Achievement Check - McGraw-Hill Ryerson

CONTENTS 

CHAPTER 1 

Linear Systems 

MODELLING MATH: Comparing Costs and Revenues . . . . . . . . . . . . . . . . . . . 1 

GETTING STARTED: 

Social Insurance Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 

Review of Prerequisite Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

1.1 Investigation: Ordered Pairs and Solutions . . . . . . . . . . . . . . . . . . . . . 4 

1.2 Solving Linear Systems Graphically . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

1.3 Solving Linear Systems by Substitution. . . . . . . . . . . . . . . . . . . . . . . 16 

1.4 Investigation: Equivalent Equations. . . . . . . . . . . . . . . . . . . . . . . . . . 24 

1.5 Solving Linear Systems by Elimination . . . . . . . . . . . . . . . . . . . . . . . 26 

TECHNOLOGY EXTENSION: Solving Linear Systems . . . . . . . . . . . . . . . . . . 34 

1.6 Investigation: Translating Words Into Equations. . . . . . . . . . . . . . . . 36 

1.7 Solving Problems Using Linear Systems . . . . . . . . . . . . . . . . . . . . . . 38 

RICH PROBLEM: Ape/Monkey Populations . . . . . . . . . . . . . . . . . . . . . . . . . . 48 

Review of Key Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 

Problem Solving: Use a Data Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 

Problem Solving: Model and Communicate Solutions. . . . . . . . . . . . . . . . . . . . . 58 

Problem Solving: Using the Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 

Chapter 1, Student Text Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 1–1 

Chapter 1, Achievement Check Rubrics . . . . . . . . . . . . . . . . . . . . . . . . . R 1–1 

Chapter Introduction 1

CHAPTER 

1 Linear Systems 

Chapter Materials 

• Teacher Resource Master 2 

(0.5-cm grid paper) 

• graphing calculators 

• rulers 

Optional: 

• almanacs 

• atlases (Canadian and world) 

• calculators 

• cubes 

• encyclopedias 

• graphing calculators that are 

programmed to solve linear 

systems, such as the TI-92 

• international time zone maps 

• Internet access 

• MATHPOWER 10, Ontario 

Edition, Assessment and 

Evaluation Resource Kit 

• reference books on the solar 

system, endangered species, 

primates, mining, metals, and 

metallurgy 

• road map of United States 

• university calendars 

Chapter Expectations 

Overall and Specific Expectations 

By the end of the chapter, students 

should be able to model and solve 

problems involving the intersection 

of two straight lines. [AGV.01] 

Specifically, in this chapter, 

students will 

• determine the point of intersection 

of two linear relations 

graphically, with and without the 

use of graphing calculators or 

graphing software, and interpret 

the intersection point in the 

context of a realistic situation. 

[AG1.01] 

• solve systems of two linear equations 

in two variables by the 

algebraic methods of substitution 

and elimination. [AG1.02] 

• solve problems represented by 

linear systems of two equations 

in two variables arising from 

realistic situations, by using an 

algebraic method and by interpreting 

graphs. [AG1.03] 

Chapter Assessment 

A variety of assessment opportunities 

is provided throughout this chapter: 

• Performance tasks called 

Achievement Checks are found 

throughout the student text. 

Suggested strategies for using 

these tasks are provided in the 

accompanying pages of this 

teacher’s resource. Also, a rubric 

has been provided for each. 

These are in blackline 

master form and are found on 

pages R 1-1 to R 1-3 of this 

teacher’s resource. 

• Each section of this teacher 

resource includes assessment 

strategies under the heading 

Assessment. These strategies are 

in the form of ideas and suggestions 

for journal entries, portfolio 

items, written assignments, 

interview questions, observation 

checklists, self-assessment checklists, 

and presentations. These 

can be used to assess the learning 

for the particular section in 

which they are found. 

•A Chapter Test is provided at the 

end of each chapter in the text. 

Related Resources 

You will find additional material 

designed to support this chapter 

in the following resources: 

1. MATHPOWER 10, Ontario 

Edition, Practice Masters CD-ROM 

This CD-ROM can be used to 

create practice masters for each 

numbered section in the text, as 

well as additional practice masters 

for reviewing prerequisite skills.


Edition, Solutions 

This resource provides worked 

solutions for most of 

the questions in the chapter. 


Edition, Computerized Assessment 

Bank 

This is a bank of practice and 

test items, organized by numbered 

section in the chapter. 



Evaluation Resource Kit 

This resource provides general 

support for assessing and evaluating 

performance in mathematics. 

Assessment Masters for 

collecting assessment data and 

record keeping are also included. 

Chapter Mental Math 

The focus strategies in Chapter 1 

are Multiplying by Multiples of 5 and 

Dividing by Multiples of 5. There are 

practice questions provided in most 

numbered sections of this teacher’s 

resource. Students could set aside a 

“Mental Math” section in their 

notebooks to which they can add as 

the chapter and year progress. 

Technology 

In the numbered (core) sections of 

Chapter 1 in the student text, students 

use the following graphing 

calculator features in Section 1.2 

and throughout the chapter: 

• the Y Editor, to enter equations 

• the standard viewing window, to 

display graphs 

• the Intersect operation, to determine 

the coordinates of the point 

of intersection of two graphs 

• the Fraction function, to convert 

approximate decimal coordinates 

to exact fraction equivalents 

Instructions for using these features 

are found in Appendix B of both 

the text and this teacher’s resource. 

Technology Extensions 

Additional technology ideas are 

provided in the student text and 

this teacher’s resource. 

In the student text, students 

have an opportunity to learn about 

the following TI-83 graphing 

calculator features: 

• Programming, on page 34 

• Solving Systems of Equations 

(TI-92), on page 35 

• The Stat List Editor and Linear 

Regression instruction, on page 49 

In this teacher’s resource, there 

are instructions on the following 

TI-83 graphing calculator features: 

• Negative Versus Subtraction Key, 

on pages 12 and 53 

• Graph Style, on page 13 

• Square Window, on page 13 

• Selecting and Deselecting Equations, 

on page 13 

• Editing Expressions, on page 13 

• Draw Function, on page 32 

• Fraction Function on page 53 

• Window Settings, on page 53 

Modelling Math 

Comparing Costs 

and Revenues 

The questions on student text 

page 1 are designed to prepare 

students for the related Modelling 

Math problems found throughout 

the chapter. 

Chapter Introduction 1

Getting Started 

Social Insurance Numbers 

Expectations 

Students will investigate the algebraic 

formula used to create social 

insurance numbers. 

Teaching Suggestions 

Arrange students in small groups to 

work through the questions. 

Question 1 

Students should realize that the 

result of step 5 will be 0 because 

any multiple of 10 subtracted from 

the next highest multiple of 10 

results in a difference of 10. And 

they should surmise that, since the 

check digit is a single digit, the 

2-digit number 10 translates to 0. 

Questions 2 and 3 

Students could use the Round Table 

cooperative learning strategy. (The 

students are divided into several 

groups. The students in a group 

take turns doing one of the steps 

involved in creating or checking 

SINs. Group member 1 does step 1 

on a piece of paper, and then hands 

the paper to group member 2 to 

record step 2, and so on until all 

steps have been completed. The 

recording sheet can then be passed 

around the group once again so 

that each group member can check 

one step — a step that he or she did 

not complete the first time around.) 

Questions 4 and 5 

These questions will be challenging 

for many of the students. 

Sample Solution 

Page 2, question 4 

Devise a SIN for which the check 

digit is 7. 

Work backward through the steps: 

Step 5: 7 

Step 4: To end up with a check 

digit of 7, the result of step 4 must 

be a number that, when subtracted 

from the next highest multiple of 

10, results in 7, such as 53. To get 

53, you must have two numbers, 

one from step 1 and one from step 

2, that add to 53. Possible numbers: 

26 27 53 

Step 3: Find six digits that add to 26: 

1 2 8 6 1 8 26 

Step 2: Arrange the six digits into 

four even numbers, each 18 or less: 

12, 8, 6, 18 

Divide each number in half: 

12 2 6 

8 2 4 

6 2 3 

18 2 9 

q is 6, s is 4, u is 3, and w is 9 

Step 1: Find four one-digit numbers 

that add to 27: 

4 6 8 9 27 

p is 4, r is 6, t is 8, and v is 9 

Arrange the digits as pqr stu vw7: 

466 483 997 

Check: 

Step 1: 4 6 8 9 27 

Step 2: 

2 6 12 

2 4 8 

2 3 6 

2 9 18 

Step 3: 1 2 8 6 1 8 26 

Step 4: 27 26 53 

Step 5: 60 53 7 

The SIN 466 483 997 has a check 

digit of 7. 

2 Chapter 1, Linear Systems

Common Errors 

• Students make careless errors 

when following steps and doing 

the necessary calculations for 

questions 2 and 3. 

R x Students should show all their 

calculations in a step-by-step 

record. They can ask a group member 

or partner to check each step, 

either after each step is completed 

or after the question is finished. 

Math Journal 

Why do you think SINs have check 

digits 

Assessment 

Group Presentation 

Each group should create a valid 

SIN and then be prepared to 

explain to the class what they did 

to create the number. 

Review of Prerequisite Skills 

Materials 




Expectations 

Students will review the following 

prerequisite skills: 

• simplifying algebraic expressions 

• solving first-degree equations 

• graphing equations using a variety 

of techniques 

• adding and subtracting polynomials 

Using the Review 

It is suggested that students complete 

the first and last parts of each question, 

for example, parts a) and h) 

of question 1. They can then check 

their answers with those at the back 

of the text. For any questions that 

they had difficulty with or got 

wrong, students can complete more 

parts of the same question and/or 

refer to Appendix A at the back of 

their texts for additional practice. 

The following cooperative learning 

strategies might be used for 

arranging students to complete this 

Review of Prerequisite Skills for 

Chapter 1: 

Pairs Drill: Students work in pairs 

on the review questions. Partners 

alternate questions, and then 

exchange and check each other’s 

answers and solutions. 

Pairs Check: A group of four students 

divides into two pairs. In each pair, 

one student does a review question, 

while the partner coaches. The partners 

then switch roles for the next 

question. The group of four reconvenes 

after the questions in each 

part of the Review have been completed. 

They then discuss questions 

that caused difficulty or had multiple 

possible solutions or answers. 

Getting Started 3

1.1 Investigation: Ordered Pairs and Solutions 

Expectations 

Students will 

• substitute given values into an 

equation in two variables to 

determine a solution. 

• substitute given values into a system 

of two equations in two variables 

to determine a solution. 

• verify solutions by substitution. 

Related Grade 9 

Expectations 

Students 

• solved first-degree equations 

using an algebraic method. 

• identified the properties of the 

slopes of line segments with 

respect to parallelism. 

Prerequisite Assignment 

1. Calculate. 

a) 6 (3) [9] 

b) 3(5) (7) [22] 

c) 7(0.5) 3(2.5) [11] 

2. Solve each equation. 

a) x 6 12 [x 6] 

b) 3y 5 17 [y 4] 

c) 15 5x 10 [x 5] 

3. Write an equivalent equation for 

each equation in question 2. 

a) [e.g., 2x 12 24] 

b) [e.g., 9y 15 51] 

c) [e.g., 3 x 2] 

4. Write an equation for a line that 

is parallel to the line for each 

equation. 

a) y 3x 5 [e.g., y 3x] 

b) y 1 2 x [e.g., y 1 x 5] 

2 

c) y 4x 6 

[e.g., y 4x 10] 

Mental Math 

Multiplying by Multiples of 5 

43 5 43 (10 2) 

(43 10) 2 

430 2 

215 

Application 

1. Calculate mentally. 

a) 13 5 [65] 

b) 70 5 [350] 

c) 83 5 [415] 

d) 130 5 [650] 

e) 162 5 [810] 

f) 240 5 [1200] 

g) 412 5 [2060] 

h) 4.4 5 [22] 

i) 10.8 5 [54] 

j) 120.8 5 [604] 

2. Create ten more mental math 

questions for which the strategy 


could be used in your calculation. 

Give your questions to a 

classmate to solve. 


Students can work with a partner 

or in small groups on all three 

explorations. 

Investigation 1 

As a class, read through the introduction 

to review how to verify 

solutions by substitution. Note that 

students might be more familiar 

with the following L.S./R.S. set up: 

For the equation 2x y 9 and 

the solution (1, 11): 

L.S. 

R.S. 

2x y 9 

2(1) 11 

2 11 

9 



Read through the introduction as a 

class to introduce students to the 

new term system of equations. 

Students can write the new term 

and a definition in their notebooks. 

