5K , then what would be the
equation of the plane that generated these traces? How would the situation have changed?
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Wrap-Up Statement:
In this section you learned how to apply your knowledge of plane to a variety of real world
applications.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 4. Select your response to
each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
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I understand all of the concepts covered in the section, “3D
Graphs in the Real World.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
When given a written description of the real world application,
I can determine the equation of the plane without any
assistance.
I can determine the intercept of the vertical axis and understand
what it represents in that specific application.
I understand what a specific ordered triple represents in a
specific application.
I can visualize the orientation of the plane based on my
understanding of the traces and the specific application.
I understand how the situation and equation of the plane
change based on changes to the two traces.
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Putting It Together Part 2
In the last six sections of this unit, you have learned:
 how to plot ordered triples in 3-space,
 identify which octant an ordered triple is located in,
 that equations of the form Ax  By  Cz  D create planes in 3-space,
 how to graph equation of the form Ax  By  Cz  D in 3-space with and without
technology,
 determine if an ordered triple lies on a specific plane,
 determine the equation of a plane using its graph, and
 to work with a variety of real world applications involving planes in 3-space.
The questions that follow will allow you to review these concepts learned in the six previous
sections.
Questions:
1. Determine which octant each of these ordered triple is found in.
(a) (2, 5, 9)
_______
(b) (2, 4, -1)
_______
(c) (-4, 1, -5)
_______
(d) (8, -2, -1)
_______
(e) (-4, -9, -3)
_______
(f) (-7, -1, 8)
_______
(g) (5, -6, 2)
_______
(h) (-9, 3, 7)
_______
2. Determine if each of the following ordered triples lie on the plane 4 x  3 y  2 z  12 .
(a) (5, 10, -2)
(b) (-3, 2, 15)
3. Determine the missing coordinate for each of the ordered triples assuming that all of the
points are found on the plane 2 x  5 y  4 z  60 .
(a) (x, 8, -8)
(b) (8, y, -6)
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4. For each of the following equations:
 rearrange the equation so that z is expressed in terms of x and y,
 determine the equation of the y-z trace,
 determine the equation of the x-z trace, and
 graph the plane on the 3D coordinate graph paper.
(a) 4 x  2 y  5z  5
(b) 4 x  y  z  1
z
z
y
x
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5. Determine the equation of each plane. The resulting equation must be in the form
Ax  By  Cz  D where A, B, C, and D are integers.
(a)
(b)
z
z
(0, 0, 3)
(5, 0, 5)
(0, 5, 1)
(5, 0, 3)
y
y
x
x
(0, 0, -4)
(0, 4, -5)
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6. Match each graph to its appropriate equation.
Graph A
Graph B
z
Graph C
z
z
y
y
y
x
x
x
Matches Equation _____
Matches Equation _____
Matches Equation _____
Equations:
Equation (i): z  mx  ny  p , m > 0, n < 0, p > 0
Equation (ii): z  mx  ny  p , m > 0, n = 0, p < 0
Equation (iii): z  mx  ny  p , m < 0, n < 0, p = 0
7. Kadeer bought an old house that needs some renovations to the
plumbing and electrical systems. He has budgeted $4000 to cover the
labor costs of a plumber and an electrician. The amount of money
(M) left over after the paying for plumber and electrician is dependent
upon the number of hours each of these trades people worked. The
electrician charges $45 per hour. The plumber charges $55 per hour.
Let E represent the number of hours the electrician works. Let P
represent the number of hours the plumber works.
M
P
E
(a) What is the equation of the plane that describes the amount of money left over in terms
of the number of hours the electrician and the number of hours the plumber work?
(b) What is the M-intercept and what does it represent in this situation?
(c) What is the equation of the E-M trace?
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(d) What is the equation of the P-M trace?
(e) If the electrician worked for 20 hours and the plumber worked for 28 hours, how much
money would Kadeer have left from his $4000 budget?
(f) Complete the ordered triple (14, P, 1720) and explain what it represents in this situation.
(g) Which one of these graphs best represents this situation?
(i)
(ii)
M
(iii)
M
P
E
M
P
E
P
E
(h) If the E-M trace was M  50E  3500 and the P-M trace was M  48P  3500 , then
what would be the equation of the plane that generated these traces? How would the
situation have changed?
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Wrap-Up Statement:
In this section you reviewed a variety of concepts you learned in the previous six sections of
this unit. All of the concepts were concerned with planes of the form Ax  By  Cz  D .
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 7. Select your response to
each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
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I understand all of the concepts covered in the section, “Putting
It Together.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I am able to select the appropriate strategies/procedures when
given a variety of questions concerned with planes.
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Systems of Equations with Three Variables
In the first section of this unit, you worked with linear equations with two variables. The
equations were of the form Ax  By  C or y  mx  b , and they formed straight lines. We
referred to them as linear functions. When we were asked to solve a system of linear equations,
we were provided with two linear functions and had to figure out where these functions
intersected each other. We could work out this point of intersection graphically, by substitution,
or using elimination. There were occasions, when we could not find one single point of
intersection (i.e. a unique solution) between the two linear functions. This occurred when the
lines were parallel (inconsistent equations - no solution) or when the two equations represented
the same linear function (equivalent equations - infinite number of solutions).
In the second section of this unit, we learned how to graph equations with three variables in
three-space. These equations were of the form Ax  By  Cz  D and they formed planes.
In this, the third and final section of the unit, we will learn how to solve a system of equations in
three variables. If we have to solve for three variables, we must be provided with three
equations. Since we are dealing with three variables and three equations, it is often referred to as
a 3 by 3 system of equations. From a graphical
perspective, the three equations represent three
Point of
planes and we are trying to find the point of
Intersection
intersection of the three planes. The diagram on
the right illustrates how three planes can intersect
at one point. Understandably, finding a point of
intersection between three planes graphically is
almost impossible to do by hand; computers are
generally used to try to illustrate this process. We
will, however, learn how to determine points of
intersection algebraically using 3 by 3
elimination.
We will start with three equations and three unknowns (variables). Using elimination, we will
simplify the question to two equations and two unknowns. Using elimination again, we will
simplify the question to one equation and one unknown. This will make more sense when you
look at the examples that have been provided.
Example 1:
Solve the following system of equations.
2 x  3 y  4 z  10
x  y  3z  7
2x  4 y  z  4
Answer #1 (Suzanne’s Solution)
Suzanne correctly answered this question. She started by eliminating the unknown x using
the first and second equations, then using the second and third equations.
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2 x  3 y  4 z  10 