Students should be able to use 

mental math for many of the 

calculations in questions 1 and 2. 

Again, as in Investigation 1, 

students can “show their work” 

by using the L.S./R.S. recording 

format to explain why each ordered 

pair is the solution. 


Read through question 1 together 

and then ask students to explain 

what each equation represents. For 

example, C 10n 25 can be 

interpreted or read as “The cost of 

placing an ad in the Daily Gleaner is 

$10 times the number of days plus 

a fixed cost of $25.” Ask questions 

such as the following: 

• How do the rates differ for the two 

papers 

(The Daily Gleaner has a higher 

fixed cost but a lower daily rate.) 

• In which paper would you choose to 

advertise Explain. 

• For a one-day ad, which is less 

expensive (Daily Standard) 

• For a five-day ad, which is less 

expensive (Daily Gleaner) 

Challenge students to 

• explain why n, the number of 

days, is the independent variable 

or “x” and C, the cost, is the 

dependent variable or “y.” 

• predict what the graph of each 

relation will look like and compare 

their appearances. 

Students will have no difficulty 

with questions 2 and 3, if they have 

completed Prerequisite Assignment— 

on page 4 of this teacher’s resource— 

questions 3 (equivalent equations) 

and 4 (equations of parallel lines). 

Assessment 

Journal 

Explain what the ordered pair that 

satisfies the system of equations 

in Investigation 3, question 1, 

represents. 

[For a three-day ad, both the Daily 

Gleaner and the Daily Standard cost 

the same amount, $55.] 

Scoring Use the following rubric 

to assess for Understanding: 

4 – competent (student shows 

complete understanding) 

3 – satisfactory (student shows 

partial, possibly incomplete 

understanding) 

2 – inadequate (the student does 

not understand or is confused) 

1 – no response 

Journal Tip 

Don’t mark every journal every day. 

Select five to ten journals randomly 

to mark each day, so that everyone 

will get feedback after a few days. 



Edition, Practice Masters CD-ROM: 

1.1 Investigation: Ordered Pairs 

and Solutions 





Bank: 

1.1 Investigation: Ordered Pairs 

and Solutions 

1.1 Investigation: Ordered Pairs and Solutions 5

1.2 Solving Linear Systems Graphically 

Materials 




• rulers 

Expectations 

Students will 


of two linear relations graphically, 

with and without the use of 

graphing calculator or graphing 

software, and interpret the intersection 

point in the context of a 

realistic situation. [AG1.01] 


linear systems of two equations, 

in two variables, arising from 

realistic situations, by interpreting 


Prerequisite Grade 9 

Expectations 

Students 

• identified the geometric significance 

of m and b in the equation 

y mx b. 

• identified the properties of slopes 

of line segments with respect to 

parallelism. 

• determined the point of intersection 

of two linear relations by 

graphing, and interpreted the 

intersection point in the context 

of an application. 


Choose parts a, b, c, and/or d of each 

question. 

1. Write each equation in the form 

y mx b. 

a) x y 5 [y x 5] 

b) 5x 2y 10 

y 5 2 x 5 

c) 4x y 16 [y 4x 16] 

d) 6x 2y 1 0 

y 3x 1 2 

2. Find a solution for each equation 

in question 1. 

a) [e.g., (0, 5)] b) [e.g., (0, 5)] 

c) [e.g., (1, 20)] d) e.g., 2, 51 2 

3. Find the x- and y-intercepts for 

the graph of each equation in 

question 1. 

a) [(0, 5) and (5, 0)] 

b) [(0, 5) and (2, 0)] 

c) [(0, 16) and (4, 0)] 

d) 0, 1 2 and 1 6 , 0 

4. Sketch the graph for each equation 


5. Write the equation for a line 

parallel to each equation in 

question 1. 

a) [y x 10] 

b) y 5 2 x 

c) [y 4x 10] 

d) [y 3x 2] 

6. Write the equation for a line 

perpendicular to each equation 


a) [y x 10] 

b) y 2 5 x 

c) y 1 4 x 10 

d) y 1 3 x 

7. Write an equivalent equation for 

each equation in question 1. 

a) [2y 2x 10] 

b) [2y 5x 10] 

c) [3y 12x 48] 

d) [6y 18x 3] 


Mental Math 


43 50 43 (100 2) 

(43 100) 2 

4300 2 

2150 

Application 


a) 15 50 [750] 

b) 90 50 [4500] 

c) 63 50 [3150] 

d) 140 50 [7000] 

e) 122 50 [6100] 

f) 260 50 [13 000] 

g) 324 50 [16 200] 

h) 2.8 50 [140] 

i) 20.6 50 [1030] 

j) 100.4 50 [5020] 







Investigation Answers 

Investigation: Use a Graphing 

Calculator 

1. a) y 7 x, y 1 x 

c) (3, 4) 

2. a) 8: (0, 7), (1, 6), (2, 5), (3, 4), 

(4, 3), (5, 2), (6, 1), and (7, 0) 

b) an infinite number 

3. a) (3, 4) 

b) verify by substituting the 

values of 3 for x and 4 for y into 

each equation 

4. a) 3 b) 4 

5. a) (3, 5) b) (6, 1) c) (3, 3) 


Investigation: Use a Graphing 

Calculator 

(graphing calculators or grid paper) 

Read the opening paragraph of the 

Investigation as a class. Arrange 

students in pairs or small groups to 

work through the questions. 

For question 1, part a), encourage 

students to solve each equation 

in the form y mx b so that 

they can determine the slope and 

y-intercept in order to predict the 

appearance of the graph. They 

can use what they know about the 

role of m and b in the equation 

y mx b. For example, the 

graph for y x 7 will have a 

negative slope (it will slope down 

from left to right) and will intersect 

the y-axis at 7. 

NOTE: The coloured-type references 

to technology in the text 

for parts b) and c), in this case Y 

Editor, standard viewing window, 

and Intersect operation, indicate 

that instructions for these procedures 

are found in Appendix B at 

the back of the text and in Appendix 

B of this teacher’s resource. 

The specific instructions for using 

the graphing calculator to find the 

point of intersection are as follows: 

1. Prepare the calculator for 

graphing: To check the Mode 

settings, press k. The default 

setting is the first entry in each 

row. Active settings are highlighted. 

If the active setting in 

any row is not the default setting, 

use the arrow keys to move 

the cursor to each default setting 

and press e to activate it. 

To check the Format settings, 

press O y. If the active setting 

is not the default setting, 

use the arrow arrows to move 

the cursor to each default setting 

and then press e to activate it. 

1.2 Solving Linear Systems Graphically 7

To clear any active statistical 

plots or equations in the Y 

Editor, press x. If any of 

Plot1, Plot2, or Plot3 across 

the top of the screen is highlighted, 

it is active and must be 

cleared. Use the arrow keys to 

move the cursor to each highlighted 

plot and press e to 

deactivate it. If there are any 

equations in Y1, Y2, and 

so on, use the arrow keys to 

move the cursor to anywhere 

on the right side of each equation 

and press b. 

The calculator is now ready 

to graph the two equations in 

the Y Editor. 

2. To enter the two equations into 

the Y= Editor, press: 

NuM7 e for y x 7 

u 1 e for y x 1 

Y Editor 

3. To display the graphs of the system 

of equations in the standard 

viewing window, press y 6. 

Note that the standard viewing 

window has the following settings: 

Standard Viewing Window 

(See Technology Extension: 

Graphing Calculator, Graph Style 

and Square Window on page 13 

of this teacher’s resource for 

instructions for displaying each 

graph using a different style and 

for displaying the two graphs so 

that they actually appear perpendicular 

in the display.) 

4. To find the coordinates of the 

point of intersection using the 

Intersect operation, press: 

Or5 e ee 

Coordinates of the Intersection Point 

Note that students are asked to 

use the standard viewing window 

of the graphing calculators. 

This displays all four quadrants 

equally with Xmax 10, Ymax 

10, Xmin 10, and Ymin 

10. You might discuss with 

the students why displaying only 

the first quadrant (Xmin 0 

and Ymin 0) would be a 

preferable representation of the 

problem situation. (Both y and x 

are whole numbers so there cannot 

be a negative number of 

medals.) You might also ask why 

both relations are considered 

discrete and therefore would 

more accurately be represented 

by a discrete graph (a series of 

separate points) rather than a 

line. (Both x and y are whole 

numbers so points such as 

(6.5, 7.5) have no meaning.) 


For question 2, elicit why there 

is not an infinite number for solutions 

for x y 7. (Both y and x 

are whole numbers that add to 7.) 

Teaching Examples 

(graphing calculators or grid paper) 

Students can read through each 

teaching example and its solution, 

and then move on to the Practice, 

and Applications and Problem 

Solving questions. 

Alternatively, students could read 

through each example and solution, 

and then a similar example can be 

assigned for them to solve. They 

can use the solution in the text for 

guidance as required. 

Depending on how the examples 

are presented, students can work in 

pairs, small groups, or individually. 

Example 1, Solution 1 

Students can refer to Appendix B in 

their texts for instructions for using 

the Y Editor, the standard viewing 

window, and the Intersect 

operation. (See the Teaching 

Suggestions on pages 7 and 8 of this 

teacher’s resource for instructions for 

finding the point of intersection on 

the TI-83 Plus or refer to Appendix B 

of this teacher’s resource.) 


Elicit another way to graph using 

paper and pencil. For example, 

determine the x- and y-intercepts 

for each graph and then plot and 

join the two points. 

Consider assigning parts of 

question 1 on text page 12 after 

reviewing Example 1. 


Students might enter the equation 

y 1 x 5 into the Y Editor as 

2 

Y2 1 X 5 or as Y2 .5X 5. 

2 


Consider assigning parts of question 

2 on text page 12 after reviewing 

Example 2. 

Example 3 

The coloured-type reference to Frac 

function in this example indicates 

that graphing calculator instructions 

are found in Appendix B of the 

text. Instructions are also found in 

Appendix B of this teacher’s resource. 

To use the Fraction function to convert 

the approximate decimal values 

of the intersection point coordinates 

to exact fractions, follow these steps 

immediately after the coordinates of 

the intersection point are displayed: 

1. To convert the X coordinate to 

a fraction, press: 

Okui1 e 

2. To convert the Y coordinate to a 

fraction, press: 

a 1 i 1 e 



reviewing this example. 

Example 4 

Students can rearrange the equations 

in part b) into the y mx b form 

before graphing. Challenge students 

to predict the appearance of the 

graphs for parts a) and b) using 

what they know about the role of m 

and b in the equation y mx b 

and parallelism. They can then graph 

using a paper-and-pencil method, 

graphing calculators, or graphing 

software to check their predictions. 

After completing this example, 

students can read the chart at the top 

of student text page 11 and then 

copy the chart into their notebooks. 

Consider assigning parts a), d), 

and g) of question 3 on text page 

12 after reviewing this example. 


Example 5 

Students can graph each system of 

equations on their graphing calculators. 

(See Technology Extension, 

Graphing Calculators, Selecting and 

Deselecting Equations on page 13 of 

this teacher’s resource for instructions 

on how to enter multiple 

equations into the Y Editor but 

display only selected ones.) 




Key Concepts 

Students can copy the key concepts 

for this section into their notebooks. 

Communicate Your Understanding 

Answers 

1. (3, 4) because the graphs of the 

two equations intersect at that 

point 

2. Sample answers: graphing 

both equations on a graphing 

calculator and using the 

Intersect operation to find 

the point of intersection; or 

graphing manually using the 

intercepts for each equation and 

then determining the point 

of intersection 

3. Each linear relation in a system 

of linear equations is represented 

graphically by a straight line, 

and two straight lines cannot 

intersect at exactly two points. 

4. There are no solutions to the 

linear system because the relations 

are represented graphically 

by parallel lines with different 

y-intercepts. Therefore, the 

lines do not intersect. 

Using Communicate Your 

Understanding 

Arrange students in small groups 

or in pairs. Students should work 

through these questions with minimal 

assistance. Responses can be 

in writing and/or given orally. 

1. Ensure that students understand 

what “Justify your answer” 

means. They may suggest substituting 

the values of the solution 

into both equations to 

“prove” or “justify” that the 

solution satisfies both equations. 

2. Students should provide step-bystep 

instructions for graphing 

the given system of equations, 

on a graphing calculator, using 

graphing software, or graphing 

manually. Each student should 

exchange his or her instructions 

with another student in order to 

test the accuracy and completeness 

of the instructions. 

3. You might ask students to 

describe two relations that 

would have exactly two solutions 

and then sketch the graph of the 

system. (For example, a system 

of equations that includes a linear 

and a non-linear relation 

could have two solutions.) 

4. Students can discuss this question 

in their groups before writing 

their responses in their notebooks. 

Suggest that students visualize 

the two graphs first. Some 

students would benefit from 

actually graphing the equations. 

Practice, and Applications and 

Problem Solving 

(graphing calculators and grid papers) 

Students are expected to determine 

the point of intersection graphically, 

with and without the use of technology. 

If students are using graphing 

calculators, ensure that all students 


Question(s) 

Students can refer to 

the following in the text: 

1 and 8 Example 1 on page 7 

2, 7, and 9 

3 

4 and 5 

6 

Example 2 on page 8 

Examples 2 and 4 on 

pages 8 and 10 



complete at least one question in 

the Practice section manually. 

In question 7 in Applications 

and Problem Solving, there is 

no clearly independent variable. 

Therefore, the equations can be 

solved for either variable, a or g, as 

long as both equations are solved 

for the same variable. For question 

9, students might need a review of 

the term “vertex of an angle.” For 

questions 10 to 12, the Intersect 

operation can still be used for finding 

the intersection points of three 

or more linear equations on the 

graphing calculator, but the arrow 

keys must be used to move the cursor 

to the appropriate graph when 

prompted each time with First 

curve Second curve The equation 

for the graph on which the 

cursor is currently located is displayed 

in the top left corner. 

The cursor is located on the graph of 

y 2x (Y1 2X). 

The following table will help 

you direct students to the related 

support material in the student 

text for many of the questions in 


Problem Solving: 



Write an equation that forms a system 

of equations with x y 4, 

so that the system has 

a) no solution 

b) infinitely many solutions 

c) one solution 

a) For a system of equations to 

have no solution, the graphs of the 

equations must be parallel and distinct, 

that is, they must have different 

y-intercepts. 

Find the slope (m) and y-intercept (b) 

of x y 4: 

x y 4 

y x 4 y mx b 

m is 1 and b is (0, 4) 

Write an equation with a slope of 1 

and a y-intercept other than (0, 4): 

y x 10 

b) For a system of equations to have 

infinitely many solutions, the equations 

must be equivalent. Write an 

equation that is equivalent to 

x y 4: 

2(x y) 2(4) 

2x 2y 8 

c) For a system of equations to 

have one solution, the graphs 

of the two equations must intersect. 

Write an equation for a line that is 

perpendicular to x y 4: 

For the graph of an equation to be 

perpendicular to the graph of 

another equation, the slopes must 

be negative reciprocals. 

1 is the negative reciprocal of 1 

because 1 1 1. 


Write an equation with a slope 

of 1: y x 4 

Common Errors 

• Students use the subtraction key, 

L and the negative key N incorrectly 

on their graphing calculators. 

R x When these keys are used incorrectly, 

an ERROR menu will appear. 

Students should select 2:Goto from 

the menu to return to the Home 

screen. The cursor will be at or near 

the location of the error. Students 

can then use the editing keys to correct 

the expression. (See Technology 

Extension, Editing Expressions, on 

page 13 of this teacher’s resource.) 

Students should also be aware that 

the negative sign has a different 

appearance than the subtraction 

key in the display. 

• Students incorrectly and/or 

incompletely solve equations with 

negative numerical coefficients. For 

example, for question 3, part k): 

x 2y 2 0 

2y x 2 

2y x 2 

 

2 2 2 

y 1 2 x 1 

R x Encourage students to show 

every step of the solution and to 

include the “implied” numerical 

coefficient of 1, where necessary. 

For question 3, part k): 

1x 2y 2 0 

1x 1x 2y 2 2 0 1x 2 

2y 1x 2 

Assessment 

Written Assignment 

2 

 

y 

2 

1 x 2 

 

2 

y 1x 

2 

 

2 

2 

y 1 2 x 1 

Assign the following: 

a) Create and solve a realistic 

problem for which you would need 

to find the point of intersection 

of a system of two linear relations. 

(Some students will use text question 

as a model; others will be 

more creative.) 

b) Explain how the point of intersection 

is the solution to your problem. 

(Students should explain how the 

graph models the realistic situation.) 

c) List several other things that 

your graph tells you about the 

problem situation. 

(Students can comment on the 

meaning of intercepts, slopes, and 

any restrictions on the variables.) 

Scoring Mark analytically out of 10: 

part a) 2 – clarity of problem 

1 – correct solution 

2 – explains method of 

solution 

part b) 1 – explains intersection 

point 

part c) 4 – other features explained 

Journal 

Compare the two methods of finding 

the point of intersection graphically: 

pencil-and-paper and graphing 

calculator. List advantages and disadvantages, 

using examples to show 

how to solve systems of equations. 

Scoring Use the following 

criteria: 

Does the student 

• include an example of a system for 

which pencil and paper would be 

preferable and one where a graphing 

calculator would be preferable 

• use correct mathematical terminology 

and form 

• record the correct calculator 

keystrokes 

• show correct solutions to both 

examples 


The graph of Y1 2X 16 appears 

as a thicker line. 

Square Window If you graph 

perpendicular lines in the standard 

viewing window, they will not 

appear perpendicular because 

∆x ∆y. To create a square viewing 

window where ∆x ∆y, select 

ZSquare from the ZOOM menu 

by pressing y 5. 

Selecting and Deselecting Equations 

You can enter multiple equations 

into the Y Editor but display 

only selected graphs in the graphing 

window. To select or deselect an 

equation, use the arrow keys to 

move the cursor onto the sign of 

the equation and then press e. 

Journal Tip 

Make sure all journal entries are 

dated so that growth can be 

assessed. 




1.2 Solving Linear Systems 

Graphically 





Bank: 


Graphically 

Technology Extension 

Graphing Calculators 

Graph Style Use the Graph Style 

icon to the left of each equation in 

the Y Editor to differentiate 

between two graphs that are to be 

displayed in the window at the same 

time. To do this, use the arrow keys 

to move the cursor onto the Graph 

Style icon to the left of one of the 

equations and then press e once 

to select a thick line. Press f to 

see how the two graphs differ. 

For example, for question 17, 

part a): 

Only the graph for Y2 4X 59 will be 

displayed. 

Editing Expressions Explore the editing 

keys on your graphing calculator. 

On the TI-83 Plus, pressing: 

• the left and right arrow keys 

moves the cursor to any location 

in the expression. 

• d deletes the character directly 

under the cursor. 

• Odinserts a character to the 

left of the cursor. 

• ODmoves the cursor to the 

beginning of the expression. 

• OBmoves the cursor to the 

end of the expression. 

• b deletes the line upon which 

the cursor is located if there is 

text on the line. 

• b deletes the entire Home screen 

if there is no text on the line. 



Comparing Costs and 

Revenue 

(graphing calculators or grid paper 

and rulers) 

Expectations 

Students will 


of two linear equations 

graphically. [AG1.01] 







If students use a graphing calculator, 

the Window settings will have 

to be set at Xmin 0, Xmax 

1000, Ymin 0, and Ymax 

1000 in order to display both 

graphs in the first quadrant of the 

graphing window and be able to 

use the Intersect operation. 

Career Connection 

Wildlife Biology 


and rulers) 

Expectations 

Students determine the point of 

intersection of two linear equations 




or in pairs. 

Question 1 

To determine the point of intersection 

using a graphing calculator, 

students will follow these steps: 

Step 1: Decide which variable is 

the independent variable, that is, 

x, and which is the dependent 

variable, or y. This is arbitrary 

for this problem. 

Step 2: Solve each equation for the 

dependent variable, for example: 

n s 130 000 and n 25s 

Step 3: Enter the equations into 

the Y Editor. 

Step 4: Determine the Window 

settings. (Xmin 0, Xmax 

130 000, Ymin 0, and Ymax 

130 000) 

Step 5: Display the graph. 

Step 6: Use the Intersect operation 

to find the point of intersection. 

(125 000, 5000) 

Question 2 

Students can inquire in the guidance 

department of their school for 

information on Wildlife Biology as 

a career. They could also visit the 

following Web site, “Exactly How 

Is Math Used in Technology” to 

investigate how math is applied to 

technology in a variety of careers, 

including wildlife biology: 

www.scas.bcit.bc.ca/scas/math/ 

examples/table.htm 

For example, students can find 

out how linear algebra is applied in 

forestry and wildlife careers. 


Achievement Check 

Expectations 

This performance task is designed 

to address the following expectations 

from the Ontario Curriculum: 

Can the student 

• interpret the intersection point in 

the context of a realistic solution 

[AG1.01] 





graphs [AG1.03] 

Sample Answer 

If I were the only one in my family 

using the Internet for e-mails and 

occasional Internet use, I would 

choose Plan A because usage would 

probably be less than 8 h monthly. 

Plan A is less expensive than both 

Plan B and Plan C for less than 8 h. 

I would change to Plan B if my 

usage increased to more than 8 h but 

was less than 22 h, and to Plan C if 

my usage increased to 22 h or more. 

If my family were using the 

Internet, I would choose Plan B 

because usage would probably be 

between 8 h and 22 h monthly. 

Plan B is less expensive than Plan A 

for more than 8 h, and Plan B is 

less expensive than Plan C for less 

than 22 h. I would change to Plan 

A if usage decreased to less than 8 h 

monthly and to Plan C if usage 

increased to more than 22 h monthly. 

If our usage were about 1 h 

daily, I would choose Plan C 

because monthly usage would be 

more than 22 h. Plan C is less 

expensive than both Plan A and 

Plan B for more than 22 h. I would 

change to Plan A if my usage 

decreased to less than 8 h monthly 

and to Plan B if my usage decreased 

to less than 22 h but more than 8 h. 

Note that a level 4 answer might 

also include: 

• an indication of what the intersection 

points mean. For example, 

at 8 h, Plan A is the same 

price as Plan B ($26.95), so at 8 h 

either plan could be chosen. 

• the need to consider average use. 

For example, one should monitor 

use to determine the average 

number of hours on-line 

monthly. 

• the fact that a flat rate can be 

attractive. For example, Plan C 

might be chosen because it has a 

flat rate that can be budgeted for 

and the usage does not have to 

be monitored. 

Assessment 

The following categories of the 

Achievement Chart of the Ontario 

Curriculum can be assessed using 

this performance task: 

• Knowledge/Understanding 

• Application 

• Communication 

There is a rubric provided for this 

task. It is in blackline master form 

and is found on page R 1-1 in this 

teacher’s resource, following the 

teaching notes for Chapter 1. 


1.3 Solving Linear Systems by Substitution 

Materials 


(0.5-cm grid paper) or graphing 

calculators 

Optional: 

• almanacs or atlases 

• cubes 

Expectations 

Students will 


in two variables by the algebraic 

method of substitution. 

[AG1.02] 





algebraic method. [AG1.03] 


Expectations 

Students 

• manipulated first-degree polynomials 

to solve first-degree equations. 

• identified the properties of the 




1. Simplify and solve each equation. 

a) 2(6 4y) 1 3y [y 1] 

b) 6x 3(2x 3) 9 

[no solution] 

c) 2(6 y) 2y 8 0 

[y 1] 

d) 5x 3(7x) 2 

x 1 

13 

2. Solve each equation for each 

variable. 

a) 2x 3y 1 

x 1 2 3 y 

; y 2 x 1 

2 3 3 

b) x 4y 6 

x 6 4y; y 3 2 4 x 

c) 7x y 0 

x y 

; y 7x 

7 

d) 2x 2y 7 

x 7 2 y; y 7 2 x 

Mental Math 


45 5 45 (10 2) 

(45 10) 2 

450 2 

225 

67 50 67 (100 2) 

(67 100) 2 

6700 2 

3350 

Application 


a) 57 5 [285] 

b) 208 5 [1040] 

c) 320 5 [1600] 

d) 550 5 [2750] 

e) 4.6 5 [23] 

f) 35.8 5 [179] 

g) 25 50 [1250] 

h) 308 50 [15 400] 

i) 520 50 [26 000] 

j) 4.5 50 [225] 








Investigation: Use the Equations 

1. a) 16l 480 b) l 30 

2. a) Substitute 30 for l into one 

of the equations and solve for t. 

b) t 450 

3. (30, 450) or (450, 30) 


4. a) l 480 t or t 480 l 

b) t 15(480 t); 

450 or 480 l 15l; 30 

c) 30 or 480 

d) yes 

5. a) 450 

b) 30 

6. a) (5, 10) 

b) (8, 1) 

c) (3, 4) 



(optional: graphing calculators or grid 

paper and rulers) 

Arrange students in pairs or small 

groups to work through the 

Investigation. 