2 x  3 y  4 z  10
 ( 2 )
x  y  3z  7 
  2 x  2 y  6 z  14
5 y  2 z  4
( 2 )
x  y  3z  7 
 2 x  2 y  6 z  14
2x  4 y  z  4 

2x  4 y  z  4
 2 y  5 z  10
By eliminating the x, she has the question down to two equations with two unknowns (y and
z). She now decides to eliminate the z.
5
5 y  2 z   4 
25 y  10 z  20
( 2 )
 2 y  5 z  10 
 4 y  10 z  20
29 y  0
29 y 0

29 29
y0
Now that she has solved for y, she goes back and solves for z and x.
2 x  3 y  4 z  10
2 x  30   42   10
2 x  8  10
2 x  10  8
2x 2

2 2
x 1
5 y  2 z  4
50   2 z  4
 2z  4

2 2
z2
The solution is x = 1, y = 0 and z = 2. From a graphical perspective, the three planes intersect
at the point (1, 0, 2).
Answer #2 (Brian’s Solution)
Brian correctly answered this question. He started by eliminating the unknown z using the
first and second equations, then using the first and third equations.
3
2 x  3 y  4 z  10 
6 x  9 y  12 z  30
( 4 )
x  y  3z  7 
  4 x  4 y  12 z  28
2 x  13 y  2
2 x  3 y  4 z  10 

2 x  3 y  4 z  10
( 4 )
2 x  4 y  z  4 
  8 x  16 y  4 z  16
 6 x  19 y  6
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By eliminating the z, he has the question down to two equations with two unknowns (x and
y). He now decides to eliminate the x because it is far easier than eliminating the y.
3
2 x  13 y  2 
6 x  39 y  6
 6 x  19 y  6 
  6 x  19 y  6
58 y  0
58 y 0

58 58
y0
Now that he has solved for y, he goes back and solves for x and z.
2 x  3 y  4 z  10
21  30   4 z  10
2  4 z  10
4z  8
z2
2 x  13 y  2
2 x  130   2
2x  2
x 1
The solution is x = 1, y = 0 and z = 2. From a graphical perspective, the three planes intersect
at the point (1, 0, 2).
As illustrated by Suzanne’s solution and Brian’s solution, there isn’t just one correct way to
answer these 3 by 3 systems of equations questions.
Example 2:
Solve the following system of equations and check your answers.
4 x  3 y  z  4
3x  2 y  2 z  10
2 x  y  3z  16
Answer:
( 2 )
4 x  3 y  z  4 
 8 x  6 y  2 z  8
3x  2 y  2 z  10 
 3x  2 y  2 z  10
 5 x  8 y  18
( 3)
4 x  3 y  z  4 
  12 x  9 y  3z  12
2 x  y  3z  16