Ask students to explain what 

each equation represents. For 

example, t l 480 means that 

the number of Siberian tigers plus 

the number of Amur leopards is 

480 altogether. Elicit other equivalent 

equations that could be used 

to represent the situation, that is, 

t 480 l, l 480 t, and 

t 

l t 15 or . 

1 5 

In question 3, students will have 

to make an arbitrary decision as to 

which variable is the independent 

variable (x) and which is the dependent 

variable (y) in order to express 

the solution as an ordered pair; 

that is, the ordered pair can be 

expressed as (t, l) or as (l, t). In this 

relationship, one variable is not 

“dependent” on the other. 

In question 4, students are asked 

to solve the system of equations by 

substitution the other way. That is, 

they solve equation (1) for either 

variable and then substitute the 

resulting expression into equation (2). 

Alternatively, students could 

approach question 4 by solving 

equation (1) for l and then substituting 

the resulting expression for l 

as the value of t in equation (2). 

After solving the equation to determine 

the value of t, they can substitute 

the value for t into either 

equation to find the value of l. 

The key concept in question 4 

is that there is more than one way 

to solve a system of equations by 

substitution. Note that, because of 

the complexity of solving some 

equations, that is, having to work 

with fractions and/or negative 

numbers, there is often a preferred 

way. (See the teaching suggestions 

for Example 1 on page 18 of this 

teacher’s resource.) 

In question 6, students can 

solve each system of equations two 

different ways in order to check the 

solution. Some students might prefer 

to check their solutions by 

graphing. For example: 

For question 6, part a): 

If students use the standard viewing 

window for question 6, part a), 

they will discover that the Intersect 

operation will not work because the 

point of intersection is not within 

the Window settings. The Ymax 

setting must be adjusted to a value 

greater than 10. 

1.3 Solving Linear Systems by Substitution 17


(optional: graphing calculators) 


teaching example and its solution. 

Alternatively, each example could 

be recorded on the board or an 

overhead for students to solve without 

referring to the solution in the 

text. They can then check their 

solutions against the one in the text. 




Example 1 

Challenge students to solve the system 

of equations by first solving 

equation (1) for y instead of for x. 

The resulting, more complex, solution 

may convince them of the practicality 

of solving for a term with a 

numerical coefficient of 1 first. 

For example: 

x 4y 6 (1) 

2x 3y 1 (2) 

Solve (1) for y: 

x 4y 6 

4y 6 x 

y 6 x 

 

4 

y 6 4 x 

4 

y 3 2 x 

4 

Substitute the expression for y into (2): 

2x 3y 1 

2x 3 3 2 x 

4 1 

2x 9 2 3 x 

1 

4 

2 3 4 x 1 9 2 

1 1x 11 

4 2 

4 

1 1 1 1x 11 4 

1 

4 2 1 

x 2 

Substitute the value for x into (1): 

x 4y 6 

2 4y 6 

4y 4 

y 1 

Consider assigning selected parts 

of question 3, parts a) to n), on 

text page 21 after reviewing this 

example. 

Example 2 

If students would like to solve 

the system of equations on their 

graphing calculators as shown at 

the top of page 19, they can refer 

to Appendix B at the back of their 

books for instructions on using the 

Fraction function to change the 

resulting decimal coordinates to 

exact fractions. To use the Fraction 

function to convert the approximate 

decimal values of the intersection 

point coordinates to exact 

fractions, follow these steps immediately 

after the coordinates of the 

intersection point are displayed: 



Okui1 e 

2. To convert the Y coordinate to 


a 1 i 1 e 




Examples 3 and 4 

Ask students to explain why it 

might be advisable to begin by 

rearranging the equations in the 

slope-intercept form (y mx b). 

(Students may realize that, if they 

can determine that the graphs are 

parallel and distinct (the m values 

are the same but b is different), 

they will know that there is no 


solution. Or, if they can determine 

that the two equations are equivalent, 

they will know that there are 

infinitely many solutions.) 

Consider assigning parts o) and 

q) of question 3 on student text 

page 21 after reviewing this example. 

Key Concepts 


into their notebooks. 


Answers 

1. If the two equations in the system 

of equations have a solution, 

the expression 3x 8 can 

be substituted for y in x y 4 

because, at the point of intersection, 

the value of y must be the 

same in both equations. 

2. Sample answer: 

Step 1: Solve the first equation, 

y 3x 8, in terms of x. 

Step 2: Substitute the expression 

for x in the second equation 

and then solve for y. 

Step 3: Substitute the actual 

value for y into either equation 

to find the actual value for y. 

Step 4: Check the solution by 

substituting it into both equations. 

3. a) The equations are equivalent 

and therefore there are infinitely 

many solutions. 

b) The equations are parallel 

and distinct, and therefore there 

is no solution. 


Understanding 


or pairs to work cooperatively on 

answering the questions. They can 

then write their responses in their 

notebooks and/or be prepared to 

present them to a larger group or 

to the class. 

1. Have students explain why the 

process of substitution is not 

valid for parallel lines and therefore 

results in a solution that is 

not true. (There is no point of 

intersection; therefore, the value 

of x or y will never be the same 

for both equations.) 

2. Students can test their descriptions 

for accuracy and completeness 

by reading each step aloud 

to their partners while their 

partners solve the system of 

equations following the steps. 

Practice, and Applications 

and Problem Solving 


and rulers) 

Before students begin work on the 

equations, select several parts from 

question 3 and elicit which equation 

should be solved first and for 

what variable and why. For example, 

for question 3, part a), 2x y 6 

should be solved first for y because y 

has a numerical coefficient of 1. 

As well, remind students that they 

might rearrange both equations in 

the form y mx b to determine 

if they are parallel (parts o) and s)) 

or equivalent (part g)). 

In question 4, students are 

expected to express the exact solution 

to each system, that is, using 

a fraction rather than an approximated 

decimal. For parts a), c), g), 

and h), students can express the 

solution as an exact decimal or 

fraction, for example, for part h), 

3 4 , 1 2 

or (0.75, 0.5). 

For question 5, students can 

graph manually or use a graphing 

calculator. 




support material in the student text 

for many of the questions in the 



Question(s) 



3 Examples 1 , 3, and 4 

on pages 17, 19, and 20 

4 

5 – 8 



Sample Solutions 


Theatre tickets 

a) Interpret each equation in words: 

The equation a s 550 means 

that the number of adults plus the 

number of students is 550. 

The equation 20a 12s 9184 

means that twenty dollars times the 

number of adults plus twelve dollars 

times the number of students is $9184. 

b) Solve the system to find the number 

of adult and student tickets sold: 

If a s 550, then a 550 s. 

If 20a 12s 9184 and a 550 s, 

then 

20(550 s) 12s 9184 

s 227 

If s 227 and a s 550, then 

a 227 550 

a 323 

Check a 323 and s 227 for 

a s 550: 

L.S. 

R.S. 

a s 550 

323 227 

550 

Check a 323 and s 227 for 

20a 12s 9184: 

L.S. 

R.S. 

20a 12s 9184 

20(323) 12(227) 

9184 

There were 323 adult tickets sold 

and 227 student tickets sold. 


Coordinate geometry 

The three lines x y 1 0, 

2x y 4 0, and x y 5 0 

intersect to form a triangle. Find 

the coordinates of the vertices of 

the triangle. 

x y 1 0 (1) 

2x y 4 0 (2) 

x y 5 0 (3) 

To find each of the three vertices, 

solve the system (1) and (2), solve 

the system (1) and (3), and then 

solve the system (2) and (3). 

Enter all three equations into the 

Y Editor: 

Solve (1) and (2): 

Use the Intersect operation to find 

the solution for Y1 X 1 and 

Y2 2X 4. 




the solution for Y2 2X 4 

and Y3 X 5, using the arrow 

keys to move the curser to the 

graph for Y3 X 5 for the 

First curve and Y2 2X 4 

for the Second curve. 

The coordinates of the three vertices 

are (1, 2), (3, 2), and (9, 14). 


What value of m gives a system 

with no solution 

x(m 1) y 6 0 

2x y 3 0 



the solution for Y1 X 1 and 

Y3 X 5, using the arrow 

keys to move the curser on to the 

graph for Y3 X 5 for the 

Second curve: 

Rearrange both equations in the 

slope-intercept form, y mx b: 

x(m 1) y 6 0 

y (m 1)x 6 

2x y 3 0 

y 2x 3 

For a system of equations to have 

no solution, the graphs of the equations 

must be parallel and distinct, 

that is, the slopes (m) must be equal 

and the y-intercepts (b) different: 

To make the graphs of the equations 

parallel, m 1 must be 

equal to 2: 

m 1 2 

m 1 


The graphs of the equations are 

distinct because the y-intercepts are 

different, (0, 6) and (0, 3). 

The value of m 1 results in a 

system with no solution. 

Common Errors 

• Students have difficulty manipulating 

polynomial expressions to 

solve equations. 

R x Remind students to 

• include every step in the solution, 

• record the numerical coefficient 

of 1 for all applicable terms, and 

• check their solutions with 

another student. 

Assessment 

Portfolio/Pairs 

Assign the following: 

a) Write and solve a system of 

equations that is best solved 

i) using substitution rather than 

by graphing. 

ii) by graphing rather than by 

substitution. 

iii) by graphing manually rather 

than by using a graphing 

calculator. 

b) Explain your rationale for 

each system. 

(Students should plan their answers 

together but each should write an 

individual report to go into the 

student’s portfolio.) 

Scoring Use the following rubric 

to evaluate for Understanding and 

Communication: 

4 – competent (student shows complete 

understanding; student 

communicates clearly, using 

correct mathematical form 

and language) 

3 – satisfactory (student shows 

incomplete understanding; 

student either communicates 

clearly or uses correct mathematical 

form and language) 

2 – inadequate (the student does 

not understand or is confused; 

student is unclear in communicating 

and uses incorrect form 

and language) 

1 – no response 

Portfolio Tips 

Include a note in the portfolio 

accompanying each item that 

explains the purpose of the item. 

Always record a date on the item in 

order to assess progress over time. 





by Substitution 




Edition, Computer Assessment 

Bank: 


by Substitution 

Extension 

(almanacs or atlases) 

Create your own mountain problems 

similar to question 6 on student 

text page 22. Look up your own 

data or use the following data: 

McKinley (North America) 6190 m 

Kibo (Africa) 

5890 m 

[e.g.: The difference between the 

heights of McKinley and Kibo is 300 m. 

McKinley is 1.05 times the height of 

Kibo. Find the heights of the two 

mountains.] 




and Revenues 



Expectations 

Students will 


in two variables by graphing 

[AG1.01] or by the algebraic 

method of substitution. [AG1.02] 








Part a) is straightforward. Students 

solve the system, either algebraically 

or by graphing. If students use a 

graphing calculator, they must ensure 

that the point of intersection is displayed 

within the Window settings: 

The point of intersection (4, 270) means 

that, at 4 h, both plumbing companies 

cost the same, $270. 

If students have approached part a) 

algebraically, they might find part b) 

easier if they graph the equations 

or, at least, rearrange the equations 

in the y mx b form so that they 

can visualize the graphs: 

C ABC 50h 70 

C Q 55h 50 

The graph for ABC begins at 

(0, 70) and the graph for Quality 

begins at (0, 50), so ABC is more 

expensive up to the point of intersection, 

(4 h, $270), or for less than 

4 h. The graph for Quality has a 

slope of 55, whereas the graph for 

ABC has a slope of 50, so, after the 

point of intersection, or after 4 h, 

Quality would be more expensive. 

For part c), students can write 

and solve an equation: 

If C Q C ABC 30, then 

(50 55h) (70 50h) 30 

h 10 

Therefore, at 10 h, Quality is $30 

more than ABC. 

Logic Power 

(optional: cubes) 

Answer: 

1. There are 34 cubes in the 

stack. 

2. a) There would be 5 cubes 

with 4 green faces. 

b) There would be 9 cubes 

with 1 green face. 


1.4 Investigation: Equivalent Equations 

Materials 



Expectations 

Students will 




• create equivalent systems of 

equations. 


Expectations 

Students 

• graphed lines by hand, using a 

variety of techniques, for example, 

using the intercepts. 

• added polynomials. 


1. Complete the ordered pair solution 

for each equation. 

a) x y 3 (x, 4) [(7, 4)] 

b) x 2y 4 (10, y) [(10, 3)] 

c) x y 5 (0, y) [(0, 5)] 

2. a) Explain how you would graph 

the following equations manually. 

x 2y 4 5x 10y 20 

b) Graph the equations. What 

do you notice Explain why. 

[They are both represented by the 

same graph because, when the 

second equation is simplified, it 

is the same as the first equation.] 



groups. 


This investigation prepares students 

for Investigation 2 by reviewing 

equivalent equations. Students 

should recognize that equivalent 

equations are the same equation, 

and, as such, have the same solution 

and the same graph. 


(grid paper) 

Students can graph each system in 

questions 1 and 2 using a different 

graph style. They will discover that 

both systems have the same solution, 

that is, both systems of equations 

intersect at the same point. 

y 

7 

6 

5 

4 

3 

1 

1 0 

1 

2 

3 

x y 7 

y 2 

(5, 2) 

1 2 3 4 6 7 

x y 3 

x 5 

x 

In question 3, part a), students 

create an equation equivalent to 

each equation in system A in order 

to create an equivalent system of 

equations called system C. 

In question 4, students should 

distinguish between how systems 

A and C relate compared with how 

systems A and B or systems B and C 

relate. That is, systems A and C are 

equivalent because they are made up 

of equivalent equations and, therefore, 

have the same graphs and solution. 

Systems A and B (and systems 

B and C) are equivalent because 

they have the same solution; however, 

they consist of different graphs 

and non-equivalent equations. 

Question 5 provides students 

with an opportunity to create a system 

of equations equivalent to a 


given system of equations. Ensure 

students are aware that the system 

must consist of two-variable equations 

that are non-equivalent to 

the given system. 


(grid paper) 

In questions 1 to 4, students add 

the equations in a given system of 

equations to create a third equation 

and then use different combinations 

of the three equations to create 

equivalent systems of two equations. 


determine if the three systems are 

equivalent by graphing or by finding 

a solution for one system and 

then checking to see if it is the 

solution to the other two systems. 


Page 24, Exploration 2, question 5 

Step 1: Determine the solution to 

the given system of equations: 

x 2 y 1 

The solution is (2, 1). 

Step 2: Create two different equations 

with a solution of (2, 1): 

Substitute (2, 1) for x and y into the 

slope-intercept form. 

y mx b 

1 m(2) b 

Substitute any value for m, and 

determine b. 

For m 6: 

1 m(2) b 

1 6(2) b 

b 13 

If m 6 and b 13, then equation 

1 is y 6x 13. 

Repeat for the second equation: 

For m 5: 1 m(2) b 

1 5(2) b 

b 9 

If m 5 and b 9, then equation 

2 is y 5x 9. 

An equivalent system of equations is 

y 6x 13 and y 5x 9. 

Assessment 

Group Presentation 

Students can work collaboratively 

to prepare a short presentation to 

describe at least two methods for 

creating a system of equations that 

is equivalent to a given system. 

[Method 1: Create an equivalent 

equation for each equation. 

Method 2: Determine the solution of 

the given system, and then create 

two non-equivalent equations with 

the same solution. 

Method 3: Add the two equations in 

the given system to create a third 

equation. Use different combinations 

of two of the three equations.] 

Scoring The class can mark 

this after brainstorming a list of 

assessment criteria and developing 

a rubric. The criteria and rubric 

should include references to correctness, 

completeness, cooperation, 

clarity of presentation, and creativity. 

Assessment Tip 

Plan to assess in a variety of ways 

within each chapter. This allows 

more students to demonstrate their 

learning success. 




1.4 Investigation: Equivalent 

Equations 





Bank: 

1.4 Investigation: Equivalent 

Equations 

1.4 Investigation: Equivalent Equations 25

1.5 Solving Linear Systems by Elimination 

Materials 

Optional: 




• rulers 

Expectations 

Students will 



method of elimination and 

substitution. [AG1.02] 







Expectations 

Students 

• manipulated first-degree polynomial 

expressions to solve firstdegree 

equations. 

• identified the proprieties of the 




1. Find the lowest common multiple 

of each pair of numbers. 

a) 2 and 5 [10] 

3. Solve each equation for y. 

b) 3 and 4 [12] 

a) x 2y 3 c) 6 and 7 [42] 

y x 

3 2 2 

d) 1 6 , 1 5 [30] b) 4x 6y 8 y 2x 4 

3 3 

d) 2 and 8 [8] 

2. Find the lowest common 

b) 2x 3y 4 

y 2x 4 

3 3 

denominator of each pair 

of fractions. 

c) 4x 3y 15 

y 4x 5 

3 

a) 1 3 , 1 2 [6] 

4. Create an equivalent equation 

for each equation in question 3 

b) 1 3 , 1 4 [12] 

and then solve for y. What do 

you notice 

1 

c) 1 2 , 1 4 [12] 

a) 3x 6y 9 y x 

3 2 2 

c) 2x 3 y 

1 5 

 

2 2 y 4x 5 

3 

5. Identify which pairs of equations 

below are equivalent equations 

and which have graphs that are 

parallel and distinct. 

a) x 2y 3 

6y 9 3x 

[equivalent] 

b) 2x 3y 4 

y 2x 5 

3 

[parallel and distinct] 

c) 4x 3y 15 

4x 30 3y 

[parallel and distinct] 


Mental Math 

Dividing by Multiples of 5 

95 5 95 10 2 

9.5 2 

19 

Application 


a) 85 5 [17] 

b) 14 5 [2.8] 

c) 61 5 [12.2] 

d) 39 5 [7.8] 

e) 42 5 [8.4] 

f) 105 5 [21] 

g) 123 5 [24.6] 

h) 135 5 [27] 

i) 101 5 [20.2] 

j) 214 5 [42.8] 



Dividing by Multiples of 5 could 

be used in your calculation. 

Give your questions to a classmate 

to solve. 



1. a) (x y) (x y) 2x 

b) 60 8 68 

c) 2x 68 

x 34 

2. a) by substituting 34 for x in 

either equation (1) or (2) 

and solving for y 

b) y 26 

3. (34, 26) 

4. x y 60 → 34 26 60 

x y 8 → 34 26 8 

5. disc 1 has 34; disc 2 has 26 

6. a) (14, 3) 

b) (9, 2) 

c) (4, 1) 






groups to work through the 


Students should recall from 1.4 

Investigation: Equivalent 

Equations that, in adding the two 

equations in a system of equations, 

they create a third equation that has 

the same solution. If all three equations 

were graphed, they would 

intersect at the same point. This is 

the premise behind solving by elimination—if 

you add (or subtract) 

two equations, you can use the 

resulting equation to find a solution. 

In this case, one of the variables 

has been eliminated from the 

resulting equation so that the value 

can be substituted back into either 

of the original equations to solve 

for the other variable. 

After completing question 5, 

students can solve the system of 

equations in the Investigation by 

substitution and by graphing. They 

can then compare the three methods. 

Solving by Substitution: 

x y 60 and x y 8 

If x y 60, then y 60 x. 

If y 60 x and x y 8, then 

x (60 x) 8 

x 60 x 8 

2x 68 

x 34 

If x 34 and x y 60, then 

34 y 60 

y 26 

1.5 Solving Linear Systems by Elimination 27

Solving Graphically (Graphing 

Calculator): 


Y1 X 60 Y2 X 8 

(Window settings: Xmin0, 

Xmax50, Ymin0, Ymax60) 

Students could also solve the system 

of equations by subtracting the 

equations in order to eliminate the 

variable x from the third equation. 

Solving by Subtracting: 


(x y) (x y) 60 8 

2y 52 

y 26 

If x y 60 and y 26, then 

x 26 60 

x 34 

For question 6, part a), students 

could add or subtract the 

equations to solve by elimination. 

In parts b) and c), they must add 

the equations. 


(optional: graphing calculators) 



Alternatively, each example 

could be recorded on the board or 

on an overhead for students to 

solve without referring to the solution 

in the text. They can then 

check their solutions against the 

one in the text. 

A third approach would be to 

have students read through each 

example and solution, and then a 

similar example from the Practice 

questions on student text pages 30 

and 31 can be assigned for them to 

solve. They can use the solution in 

the text for guidance as required. 




Example 1 

This example will be straightforward 

if students have completed 

the Investigation. Students simply 

add the equations to eliminate one 

of the variables and create a third 

equation. This equation is then 

used to solve the system. 

Students might find the following 

exercise useful in understanding 

the relationship between the two 

original equations (3x 2y 19 

and 5x 2y 5) and the equation 

arrived at by adding the two equations 

(x 3). 

Students can graph the three 

equations by following these steps: 

1. Press x and enter the two 

equations 3x 2y 19 and 

5x 2y 5 in slope-intercept 

form. 

Note that a thick graph style has 

been selected for Y1. (See 

Technology Extension, Graph Style, 

on page 13 of this teacher’s 

resource.) 

To display the graph in the standard 

viewing window, press y 6. 


2. To graph the equation x 3, 

using the vertical line function 

of the Draw menu, press: 

OkbOm4 3 e 

Students will discover that the 

three equations form a system of 

equations, that is, they all intersect 

at the same point. This is why the 

third equation, x 3, can be used 

to find the solution for the other 

two equations. 

NOTE: To clear the x 3 line 

from the display, press Om1. 




Example 2 

In this example, equivalent equations 

must first be created such 

that, when subtracted, one of the 

variables is eliminated from the 

resulting equation. 

As the solution shows, either 

variable can be eliminated. For systems 

of equations such as these, 

students need to select the variable 

to eliminate and then identify the 

lowest common multiple of the 

numerical coefficients. 

You might challenge students to 

solve the system by substitution. 

They will discover that the process 

is much more involved, complicated 

by having to work with fractions. 




Example 3 

In this example, the equations are 

first rearranged so that like terms 

are in the same column. Then, an 

equivalent equation without decimals 

is created for each, by multiplying 

by a power of 10. Note that a 

power of 10 must be used such that 

all decimals are cleared. For example, 

for an equation like 0.6x 0.3y 

0.24, each term must be multiplied 

by 100 to clear both decimal 

places in 0.24 (60x 30y 24). 

Once the equations are arranged 

and all decimals are cleared, the 

system of equations can be solved 

by elimination. 




Example 4 

In this example, the lowest common 

denominator for each equation 

must be identified in order to 

create equivalent equations without 

fractions. Once the fractions have 

been cleared, the system can be 

solved by elimination. 




Key Concepts 


into their notebooks. 

Another key concept that underlies 

why the process of elimination 

works is that, when two equations 

are added, the resulting equation has 

the same solution. That is, all three 

equations intersect at the same point. 

Therefore, at the point of intersection, 

x and y have the same value. 


Answers 

1. To eliminate y: 

Step 1: Multiply the first equation 

by 2. 

Step 2: Add the two equations. 


Step 3: Solve the resulting 

equation for x. 

Step 4: Substitute the value for 

x into either of the original 

equations and solve for y. 

2. In order to obtain a common 

numerical coefficient for the 

variable that is to be eliminated 

through adding or subtracting 

the two equations 

3. Answers will vary. Some possible 

answers are: 

a) elimination, because the 

variable y can be eliminated by 

simply adding the two equations 

b) elimination, because the variable 

x can be eliminated by simply 

adding the two equations 

OR graphing with a graphing 

calculator because the equations 

are in the slope-intercept form, 

ready for entering into the 

Y Editor 

c) substitution, because the 

second equation can be solved 

for y easily 

d) elimination, because substitution 

and graphing would 

require more steps 


Understanding 

Students can write their responses 

to these questions in their notebooks 

and/or be prepared to explain 

their answers orally to a classmate, 

a small group, or to the class. 

1. Students might describe how to 

eliminate y or eliminate x. 

Discuss why eliminating y might 

be preferable. (There are fewer 

steps involved.) 

3. The responses to each part of this 

question will vary considerably. 

The important thing here is for 

students to explain their rationale 

for choosing one method over 

another. Note that, if students are 

using graphing calculators, they 

will be more likely to choose 

graphing in part b). 





For question 2, students can check 

their solutions by substituting the 

solution into both equations. 

Alternatively, they can check their 

solutions by solving the system by 

substitution or graphically. 


solve the system by elimination or 

substitution, or graphically. Again, 

they can check solutions by solving 

the system another way. 

In questions 5 and 6, students 

might analyze the systems of equations 

first before attempting to 

solve in order to determine if there 

is no solution (if the graphs of the 

equations are parallel and distinct, 

the m values are equal and b values 

are different as in question 5, 

part d), and question 6, part i)), 

or if there are multiple solutions 

(if the equations are equivalent as 

in question 5, part f), and question 

6, part g)). 