2 x  y  3z  16
 10 x  8 y  28
 5 x  8 y  18 
  5 x  8 y  18
( 1)
 10 x  8 y  28 
 10 x  8 y  28
5 x  10
x  2
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 10 x  8 y  28
 10 2  8 y  28
20  8 y  28
 8y  8
y  1
3 x  2 y  2 z  10
3 2   2 1  2 z  10
 6  2  2 z  10
 4  2 z  10
2 z  14
z7
The solution is x = -2, y = -1 and z = 7. From a graphical perspective, the three planes
intersect at the point (-2, -1, 7).
To check the answers, determine if the values of x, y and z satisfy all three of the original
equations.
4 x  3 y  z  4
4 2  3 1  7   4
 8   3  7  4
 4  4
3x  2 y  2 z  10
3 2  2 1  27   10
 6  2  14  10
10  10
2 x  y  3z  16
2 2   1  37   16
 4   1  21  16
16  16
Since the answers satisfy all three equations, we know the answers are correct.
Questions:
1. Solve the following systems of equations. Take your time with these questions. A small
careless mistake will mess things up in spite of you understanding how to complete the
question.
(a) 3x  2 y  z  2
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(b) 2 x  4 y  z  1
x  2 y  2 z  4
 3x  5 y  2 z  7
(c) 6 x  3 y  z  15
2x  5 y  2z  3
4x  y  z  3
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(d) 3x  6 y  2 z  4
7 x  y  5z  7
 2x  3 y  4z  6
(e) x  3 y  4 z  12
 3x  2 y  5 z  1
4 x  4 y  7 z  6
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(f) 2 x  3 y  2 z  11
 2 x  4 y  3z  22
3x  5 y  2 z  12
2. In each case you are provided with a system of equations and a student’s answers. You are
asked to determine whether the answers are correct by determining if the answers satisfy the
three equations. You are not expected to conduct 3 by 3 elimination.
(a) 2 x  4 y  5z  22
x  2 y  7 z  35
5x  y  3z  25
Student’s Answers: x = 3, y = -2, z = 4
(b) 4 x  3 y  z  10
2 x  3 y  7 z  8
3x  2 y  5z  17
Student’s Answers: x = -5, y = -4, z = -22
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Checking Answers Using a TI-83 or TI-84
Solving systems of equations with three variables using elimination is a time-consuming process.
Checking your answers by substituting into the three original equations is great for confirming
that one has the correct answers but it does not provide us with a lot of useful information when
our solutions do not satisfy all three equations. When our answers do not satisfy the equations, it
is possible that:
 all three answers are wrong.
 one answer is right, and two answers are wrong.
 two answers are right and only one answer is wrong.
Since we are unable to determine the degree to which our answers are wrong then it takes us a lot
more time to track down our mistakes in our calculations. We need a better way to check our
answers. This can be accomplished using the inverse matrix method on the graphing
calculator. A matrix is a square or rectangular array of numbers comprised of rows and
columns.
3  2
0 7 


2  2 matrix
1 5 6 
 3 4  9


11 2  4
 14 
 23


 6 
3 3 matrix
3 1 matrix
 3  2 1 9
21 4  3 0


2  4 matrix
The dimensions of a matrix are given by the number of rows followed by the number of
columns. In the fourth example shown above, the matrix is comprised of two rows and four
columns therefore the dimensions of the matrix are 2  4 .
Matrices (plural of matrix) can be manipulated to complete a variety of calculations however we
are only going to us them to check our solutions to systems of equations with three variables.
The inverse matrix method involves finding the inverse of the matrix of coefficients and
multiplying it by the matrix of constants. The resulting matrix will contain elements that
represent the answers for x, y and z in our system of equations. At this level, you are not
expected to understand why this process works.
Example 1:
Solve the following system using a graphing calculator.
3x  4 y  z  7
2 x  5 y  3z  6
 4x  y  2z  1
Answer:
Identify the matrix of coefficients and the matrix of constants.
Matrix of Coefficients
4
1
3
 2  5  3


 4 1
2 
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Matrix of Constants
7 
 6
 