In question 7, parts b) and f), 

students must multiply the first 

equation in each system by 100 

to clear all decimals. 

In question 8, part e), students 

must convert the mixed number to 

a fraction before clearing the fractions. 

(See Common Errors on page 

31 of this teacher’s resource.) 








Question(s) 



1 – 3 Example 1 on page 27 

5, 6, 10 – 12 

7 

8 


Examples 3 on page 28 



Page 33, question 18, part b) 

For what value of c will the system 

have no solution 

cy 1 5x 

9y 8 15x 

For a system of equations to have 

no solution, the graphs of the equations 

must be parallel and distinct. 

Arrange both equations in the 

slope-intercept form, y mx b: 

cy 1 5x 

y 5 c x 1 c 

9y 8 15x 

y 5 3 x 8 9 

For the graphs to be parallel and 

distinct, the slope, m, must be the 

same, and the y-intercept, b, must 

be different. 

Therefore, c 3. 

Common Errors 

• Students incorrectly clear fractions 

from equations because they 

forget to multiply the constant 

term on the right side of the equation 

by the common denominator. 

For example, for the first equation 

in question 8, part b): 

x y 

4 2 

3 

4x 3y 2 

R x Students should be encouraged 

to show each calculation required 

to create the equivalent equation. 

For example: 

3 

x 4 

y 2 

x 

12 3 12 y 

4 12(2) 

4x 3y 24 

• Students incompletely clear decimals 

from equations because they 

forget to multiply by a power of ten 

large enough to clear decimal hundredths. 

For example, for the first 

equation in question 7, part b): 

1.7x 3.5y 0.01 

(10)1.7x (10)3.5y (10)0.01 

17x 35y 0.1 


to consider every term in the equation 

before deciding what power of 

10 to multiply by. 

Assessment 

Journal in Pairs 

Ahmed has missed this lesson due 

to illness. Explain to him, in writing, 

how to solve a system of equations 

by elimination. Include a 

description of what makes a system 

‘easy’ and what makes a system 

‘hard’ to solve using this method. 

Scoring Use the six Key Concepts 

from page 30 of the text to assess 

for completeness. Mark the use of 

mathematical language using a 

communication rubric. 


Tip for Assessing Pairs 

Use the Numbered Heads strategy; 

that is, randomly label each student 

in each working pair with a “1” or a 

“2”, using a flip of a coin. After students 

have completed their journal 

entries, flip the coin again to randomly 

select which of the two 

numbers to mark, and then mark 

only the “1’s” or the 2’s. Use this as 

the mark for the both students in 

the pair. Be sure to inform students 

of your method before they begin. 

This method will encourage students 

to work cooperatively. 





by Elimination 





Bank: 

1.5 Solving Linear Systems by 

Elimination 



Draw Function Explore the Draw 

menu to create vertical and horizontal 

lines that are not possible to 

create using the Y Editor. 

Begin by clearing all equations 

from the Y= Editor. Press x and 

use b to delete any equations. 

To create the graph x 5, a 

vertical line, press: 

OkbOm4 5 e 

To create the graph y 5, a 

horizontal line, press: 

OkbOm3 5 e 

Note that these graphs do not produce 

equations in the Y Editor. 

Therefore, the TRACE function 

will not identify the equation. 

To clear lines, press O m1. 


(grid paper, rulers) 

Expectations 






in two variables by an algebraic 

method [AG1.02] 





algebraic method [AG1.03] 


Step 1: Find the vertices of the 

triangle: 

Substitute 0 for y in both equations 

to find the x-intercepts: 

2x 3y 15 and the x-axis 

intersect at (7, 0). 

4x 5y 16 and the x-axis 

intersect at (4, 0). 

Use elimination to solve the system 

2x 3y 14 and 4x 5y 16: 

The solution is (1, 4). 

The vertices of the triangle are 

(7, 0), (4, 0), and (1, 4). 


Note that a level 4 answer might 

include: 

• the idea that a triangle with either 

the same base and half the height, 

or the same height and half the 

base, will have half the area. 

• the fact that a triangle with base 

vertices (4, 0) and (7, 0) and a 

third vertex anywhere on the line 

y 2 will have half the area. 

• the fact that a triangle with a 

base length of 11 units and a 

height of 2 units (or a base 

length of 5.5 units and a height 

of 4) will have half the area and 

could have three completely 

different vertices, for example, 

(0, 0), (11, 0) and (x, 2). 

Step 2: Sketch the triangle to determine 

the dimensions of the triangle: with half the area, 11 square units: 

Step 4: Find the vertices of a triangle 

A triangle with the same base and 

half the height will have half the area. 

y 

(1, 4) 

y 

(1, 4) 

(4, 0) 

(7, 0) 

x 

(1, 2) 

(4, 0) 

(7, 0) 

x 

The base of the triangle is from 

(4, 0) to (7, 0) or 11 units. 

The height of the triangle is from A triangle with vertices (7, 0), 

(1, 0) to (1, 4) or 4 units. 

(4, 0), and (1, 2) will have half 

the area of a triangle with vertices 

Step 3: Find the area of the triangle: 

A 1 (7, 0), (4, 0), and (1, 4). 

2 bh 22 square units 

Assessment 






• Thinking/Inquiry/Problem 

Solving 








Pattern Power 

Answer: 10 

Possible Solution: 

The number in the middle of each 

triangular arrangement of numbers 

is the sum of the difference 

between pairs of numbers at the 

vertices of the diagrams. For 

example, for the first diagram: 

5 3 2, 9 3 6, and 

9 5 4; 2 6 4 12 

Therefore, for the last diagram: 

8 4 4, 8 3 5, and 

4 3 1; 4 5 1 10 


Technology Extension: Solving Linear Systems 

Materials 


• graphing calculators such as the 

TI-92 or TI-92 Plus that are 

preprogrammed to solve linear 

systems 

Expectations 

Students will solve systems of two 

linear equations in two variables 

using technology. 



groups to do either or both explorations, 

depending on the availability 

and number of calculators and 

the interest of the students. 

Exploration 1 

(graphing calculators) 

Read through and discuss the introduction 

at the top of page 34 as a 

class. This will help students understand 

how the calculator program 

operates to determine a solution, 

given the coefficients and constant of 

the two equations. This will also prepare 

them for answering question 1. 

The keystrokes required to program 

the calculator to determine 

the solution of two linear systems 

are as follows: 

Select the NEW program menu by 

pressing mBBe. 

Line 1. Enter the program name, 

SOLVESYS. 

Note that the ALPHA-lock is on 

(a flashing A) so the program name 

can be entered without having to 

press a before each letter. 

Press the appropriate letter keys 

(the green letters to the top right of 

the keys) to enter SOLVESYS and 

then press e. 

(Note that the program name can 

be anything that readily identifies 

the program’s purpose.) 

Optional step: To enter the first 

step of any program, :ClrHome, 

which clears the Home screen, 

press m B8 e. 

Line 2. To copy :Disp to the command 

line, press mB3. To turn 

on ALPHA-lock, press Oa. 

Enter the command “ENTER 

COEFFICIENTS”, including the 

double quote and pressing 0 for a 

space, and then press e. 

(Note that an abbreviation such as 

“ENTER COEFFS” can be used 

instead as long as it is clear what is 

being asked for.) 

Line 3. To select :Prompt and 

enter A,B,C,D,E,F, press: 

mB2 and then 

aiGazGam 

GavGanGa 

ce 

Line 4. To select :If and enter 

AEBD0, press m 1 a i 

anLazavO 

i 1 0 e. 

Line 5. To select :Goto and enter 1, 

press m 0 1 e. 

Line 6. Enter (CEBF)/(AEBD) 

→X, using p for the →. Press e 

when done. 

Line 7. Enter 

(AFCD)/(AEBD)→Y. 

Line 8. To select :Disp and enter 

X,Y, press m B3 a pG 

a 1 e. 

Line 9. To enter :Stop, press m 

ace. 

Line 10. To select :Lbl and enter 1, 

press m 9 1 e. 

Line 11. Select :If (m 1) and 

enter CEBF0 orAFCD0, 

using O i1 for the sign, and 

OiB2 for “or”. Press e 

when done. 


Line 12. Enter :Then by pressing 

m 2 e. 

Line 13. Select :Disp (m B3) 

and then enter “INFINITELY 

MANY” using the Alpha-lock feature 

(O a). Include the double 

quotes and press e when done. 

Line 14. Enter :Else by pressing 

m 3 e. 

Line 15. Select :Disp and then 

enter “NO SOLUTION” and 

press e. 

To store the program in memory, 

press O k. You will now be back 

at the Home screen. 

To execute the stored program, 

follow these steps: 

1. Press m and then select the program 

from the list of programs. 

2. Press e to begin. 

3. Follow the prompts and press 

e after each entry. 

The solution to Exploration 1, 

Question 2, part a) is (1, 2). 

To delete the program, press 

OM2 7, use the C key 

to select the program to be 

deleted, and then press d 2. 

To edit the program, press m, 

select the program to be edited, 

and then press Beand use 

the standard editing keys. Ensure 

that the revised program is 

stored in memory. 

Exploration 2 

(enhanced graphing calculators) 

The following keystrokes were used 

to solve the system of equations on 

the TI-92 Plus as shown in the 

example: 

1. To access the Algebra menu and 

select solve(, press 6 1. 

2. To solve equation (1) for x, press: 

4 ìL3 î [N20 G 

ìIe 

3. To solve equation (2) for y, 

using the “With” operator to 

substitute the solution from step 

2 for x, follow these steps: 

A. Press 6 1. 

B. Press 2 ì M5 î [1 6 

GîIto enter the equation 

to solve for y. 

C. Press O Üto initiate the 

“With” operator. 

D. Press the cursor pad to move 

the cursor up to highlight the 

solution for x in the display. 

E. Press e to recall the solution 

for x and insert it into the 

Entry line. 

F. Press e to solve the equation 

for y. 

4. To solve equation (2) for x, press: 

6 1 2 ìM5 î [1 6 

GìIOÜ, move the 

cursor to highlight y 4 in the 

display, and then press ee. 

NOTE: Press b to clear the Entry 

Line and 5 8 to clear the screen. 

See pages 146 and 171 of this 

teacher’s resource for details on 

Pretty Print Mode. 

Assessment 

Observation/Interview 

Students should be prepared to 

explain their answers to 

Exploration 1, question 1, and/or 

Exploration 2, question 2, to 

another student, to a small group, 

or to the teacher. 

Technology Extension: Solving Linear Systems 35

1.6 Investigation: Translating Words Into Equations 

Expectations 

Students will 

• model phrases and sentences in 

algebraic expressions and equations 

in two variables. 

• use equations to model relations. 

• use pairs of equations to model 

systems of relations. 


Express each sentence as an algebraic 

equation. Let x represent the larger 

number, and y the smaller. 

1. x 5y 10 

[e.g., One number is 10 more than 

5 times another.] 

2. x 5 y 

[e.g., One number is 5 more than 

another.] 

3. x y 9 

[e.g., The difference between two 

numbers is 9.] 

4. x 3y 8 

[e.g., One number is 8 less than 3 

times another.] 

Mental Math 


120 50 120 100 2 

1.2 2 

2.4 

Application 


a) 160 50 [3.2] 

b) 210 50 [4.2] 

c) 370 50 [7.4] 

d) 980 50 [19.6] 

e) 135 50 [2.7] 

f) 285 50 [5.7] 

g) 1150 50 [23] 

h) 45 50 [0.9] 

i) 65 50 [1.3] 

j) 8 50 [0.16] 






to solve. 


Arrange students in small groups. 

Students will need an opportunity 

to discuss the questions among 

group members. 

Introduce this section by telling 

students that the section has been 

designed to prepare them for 1.7 

Solving Problems Using Linear 

Systems, by providing opportunities 

to create algebraic models to represent 

problem situations. This is the 

first step in solving problems using 

the strategy of algebraic modelling, 

and it is the most challenging step 

for many students. 

Elicit the difference between 

expressions and equations. (An 

expression is a phrase such as “the 

sum of two numbers” or “x y” 

while an equation is a sentence such 

as “The sum of two numbers is 5.” 

or “x y 5.”) 


In this investigation, students create 

algebraic expressions. For questions 

2 to 5, students might find it easier 

if they use variables that reflect what 

they represent. 

For example, for question 2: 

Let p represent the plane’s speed, 

and w the wind speed: 

a) The plane’s speed is decreased by 

the headwind, so its speed is p w. 

b) The plane’s speed is increased by 

the tailwind, so its speed is p w. 


For example, for question 3, part a): 

Let i represent the initiation fee, 

and m the monthly charge: 

The membership cost is the 

initiation fee plus the number of 

months times the monthly charge, 

that is, i 7m. 

Investigations 2 and 3 

In these two investigations, students 

create equations to represent 

problem situations. 

Common Errors 

• Students have difficulty determining 

relationships in tables. 

R x Students can determine possible 

relationships for the first pair of 

numbers and then try them out 

on other pairs. 

For example, for Investigation 2, 

question 1, part d): 

x 

2 

0 

2 

y 

3 

1 

5 

Possible relationships for 2 and 3: 

2 1 3 

2 2 1 3 

Try each relationship on 0 and 1: 

0 1 1 NO 

0 2 1 1 YES 

Try 2 1 on 2 and 5: 

2 2 1 5 YES 

Assessment 

Quick Quiz 

Have students represent each 

problem situation using a system 

of equations. 

a) Brad has $12 more than Peter. 

Together they have $84. 

[B 12 P or B P 12; B P 84] 

b) The area of the Pacific Ocean 

is twice the area of the Atlantic. 

Their total area is 250 000 000 km 2 . 

[P 2A; P A 250 000 000] 

c) The American Falls is 2 m taller 

than the Horseshoe Falls. Their 

average height is 58 m. 

[A H 2 or A H 2; 

A H 

58] 

2 

Learning Skills Assessment 

Each group creates four problems on 

separate index cards. Corresponding 

systems of equations are placed on 

another four index cards. Groups 

exchange cards and try to match 

them. Note that groups should use 

variables x and y to disguise the 

meaning of the variables. 

Scoring Observe group process 

and learning skills of individuals. 

Teamwork can be assessed using 

E(xcellent); G(ood); S(atisfactory); 

and N(eeds Improvement). See 

Assessment Master XX (Learning 

Skills Inventory) in the Assessment 

and Evaluation Resource Kit.) 




1.7 Investigation: Translating 

Words Into Equations 





Bank: 

1.6 Investigation: Translating 

Words Into Equations 

1.6 Investigation: Translating Words Into Equations 37

1.7 Solving Problems Using Linear Systems 

Materials 


Optional: 


• reference books on mining, 

metals, and metallurgy 

Expectations 

Students will solve problems represented 

by systems of two linear 

equations in two variables arising 

from a realistic situation using an 



Expectations 

Students solved problems using the 

strategy of algebraic modelling. 


1. Write each percent as a decimal. 

a) 35% [0.35] 

b) 23.5% [0.235] 

c) 50% [0.5] 

d) 150% [1.5] 

2. Complete Practice questions 1 

to 3 on student text page 43. 

3. Complete Applications and 

Problem Solving question 4 

on student text page 44. Show 

your complete solution and 

explain why you chose the 

method you did for solving 

the system of equations. 

[l larger number 

s smaller number 

l s 255 and l s 39 

Solve by substitution: 

s 255 l, so 

l (255 l ) 39 

l 147 

If l 147, and l s 39, then 

147 s 39 

s 108 

Check: 

103 147 255 and 147 103 39] 

4. a) How do angles a, b, c, d, e, f, 

g, and h relate to each other 

a b 

d c 

e f 

h g 

[a c e g; 

b d f h; 

a d b c e h f g 

a b d c e f 

h g 180°] 

b) If angle a is 120°, what is the 

measure of each other angle 

[c e g 120°; 

b d f h 60°] 

Mental Math 


120 5 120 10 2 

12 2 

24 

120 50 120 100 2 

1.22 

2.4 

Application 


a) 170 5 [34] 

b) 250 5 [50] 

c) 330 5 [66] 

d) 98 5 [19.6] 

e) 3 5 [0.6] 

f) 340 50 [6.8] 

g) 11 50 [0.22] 

h) 245 50 [4.9] 

i) 605 50 [12.1] 

j) 17 50 [0.34] 






to solve. 



Investigation: Use a Linear System 

1. a) n c 13 

b) n c 3 

c) n c 13 and n c 3 

2. Elimination, because simply 

adding or subtracting the equations 

requires fewer steps than 

if substitution or graphing 

were used 

3. The solution is 8 natural sites 

and 5 cultural sites. 

4. a) 8 b) 5 


Investigation: Use a Linear System 


groups to complete the 


Elicit from the students examples 

of World Heritage sites in 

Canada with which they are 

familiar (for example, the Niagara 

Escarpment, Niagara Falls). 

Possible solution: 

Step 1: Model the problem 

algebraically by creating a system 

of equations: 

There are 13 World Heritage 

Sites consisting of natural and 

cultural sites: 

n c 13 

There are 3 more natural sites 

than there are cultural sites: 

n c 3 

Step 2: Solve the system 

algebraically: 

Use elimination to solve for 

one variable: 

n c 13 

(n c 3) 

2n 16 

n 8 

Substitute 8 for n into either 

equation to solve for c: 

If n 8 and n c 13, then 

8 c 13 

c 5 

Step 3: Check the solution: 

If n is 8 and c is 5, then 

8 5 13 and 8 5 3. 

The solution is: 

8 natural sites and 5 cultural sites 


(calculators) 



Alternatively, each example could be 

recorded on the board or an overhead 

for students to solve without 

referring to the solution in the text. 

They can then check their solutions 

against the one in the text. A third 

approach would be to have students 

read through each example and solution, 

and then a similar example can 

be assigned for them to solve. These 

questions can be selected from the 

Applications and Problem Solving 

questions beginning on page 44. 

Students can use the solution in 

the text for guidance, as required. 




Example 1 

You might discuss why it is preferable 

to use variables that reflect 

their meaning. In this case, t could 

have been used to represent the 

term deposit and b to represent 

the bonds. 

Alternative solution: 

Use different variables: 

t b 10 000 

0.04t 0.05b 440 

Clear the decimals: 

t b 10 000 

4t 5b 44 000 

1.7 Solving Problems Using Linear Systems 39

Use elimination: 

4t 4b 40 000 

4t 5b 44 000 

b 4000 

b 4000 

If b 4000, and t b 10 000: 

t 4000 10 000 

t 6000 

After students have reviewed this 

example, consider assigning question 

5 on student text page 44. 

(See Sample Solution on page 41 of 

this teacher’s resource.) 

Example 2 

You might consider assigning this 

example and its related questions in 


only to selected students. 

Discuss different ways to solve 

the system; for example, clear the 

decimals before solving, and/or use 

elimination. 






Note that students model their 

solutions to the problem presented 

in the Career Connection feature 

on student text page 46 upon this 

example. 

Example 3 




(See Sample Solution on page 42 

of this teacher’s resource.) 

Example 4 

Students might find a diagram useful 

to organize the given information 

and understand the problem: 

x km 

100 km/h 

W 

x km at 100 km/h 

( 

x km 

100 km/h ) h 

500 km 

y km 

80 km/h 

P 

y km at 80 km/h 

( 

y 100 km/h) h 

Distance (km) 

Speed (km/h) 

Time (h) 

It is very important for students 

to understand how the terms for 

time were arrived at for the second 

equation. The following explanation 

might help: 

If speed di stance 

, then 

time 

time d istance 

. If distance is x km 

speed 

and the speed is 100 km/h, then 

x km 

time is . 

10 0 km/h 

After students have reviewed 

this example, consider assigning 

question 6 on student text page 44. 



Key Concepts 


into their notebooks. Steps b) and 

c) are the most difficult steps in the 

process of solving problems involving 

systems of equations. Discuss 

with students techniques they can 

use to help them organize given 

information, understand the problem, 

and determine how the variables 

are related, for example, 

using tables and diagrams. 


Answers 

1. a) x y 5000; 

0.06x 0.03y 240 

b t 

b) b t 440; 5 

8 0 10 0 

2. Check back using the facts given 

in the original question. For 

example, for part a), add the 

two amounts to see if they total 

$5000; then calculate the interest 

earned on each amount at 


may seem to be correct when 

checked for the total distance travelled, 

440 km, but may not check 

for the total time taken, 5 h. 

Practice, and Applications 

and Problem Solving 

(calculators) 

Some of the questions in the 


could be assigned immediately after 

the related teaching example is presented. 

This would provide students 

with an opportunity to apply 

what they have learned to a similar 

problem situation immediately. 







Question(s) 



1, 5, and 8 Example 1 on page 39 

3, 6, 17, and 19 

7 and 13 

2, 10, and 12 




each rate; and then find the total 

interest. It should be $240. 


Understanding 

1. Students might find it helpful 

to refer to the solutions for 

Example 1 on student text 

page 39 for part a) and 

Example 4 on student text 

page 42 for part b). 

Sample Solution: 

Part a): 

$ 

Invested 

$ 

Interest 

6% 

x 

0.06x 

3% 

y 

0.03x 

Total 

($) 

5000 

240 

System of Equations: 

x y 5000 

0.06x 0.03y 240 

Part b): 

Distance 

(km) 

b 

t 

Total: 

440 km 

Speed 

(km/h) 

80 

100 

Time 

(h) 

b 

80 

t 

100 

Total: 

5 h 

System of Equations: 

b t 440 

b t 

5 

8 0 10 0 

2. It is important for students to 

check the solution against all the 

given facts in a problem. For 

example, a solution for part b) 

Use the Prerequisite Assignment 

question 4 on page 38 of this 

teacher’s resource to prepare students 

for questions 9 and 12. 



Earning Interest 

Isabel invested $8000, part at 9% 

per annum and the remainder at 4% 

per annum. After one year, the total 

interest earned was $420. How 

much did she invest at each rate 

Step 1: Determine variables: 

Let n represent the amount 

invested at 9%. 

Let f represent the amount invested 

at 4%. 


Step 2: Create a table: 

$ 

Invested 

$ 

Interest 

9% 

n 

0.09n 

4% 

f 

0.04f 

Total 

($) 

8000 

420 

Step 3: Create a system of equations: 

n f 8000 

0.09n 0.04f 420 

Step 4: Solve the system: 

Solve for n in equation 1 and clear 

decimals from equation 2: 

n 8000 f 

9n 4f 42 000 

Use substitution: 

9(8000 f ) 4f 42 000 

f 6000 

If f 6000, then n 2000. 

Step 5: Check: 

$6000 $2000 $8000 

$6000(0.04) $2000(0.09) $420 

She invested $6000 at 4% and 

$2000 at 9%. 


Driving 

Kareem took 5 h to drive 470 km 

from Sudbury to Brantford. For 

part of the trip he drove at 100 km/h 

and for the rest he drove at 90 km/h. 

How far did he drive at each speed 


Let h represent the distance driven 

at 100 km/h. 

Let n represent the distance driven 

at 90 km/h. 


Distance 

(km) 

h 

n 

Total: 

470 km 


h n 470 

h n 

5 

10 0 9 0 

Speed 

(km/h) 

100 

90 

Time 

(h) 

h 

100 

n 

90 

Total: 

5 h 


Solve for n in equation 1, and clear 

fractions from equation 2: 

n 470 h 

h 

n 

(900) (900) 5(900) 

10 0 9 0 

9h 10n 4500 


9 h 10(470 h) 4500 

h 200 

If h 200, n 270 


200 km 270 km 470 km 

200 km at 100 km/h 270 km at 

90 km/h 2 h 3 h 5h 

He drove 200 km at 100 km/h and 

270 km at 90 km/h. 


Patrol boat 

It took a patrol boat 5 h to travel 

60 km up a river against the current 

and 3 h for the return trip. Find the 

speed of the boat in still water and 

the speed of the current. 


Let b represent the speed of the 

boat in still water. 

Let c represent the speed of the 

current. 

Let b c represent the speed of the 

boat travelling with the current. 

Let b c represent the speed of the 

boat travelling against the current. 



Let f represent the volume of 5% 

solution. 

Let t represent the volume of 10% 

solution. 


Volume of 

solution (mL) 

Volume of 

acid (mL) 

% Acetic Acid 

5% 10% 9% 

f t 50 

0.05f 

0.1t 

0.09(50) 


Direction 

With 

current 

Against 

current 

km 

60 

60 

km/h 

b c 

b c 


3(b c) 60 

5(b c) 60 


b c 20 

b c 12 

2b 32 

b 16 

If b 16, then 3(16 c) 60 

c 4 

h 

3 

5 


The speed of the boat travelling 

downstream is 16 km/h 4 km/h 

or 20 km/h. A boat travelling 60 km 

at 20 km/h would take 3 h. 

The speed of the boat travelling 

upstream is 16 km/h 4 km/h or 

12 km/h. A boat travelling 60 km 

at 12 km/h would take 5 h. 