 1 
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Complete the following steps on a graphing calculator.
1. Enter the Matrix of Coefficients
MATRIX > EDIT > [A] > Enter the dimensions > Enter the elements
3 3
> QUIT
2. Enter the Matrix of Constants
MATRIX > EDIT > [B] > Enter the dimensions > Enter the elements
3 1
> QUIT
3. Take the Inverse of Matrix A and Multiply It by Matrix B
MATRIX > [A] > x-1 > MATRIX > [B] > ENTER
The answers are x = 2, y = -1, and z = 5.
Example 2:
Solve the following system using a graphing calculator.
x  3 y  4 z  11
4 x  5 y  2 z  12
 3x  y  6 z  5
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Answer:
The answers are x = 7, y = -2, and z = 3.
Questions:
Use a graphing calculator to answer each of these questions.
1. Tanya attempted to solve the following system using pencil and paper. She obtained the
answers x = 3, y = 6, and z = -1. Determine whether any or all of her answers are correct.
2 x  3 y  5 z  21
4 x  2 y  3z  1
x  y  2 z  12
2. Steve attempted to solve the following system using pencil and paper. He obtained the
answers x = 2, y = 5, and z = -3. Determine whether any or all of his answers are correct.
4 x  2 y  3z  9
6x  y  2z  1
3x  y  z  14
3. Shima attempted to solve the following system using pencil and paper. She obtained the
answers x = 1, y = 2, and z = -4. Determine whether any or all of her answers are correct.
2x  3y  z  4
5x  2 y  z  9
7 x  5 y  3z  9
4. Hiroshi attempted to solve the following system using pencil and paper. He obtained the
answers x = 2, y = -3, and z = 4. Determine whether any or all of his answers are correct.
3x  2 y  5 z  23
x  3 y  2 z  3
2 x  2 y  7 z  5
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Making Predictions
We have learned how to solve systems of three equations with three variables using elimination.
From a graphical perspective, the answers for the three variables represented the point of
intersection between three planes. In this activity, you will be given a diagram of more than one
plane and be asked to determine the following.
 What is the minimum number of equations are we dealing with?
 Is there a unique solution to the equations?
Example 1:
For each of the following, determine the minimum number of equations we are dealing with and
determine whether there is a unique solution to the system of equations.
(a)
(b)
Answers:
(a) Since we can see three distinct planes, then we know that the minimum number of planes
is three. Since the three planes intersect at one common point, then we know that there is
a unique solution.
(b) Since we can see two distinct planes, then we know that the minimum number of planes
is two. The planes do not intersect at one common point so there is no unique solution.
Questions:
1. For each of the following, determine the minimum number of equations we are dealing with
and determine whether there is a unique solution to the system of equations.
Your Answers:
(a)
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Your Answers:
(b)
(c)
(d)
(e)
2. Do all 3 by 3 systems of equations have unique solutions? Explain.
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Word Problems
Example 1:
Mr. Rhodenizer, an adult education math instructor, has given his class the following puzzle. He
tells the class that three shoppers were instructed to go to one particular grocery store and
purchase three specific items: canned soup, cookies and detergent. They were told to purchase
specific brands, sizes, and permitted to pick up as many of each item as they wished without
exceeding $40. The following table illustrates how much each shopper spent and how many of
each item they selected. Based on this information, how much did each item cost?
Shopper #1
Shopper #2
Shopper #3
Canned Soup
4
2
3
Cookies
3
2
7
Detergent
2
4
1
Total Price
27
30
32
Answer:
Let s represent the price per can of soup
Let c represent the price of the cookies
Let d represent the price of the detergent
Create three equations where each represents the amount of money a specific shopper spent.
4s  3c  2d  27
2s  2c  4d  30
3s  7c  d  32
Solve the 3 by 3 system of equations. There are a variety of ways this can be accomplished,
here is one possible way.
4s  3c  2d  27 

4s  3c  2d  27
( 2 )
3s  7c  d  32 
  6s  14c  2d  64
 2s  11c  37
2s  2c  4d  30 

2s  2c  4d  30
 ( 4 )
3s  7c  d  32 
  12s  28c  4d  128
 10s  26c  98
( 5 )
 2s  11c  37 
 10s  55c  185
 10s  26c  98 
  10s  26c  98
29c  87
c3
 2s  113  37
 2s  4
s2
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32  73  d  32
27  d  32
d 5
89
The soup costs $2 per can.
The cookies cost $3. The
detergent costs $5.
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Example 2:
Joan, Suzette, and Charlie work as telemarketers. Between the three of them, they can process
570 orders in a day. Joan processes 60 more orders in one day than Suzette. Charlie processes
30 less orders in one day than Joan. How many orders in one day does each of these individuals
process?
Answer:
Let j represent the number of orders Joan processes in one day.
Let s represent the number of orders Suzette processes in one day.
Let c represent the number of orders Charlie processes in one day.
Create three equations.
“Between the three of them, they can process 570 orders in a day.”
j  s  c  570
“Joan processes 60 more orders in one day than Suzette.”
j  s  60 can be rearranged to read j  s  60
“Charlie processes 30 less orders in one day than Joan.”
c  30  j can be rearranged to read j  c  30
Solve the 3 by 3 system of equations. There are a variety of ways this can be accomplished,
here are two possible ways.
Method 1: Pencil and Paper
j  s  c  570
j  s  60
2 j  c  630
j  c  30
2 j  c  630
3 j  660
j  220
j  s  60
220  s  60
s  160
j  c  30
220  c  30
c  190
Joan processes 220 orders.
Suzette processes 160 orders.
Charlie processes 190 orders.
Method 2: Inverse Matrix Method on Graphing Calculator
Joan processes 220 orders. Suzette processes 160 orders. Charlie processes 190 orders.
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Questions:
Questions 1 and 2 must be solved using paper and pencil. The remaining questions may be
solved using the inverse matrix method on the graphing calculator.
1. Three different jet engine parts are produced using three different machines. The following
chart shows the number of hours each machine spends on each of the engine parts and the
total number of hours each machine is available in one day. Determine how many of each
engine part can be produced in one day.
Machine 1
Machine 2
Machine 3
Time on
Engine Part A
2
1
1
Time on
Engine Part B
1
1
3
Time on
Engine Part C
2
4
1
Total Number
of Hours
12
14
16
2. A college basketball team scored a total of 90 points from a combination of 1-point foul
shots, 2-point shots, and 3-point shots. These points were accumulated by making a total of
46 baskets. The number of 2-point shots exceeded the combination of 1-point and 3-point
shots by 22. How many shots of each kind were made?
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3. Sasha is interested in changing to a new long distance phone plan that her three friends have
been raving about. The amount these individuals pay is based on the number of minutes they
made long distance calls within Canada, to the United States and overseas. Each friend
supplied the following information regarding the time spent making long distance calls and
the related long distance charges. How much does it cost per minute to phone long distance
within Canada, into the United States, and overseas. State your answers in cents per minute.
Number of Minutes Within Canada
Number of Minutes to the U.S.
Number of Minutes Overseas
Amount Paid in Cents
Friend 1
20
30
20
680
Friend 2
40
20
30
840
Friend 3
30
25
10
580
4. At a local coffee shop, a small cup of coffee costs $1.10, a medium cup costs $1.25, and a
large cup costs $1.50. On one particular morning, they sold 164 cups of coffee while
collecting $216.40 (before taxes and tips). The number of large cups of coffee sold was 8
less than the combined number of small and medium cups of coffee sold. How many cups of
each size did the shop sell on that morning?
5. Nasrin invested $9000 into three plans at interest rates of 4%, 5%, and 8% per annum (i.e.
per year). She left the money in each plan for one year. At the end of the year, her total
earnings from interest were $580. The earnings from the amount invested at 8% made $220
more than the combined earnings from the other two investments. How much did she invest
in each of the plans?
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Wrap-Up Statement:
In the last four sections you learned how to solve and check systems of equations with three
variables with and without technology, in and out of context. From a graphical perspective
you learned that the unique solution to a 3 by 3 system of equations represented the point of
intersection between three planes.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed the four sections titled “Systems of
Equations with Three Variables”, “Checking Answers Using a TI-83 or TI-84”, “Making
Predictions”, and “Word Problems.” Select your response to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
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I understand all of the concepts covered in the section,
“Systems of Equations with Three Variables.”
I do not need any further assistance from the instructor on the
material covered in this section.
I can successfully solve a system of equation with three
variables using elimination. I do not need any more practice
questions.
I understand all of the material covered in the section,
“Checking Answers Using a TI-83 or TI-84.”
I do not need any further assistance from the instructor on the
material covered in this section.
I can use a graphing calculator to check my answers to a 3 by 3
system of equations. I do not need any more practice
questions.
I understand all of the concepts covered in the section,
“Making Predictions”
I do not need any further assistance from the instructor on the
material covered in this section.
Given a diagram of planes, I can determine the minimum
number of equations we are dealing with and whether a unique
solution exists. I do not need any more practice questions.
I understand all of the concepts covered in the section, “Word
Problems.”
I do not need any further assistance from the instructor on the
material covered in this section.
I can successfully complete a contextual word problem
involving a system of equations in three variables. I do not
need any more practice questions.
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2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
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Appendix
Grid for “The Phone Plan” Activity
600
400
200
C
200
400
600
U
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3-D Coordinate Graph Paper
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3-D Coordinate Graph Paper
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Answers:
Interpretation Involving Two Linear Functions (pages 1 to 5)
1. (a) v  10t  80
(b) v  20t  120
(c) Tank A: 80 litres
Tank B: 120 litres
(d) Tank A: 10 litres per minute
Tank B: 20 litres per minute
(e) 20 litres
(f) 4 minutes
(g) between 0 and 4 minutes
2. (a) c  25n  600
(b) c  30n  400
(c) Hall A: slope = 25
Hall B: slope = 30
Represent the cost per person
(d) Hall A: c-intercept = 600
Hall B: c-intercept = 400
Represent the initial costs (rental hall, decorations, cleaning costs, staff,…)
(e) 40 people
(f) Hall B
(g) when more than 40 people attend
(h) $850
3. (a) a  5t  150
(b) a  10t  200
(c) Son: slope = -5
Daughter: slope = -10
Represent how much each pays back per day
(d) Son: a-intercept = 150
Daughter: a-intercept = 200
Represent how much they initially borrowed from their parent
(e) Son: t-intercept = 30
Daughter: t-intercept = 20
Represent the number of days it takes to pay back the loan
(f) between 10 and 30 days
(g) between 0 and 10 days
(h) At 10 days, they both owe $100
(i) 5 days
(j) $50
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1
4. (a) c  d  35 or c  0.2d  35
5
1
(b) c  d  25 or c  0.25d  25
4
(c) $55
(d) 1100 kilometres
(e) $35
(f) $0.25 or 25¢ per kilometre
(g) If he travelled 200 km, the charge would be $75 from either agency.
(h) Agency A
Solving Systems of Equation Graphically (pages 6 to 11)
1. (a) slope = 
2
, y-intercept = 5
3
4
, y-intercept = 2
3
(c) slope = 4, y-intercept = -5
(b) slope =
2. (a) solution
(b) not solution
(c) solution
3. (a) x = 3, y = 2
(c) x = -5, y = -1
(e) x = 5, y = -4
4. (a)
(b)
(c)
(d)
(b) x = 4, y = -3
(d) x = -2, y = 3
c  30h  150
c  25h  165
(3, 240)
If both painters work for 3 hours, they charge the same amount ($240).
Solving Systems by Substitution (pages 12 to 19)
1. (a)
(c)
(e)
(g)
x = 2, y = 5
x = 6, y = 3
x = 4, y = 3
m = -4, n = -2
(b)
(d)
(f)
(h)
2. (a) x = 2, y = -1
(c) p = 4, q = -1
x = 4, y = 1
x = 5, y = -2
x = -3, y = 5
r = -6, s = -5
(b) g = -1, h = 3
(d) x = 0.5, y = 2
3. v  7t  85 , v  4t  64 , 7 minutes
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4. c  10n  185 , c  8.5n  215 , 20 people
5. x  y  12 , 2 x  3 y  27 , The first number is 9. The second number is 3.
6. x  y  11 , x  2 y  5 , The larger number is 17. The smaller number is 6.
7. 2l  2w  174 , l  2w  12 , The length is 62 metres. The width is 25 metres.
8. s  l  75 , s  0.5l  15 , The larger gear has 40 teeth. The smaller gear has 35 teeth.
9. 4h  1 f  15 , 6h  3 f  24 , Hamburger: $3.50
French Fries: $1
10. c  18m  650 , c  22n  574 , 19 months
Solving Systems Using Elimination (pages 20 to 30)
1. (a) x = 5, y = -1
(b) x = -4, y = 1
(c) g = 6, h = 4
(d) m = -2, n = 3
(e) x = 1, y = 6
(f) x = 0, y = -2
(g) p = -5, q = -3
(h) x = 7, y = -4
2. (a) x = 4, y = 7
(b) p = -4, q = 3
(c) r = -2, s = 9
(d) x = 6, y = 2
(e) g = 4, h = -4
(f) x = 1700, y = 1350
(g) x = 8, y = -2
(h) p = -6, q = 4
3. c  v  39 , 7c  10v  321 , 23 cars and 16 vans
4. s  l  23 , 2s  l  4 , 14 and 9
5. s  h  41 , 2s  5h  133 , 17 hardcover books and 24 softcover books
6. c  0.18d  60 , c  0.14d  75 , 375 kilometres
7. x  y  1100 , 0.07 x  0.05 y  70.60
$780 was invested in the account that pays 7% interest per year
$320 was invested in the account that pays 5% interest per year
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8. c  b  54 , 15c  18b  879
31 chicken dinners and 23 beef dinners
9. s  l  42 , 5s  7l  236
29 short-sleeved shirts and 13 long-sleeved shirts
Inconsistent and Equivalent Equations (pages 31 to 33)
Case 1
1. No
1
1
2. y   x  5 and y   x  1
2
2
3. The lines are parallel to each other.
4. No
5. There is no solution because the two lines never intersect.
Case 2
1. No
2. y  2 x  3 and y  2 x  3
3. The two lines lie right on top of each other. They are actually the same line.
4. Yes
5. There are an infinite number of solutions because the two lines lie right on top of each other
(infinite number of points of intersection).
Questions
1. (a)
(b)
(c)
(d)
(e)
(f)
infinite number of solutions (equivalent equations)
x = 2 and y = -1
no solution (inconsistent equations)
infinite number of solutions (equivalent equations)
no solution (inconsistent equations)
x = -3 and y = -2
2. Answers will vary.
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Putting It Together Part 1 (pages 34 to 37)
1. x = 4 and y = -1
2. x  y  19 , 2 x  y  5 , The larger number is 11. The smaller number is 8.
3. x  y  8 , 2 x  3 y  81 , The larger number is 21. The smaller number is 13.
4.
p  c  75 , 0.89 p  1.19c  80.25 , 30 cans of peas and 45 cans of corn
5. 6h  2d  21, 4h  8d  29 , Hamburger: $2.75, Chili Dog: $2.25
6. v  1.5t  14 , v  2.1t  5 , 15 minutes
7. 2l  2w  68 , l  2w  1 , Length: 23 cm, Width: 11 cm
8. x  y  850 , 0.07 x  0.06 y  57.1
$610 was invested in the account that pays 7% interest per year
$240 was invested in the account that pays 6% interest per year
9
(a) no solution (inconsistent equations, parallel lines)
(b) infinite number of solutions (equivalent equations)
Introduction to Three Dimensions (pages 38 to 41)
2. (a) 1
(d) 2
(g) 7
(j) 6
(m) 4
(p) 7
3.
Point Found in the:
1st octant
2nd octant
3rd octant
4th octant
5th octant
6th octant
7th octant
8th octant
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(b)
(e)
(h)
(k)
(n)
(q)
3
8
8
5
8
5
x-coordinate
+
+
+
+
-
(c)
(f)
(i)
(l)
(o)
(r)
y-coordinate
+
+
+
+
101
5
7
1
2
3
3
z-coordinate
+
+
+
+
-
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The Phone Plan (pages 42 to 46)
1. (b) $34
2. (c) $38
3. (d) $39.90
4. (d) B  0.05C  0.10U  30
5. If Jacob makes long distance calls within Canada totaling 12 minutes and long distance calls
to the United States totaling 46 minutes, then his phone bill will be $35.20.
6. 35.50
8.
7. 29
Minutes Worth of
Canadian Long
Distance Calls
0
200
400
600
0
0
0
200
Minutes Worth of
U.S. Long Distance
Calls
0
0
0
0
100
200
300
100
Jacob’s Phone Bill
Ordered Triple
(C, U, B)
30
40
50
60
40
50
60
50
(0, 0, 30)
(200, 0, 40)
(400, 0, 50)
(600, 0, 60)
(0, 100, 40)
(0, 200, 50)
(0, 300, 60)
(200, 100, 50)
Minutes Worth of
Canadian Long
Distance Calls
400
600
200
400
600
200
400
600
Minutes Worth of
U.S. Long Distance
Calls
100
100
200
200
200
300
300
300
Jacob’s Phone Bill
Ordered Triple
(C, U, B)
60
70
60
70
80
70
80
90
(400, 100, 60)
(600, 100, 70)
(200, 200, 60)
(400, 200, 70)
(600, 200, 80)
(200, 300, 70)
(400, 300, 80)
(600, 300, 90)
9. Graph (b)
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Graphing Planes (pages 47 to 51)
1
3
(a) x-z trace: z   x  1
2
1
y-z trace: z   y  1
2
5
x4
3
2
y-z trace: z  y  4
3
(b) x-z trace: z 
3
x
4
1
y-z trace: z   y
2
(c) x-z trace: z 
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(d) x-z trace: z  3x  2
1
y-z trace: z   y  2
5
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2
x 1
3
y-z trace: z  2 y  1
(f) x-z trace: z 
(e) x-z trace: z  2 x  3
y-z trace: z   y  3
1
x2
2
3
y-z trace: z  y  2
4
(g) x-z trace: z 
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(h) x-z trace: z  4 x
y-z trace: z 
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2
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(i) x-z trace: z  x  4
y-z trace: z  y  4
The Plane Truth (pages 52 to 55)
1. (a) on the plane (8 = 8), 1st Octant
(c) on the plane (8 = 8), 8th octant
(b) not on the plane (15  8), 1st octant
(d) not on the plane (9  8), 6th octant
2. (a) z = -5
(c) x = 13
(b) y = 8
(d) y = 8
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2
(b) x-z trace: z   x  1
3
1
y-z trace: z   y  1
3
3. (a) x-z trace: z  3x  4
y-z trace: z  2 y  4
4
x 1
5
3
y-z trace: z  y  1
5
(d) x-z trace: z 
(c) x-z trace: z  3x
y-z trace: z 
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y
5
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2
(e) x-z trace: z   x  4
5
3
y-z trace: z   y  4
5
(f) x-z trace: z 
(g) x-z trace: z  3
3
(h) x-z trace: z   x  2
2
y-z trace: z  2
y-z trace: z  2 y  1
y-z trace: z  3
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x 1
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(i) x-z trace: z  0
5
y-z trace: z  y
2
From Graphs to Equations (pages 56 to 61)
1. (a)
(b)
(c)
(d)
(e)
(f)
 x  2 y  z  1 or x  2 y  z  1
 2 x  y  z  1 or 2 x  y  z  1
4 x  y  5z  15 or  4 x  y  5z  15
6 x  y  5z  20 or  6 x  y  5z  20
 35x  4 y  10 z  20 or 35x  4 y  10 z  20
3x  2 y  5z  0 or  3x  2 y  5z  0
2. Graph A matches Equation (ii)
Graph B matches Equation (iii)
Graph C matches Equation (i)
3D Graphs in the Real World (pages 62 to 70)
1. (a) B  0.03C  0.06U  25
(b) B = 28.87
If Barb makes long distance calls within Canada totaling 37 minutes and long distance
calls to the United States totaling 46 minutes, then the phone bill will be $28.87.
(c) 24 minutes
(d) B-intercept = 25
The B-intercept represents the flat fee for local calls.
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(e)
(f)
(g)
(h)
B  0.06U  25
B  0.03C  25
Graph (iii)
B  0.04C  0.07U  22
This phone plan charges $0.04 per minute for long distance calls within Canada, $0.07
per minute for long distance calls to the United States, and a flat fee of $22 for local calls.
2. (a) C  600R  120W  2400
(b) C-intercept = 2400
The C-intercept represents the minimum number of calories you need per day to survive
if you neither run nor walk.
(c) W = 3
If you run for 2 hours and walk for 3 hours, you will need 3960 calories.
(d) 4500 calories
(e) C  120W  2400
(f) C  600R  2400
(g) Graph (ii)
(h) The equation would now be C  600R  120W  2000 .
The graph would have a lower C-intercept but the slopes of the traces would remain the
same.
(i) C  500R  140W  2600
In this situation the body requires 2600 calories a day just to survive. If an individual
exercises by walking, he/she will require an additional 140 calories for every hour of
walking. If an individual exercises by running, he/she will require an additional 500
calories for every hour of running.
3. (a) V  0.06 A  0.09B  80
(b) B = 180
If Pump A operates for 240 minutes and Pump B operates for 180 minutes, then there
will be 49.4 m3 of water in the pool.
(c) V-intercept = 80
The V-intercept represents the initial amount of water in the pool before either of the
pumps is turned on.
(d) 560 minutes
(e) V  0.06 A  80
(f) V  0.09B  80
(g) Graph (iii)
(h) V  0.08 A  0.12B  75
Two different pumps are draining a swimming pool that holds 75 m3 of water. Pump A
removes water at a rate of 0.08 m3/min and Pump B removes water at a rate of 0.12
m3/min.
4. (a)
(b)
(c)
(d)
E  12.50T  14K
E  14K
E  12.50T
E -intercept = 0
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(e)
(f)
(g)
(h)
The E-intercepts represents the amount of money the couple will earn if neither of them
works. If they don’t work, they don’t earn anything.
Tyrus worked for 34 hours.
K = 33
If Tyrus works for 38 hours and Kimi works for 33 hours, their combined earnings are
$937.
Graph (i)
E  13T  14.50K
Tyrus earns $13 per hour. Kimi earns $14.50 per hour.
Putting It Together Part 2 (pages 71 to 76)
1. (a) 1st
(e) 7th
(b) 5th
(f) 3rd
(c) 8th
(g) 2nd
(d) 6th
(h) 4th
2. (a) The point (5, 10, -2) is not on the plane.
(b) The point (-3, 2, 15) is on the plane.
3. (a) x = -6
(b) y = 4
4
4. (a) x-z trace: z   x  1
5
2
y-z trace: z  y  1
5
(b) x-z trace: z  4 x  1
y-z trace: z   y  1
5. (a)  7 x  5 y  5z  20 or 7 x  5 y  5z  20
(b)  2 x  10 y  5z  15 or 2 x  10 y  5z  15
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6. Graph A matches Equation (iii)
Graph B matches Equation (i)
Graph C matches Equation (ii)
7. (a) M  45E  55P  4000
(b) M-intercept = 4000
The M-intercept represents the initial amount of money he budgeted for electrician and
plumber labor costs.
(c) M  45E  4000
(d) M  55P  4000
(e) $1560
(f) P = 30
If the electrician works for 14 hours and the plumber works for 30 hours, then Kadeer
will be left with $1720.
(g) Graph (iii)
(h) M  50E  48P  3500
Kadeer budgeted $3500 for the labor costs. The electrician charges $50 per hour. The
plumber charges $48 per hour.
Systems of Equations with Three Variables (pages 77 to 83)
1. (a) x = 1, y = -2, z = 3
(c) x = 0, y = -3, z = 6
(e) x = 5, y = -3, z = 2
(b) x = 4, y = -1, z = -5
(d) x = 2, y = -2, z = 1
(f) x = 3, y = -1, z = 4
2. (a) Correct Answers
(b) Incorrect Answers
Checking Answers Using a TI-83 or TI-84 (pages 84 to 86)
1. Tanya’s answers for y and z are correct. The answer for x should be 4.
2. Steve answered the question correctly.
3. All of Shima’s answers are incorrect. The answers should be x = -1, y = 4 and z = 6.
4. Hiroshi’s answer for y is correct. The answers for x and z should be 4 and 1 respectively.
Making Predictions (pages 87 to 88)
1. (a) Minimum Number of Equations: Two
No Unique Solution
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(b) Minimum Number of Equations: Three
No Unique Solution
(c) Minimum Number of Equations: Four
No Unique Solution
(d) Minimum Number of Equations: Three
Unique Solution
(e) Minimum Number of Equations: Three
No Unique Solution
2. Not all 3 by 3 systems of equations (i.e. 3 equations with 3 variables) have a unique solution
(i.e. a common point of intersection) as illustrated in 1(b) and 1(e).
Word Problems (pages 89 to 93)
1. 2 A  B  2C  12
A  B  4C  14
A  3B  C  16
They can produce two type A parts, four type B parts and two type C parts.
2. x  2 y  3z  90
x  y  z  46
y  x  z  22
(  x  y  z  22 )
The team made seven 1-point shoots, thirty-four 2-point shots, and five 3-point shots.
3. 20c  30u  20o  680
40c  20u  30u  840
30c  25u  10o  580
Within Canada it costs 7¢ per minutes. To the United States it costs 10¢ per minute.
Overseas it costs 12¢ per minute.
4. s  m  l  164
1.10s  1.25m  1.50l  216.40
l  8  s  m (s  m  l  8)
The shop sold 54 small cups of coffee, 32 medium cups, and 78 large cups.
5. a  b  c  9000
0.04a  0.05b  0.08c  580 0.08c  0.04a  0.05b  220
(  0.04a  0.05b  0.08c  220)
Nasrin invested $2000 into the 4% interest plan, $2000 into the 5% interest plan, and $5000
into the 8% interest plan.
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