The boat’s speed in still water was 

16 km/h and the current’s speed 

was 4 km/h. 


Vinegar solutions 

White vinegar is a solution of 

acetic acid in water. There are two 

strengths—a 5% solution and a 

10% solution. How many millilitres 

of each solution are required to 

make 50 mL of a 9% solution 


f t 50 

0.05f 0.1t 4.5 


Solve equation 1 for f, and clear 

decimals from equation 2: 

f 50 t 

5f 10t 450 


5(50 t) 10t 450 

t 40 

If t 40, then f 40 50 

f 10 


40 mL 10 mL 50 mL 

The 9% solution must contain 

0.09(50) or 4.5 mL of acid: 

0.1(40) 0.05(10) 4.5 

40 mL of 10% solution and 10 mL 

of 5% solution are required. 

Common Errors 

• Students have difficulty creating 

systems of equations to represent 

and solve problems. 

R x The following suggestions may 

help students: 

• Use variables that reflect what 

they mean, for example, d for 

distance and n for money 

invested at nine percent (9%). 

• Create a table to organize the given 

information and the variables. 

• Draw diagrams to organize the 

given information and understand 

the problem. 

• Refer to the related teaching 

example in the student text. 


Assessment 

Portfolio 

Ask students to select a problem 

and solution from the Applications 

and Problem Solving questions to 

include in their portfolios. Have 

them include brief descriptions of 

any difficulties that they might have 

encountered when solving the problem 

and where they needed help. 

Scoring Evaluate the solution based 

on criteria such as the following: 

• What is the level of difficulty of 

the problem chosen 

• How complete is the solution 

• Is there evidence that the solution 

was checked 

• Was the problem solved 

• How well was the solution communicated 

Were correct mathematical 

form and terminology 

used, where appropriate 

Dictionary 

Students can work in small groups 

or pairs to create Algebra 

Dictionaries that contain words or 

phrases matched to their algebraic, 

or mathematical counterparts. They 

can also include explanations of 

how the words or phrases relate to 

an example problem situation. For 

example, students might list the 

phrases “with the current,” “speed 

in still water,” and “speed of the 

current,” and then explain that a 

boat’s “speed in still water” must be 

added to the “speed of the current” 

if the boat is travelling “with the 

current.” Other possible words or 

phrases are: difference; total; sum; 

altogether; head wind, tail wind, 

and speed in still air; average speed, 

overtake, and equal distance; revenue, 

cost, and break even. 

Encourage students to use a 

spreadsheet or database for this so 

that additional words can be added 

and the list easily sorted. 

Scoring This can be marked analytically, 

based on the number 

of words correctly defined. 

Assessment Tip 

A quick quiz can be created, based 

on the words and examples from 

the dictionaries. 

Journal 

“I can solve problems represented 

by linear systems of two equations 

involving interest, mixtures, currents, 

and speed.” Justify this statement 

with examples. 

Scoring Use an Understanding 

rubric to evaluate for both understanding 

of concepts and performance 

or execution of algorithms.) 




1.7 Solving Problems Using 



Edition Solutions 



Bank: 

1.7 Solving Problems Using 


Extension 

Create and solve a problem similar 

to each problem shown in each of 

the four teaching examples on student 

text pages 39 to 42. 


Career Connection 

Metallurgy 

(calculators, reference books on mining, 

metals, and metallurgy, Internet access) 

Expectations 

Students will 

• solve a problem represented by a 

linear system of two equations in 

two variables arising from a realistic 

situation by using an algebraic 

method. [AG1.03] 

• investigative topics related to the 

career of metallurgy. 


Question 1 

Students might find it helpful 

to model their solutions after 

Example 2 on student text page 40. 

Possible solution: 

Step 1: Organize the given information 

in a table, using variables to 

represent the unknown values: 

Let e represent the mass of 18-karat 

gold. 

Let n represent the mass of 9-karat 

gold. 

Mass of 

jewellery (g) 

Mass of 

gold (g) 

75% 

e 

0.75e 

% Gold 

37.5% 

n 

62.5% 

150 

0.375n 0.625(150) 

Step 2: Write a system of equations: 

e n 150 

0.75e 0.375n 93.75 

750e 375n 93 750 

Step 3: Solve the systems by 

substitution: 

e 150 n 

750(150 n) 375n 93 750 

n 50 

If n 50 and e n 150, 

then e 100. 


50 g 100 g 150 g 

50(0.375) 100(0.75) 93.75 

93.75 g is 62.5% of 150 g 

50 g of 9-karat and 100 g of 

18-karat will make 150 g of 

15-karat gold that is 62.5% gold. 

Question 2 

Students could research some of 

the following metals to find out 

how they are extracted, how they 

are purified or processed, how they 

are prepared for use, and for what 

they are used: 

• aluminum • copper 

• iron 

• lead 

• magnesium • nickel 

• silver 

• tin 

• platinum 

Question 3 

Students could search the Internet 

using the search word “metallurgy.” 




and Revenues 

(calculators) 

Expectations 

Students will solve a problem represented 

by a linear system of two 

equations in two variables arising 

from a realistic situation by using 

an algebraic method. [AG1.03] 


Possible solutions: 

Parts a) and b) 

Step 1: Represent the problem 

algebraically: 

Let t represent the number of 

tickets sold. 

Write an expression for the cost 

of holding the festival based on the 

number of tickets sold: 

It will cost $2000 plus $2 per ticket 

for the cost of the cap: 

2000 2t 

Write an expression for the expected 

revenue based on the number of 

tickets sold. The festival will bring 

in $10 for every ticket sold: 

10t 

Step 2: To break even, the cost and 

revenue must be the same, so 

equate the two expressions and 

solve for t: 

2000 2t 10t 

t 250 

To break even, 250 tickets must 

be sold. 

Part c) 

To make a profit of $16 000, the 

revenue must be $16 000 greater 

than the cost. Write a new expression 

to represent the required 

revenue, and then equate the 

two equations and solve for t: 

2000 2t 16 000 10t 

t 2250 

To make a profit of $16 000, 

2250 tickets must be sold. 


five 

4 

two 

twentyfive 

twentyeight 

11 

twelve 

6 

twentytwo 

9 

fifteen 

7 

eight 

5 

eighteen 

8 

two 

3 

twentyfive 

10 

Number Power 

Answer: 

a) Magic Word Square 

eight 

Magic Number Square 

4 

11 

6 

9 

7 

5 

eighteen 

five 

twentyeight 

twelve 

twentytwo 

fifteen 

8 

3 

10 

b) The number of letters in 

each number word in the Magic 

Word Square is the same as the 

number in the corresponding 

square in the Magic Number 

Square: 

Possible Solution: 

Part a) To complete the Magic 

Number Square, students must 

first figure out how the Magic 

Squares are related by comparing 

the number word and the 

number in the middle squares of 

both Magic Squares. (The middle 

square is the only common 

square in both Magic Squares 

that is completed.) They must 

realize that the number word 

“fifteen” has 7 letters. Students 

then complete the Magic 

Number Square by counting the 

number of letters in each number 

word of the Magic Word 

Square, and then inserting that 

number in the corresponding 

square of the Magic Number 

Square. The resulting square is a 

Magic Number Square with the 

magic sum of 21. 

To complete the Magic Word 

Square, students must find numbers 

that have number words 

with the same number of letters 

as the number in the corresponding 

square of the Magic 

Number Square. However, the 

number word for the given number 

should not be used. For 

example, the word “four” should 

not be used in the top left corner 

of the Magic Word Square even 

though “four” has 4 letters. The 

number words “five” and “nine” 

should both be tried in that 

square instead. Students continue 

until they have a Magic 

Word Square, that is, each diagonal, 

row, and column has the 

same sum. In this case, the magic 

sum is 45. 


Rich Problem 

Ape/Monkey Populations 

Materials 



• rulers 

Optional: 



• encyclopedias, and reference 

books on primates 


Edition, Assessment and Evaluation 

Resource Kit, Assessment Master 

XX (Learning Skills Inventory) 

Expectations 

Students will 

• graph lines from a table of values 

and extrapolate from the graph. 



graphically, and interpret the 

intersection point in the context 

of a realistic situation. [AG1.01] 

• determine the slope of a line and 

identify it as a constant rate of 

change. 


Discuss the introduction on student 

text page 48 as a class. Arrange students 

in pairs or small groups to 

compete the explorations. 

Exploration 1 

(grid paper, rulers) 

See Sample Solution for question 1 

on this page of the teacher’s 

resource. 

Students will find a ruler helpful 

for extrapolating the graphs in 

questions 2 and 3. 

Exploration 2 

For question 5, students can use 

the formula for finding slope given 

y 

two points on a line, m 2 y1 

x 

. 

2 x1 

(See Sample Solution on this page of 

the teacher’s resource.) 

Allow time for students to discuss 

questions 1, 2, 4, 6, and 7 in 

their groups before a class discussion 

of the answers. 



Percent of the Ape/Monkey Population 

100 

(0, 94) 

(4, 90) 

90 

80 

(20, 80) 

(7, 85) 

70 

60 

50 

40 

30 

Apes 

Monkeys 

(10, 70) 

(10, 30) 

20 

(7, 15) 

(20, 20) 

10 

(0, 6) 

(4, 10) 

0 

20 15 10 5 0 

Time (million of years ago) 


a) Find the slope of the Monkey 

graph using (10, 70) and (20, 20): 

y 

m 2 y1 

x 

2 0 70 

 

2 x1 

20 

10 

5 

Find the slope of the Ape graph 

using (10, 30) and (20, 80): 

y 

m 2 y1 

x 

8 0 30 

 

2 x1 

20 

10 

5 

b) Negative and positive slope 

values are based on an x-axis 

that increases from left to right 

and a y-axis that increases from 

bottom to top. The slope would 

be positive for the Monkey 

graph and negative for the Ape 

graph if the x-axis increased 

from left to right. 


5. To determine the equation for 

the Monkey graph, press o 

B 4 O 1 G O3 G s 

B 1 2 e. 

p 5t 120 

6. Adjust the Window settings 

(Xmin 0, Xmax 100, 

Ymin 0, and Ymax 100). 

7. To determine the point of 

intersection, press O r5 e 

ee. 


To determine the equation of each 

line and the point of intersection, 

students can follow these steps: 

1. Prepare the calculator for 

graphing. (See the Teaching 

Suggestions in 1.2 Solving 

Linear Equations Graphically 

on page 7 of this teacher’s 

resource.) 

2. Prepare the calculator for lists: 

To display the Stat List Editor, 

press o e. To clear any lists, 

use the arrow keys to move the 

cursor onto each heading, L1 to 

L3, and press b e. 

3. Enter two values for time, 20 

and 10, in List 1 (L1) and the 

corresponding values for Apes, 

80 and 30, in List 2 (L2) and the 

values for Monkeys, 20 and 70, 

in List 3 (L3): 

4. To determine the equation for 

the Ape graph, press o B 

4 O 1 G O2 G sB 

1 1 e. 

p 5t 20 

Assessment 

Learning Skills Inventory 

Use Assessment Master XX in the 

MATHPOWER, Ontario Edition, 

Assessment and Evaluation Resource 

Kit, to assess learning skills. 


MATHPOWER 10, Ontario 


Cross-Discipline 

Zoology 

(Internet access, encyclopedias, primate 

reference books) 

Research the following: 

• the actual numbers for presentday 

ape and monkey populations 

• other types of anthropoids 

[tarsiers and man] 

• apes and monkeys 

[apes: gorillas, chimpanzees, 

gibbons, and orangutans; monkeys: 

marmosets, spider monkeys, and 

howlers (New World monkeys); 

and mandrills, macaques, and 

mangabeys (Old World monkeys)] 

Rich Problem 49

Review of Key Concepts 

Materials 




Expectations 

Students will review modelling and 

solving problems involving the 

intersection of two straight lines. 

[AGV.01] 

Specifically, students will review: 

• determining the point of intersection 


graphically, with and without the 

use of graphing calculators or 

graphing software, and interpret 

the intersection point in the context 

of a realistic situation. 

[AG1.01] 

• solving systems of two linear 

equations in two variables by the 

algebraic methods of substitution 

and elimination. [AG1.02] 

• solving problems represented by 






(See Review Study Guide.) 

Using the Review 

(grid paper and graphing calculators) 

Suggest to students that they do the 

first and last parts of each question 

with multiple parts, for example, 

parts a) and h) of question 1. If 

they experience difficulty with these 

questions, they can refer to the 

appropriate section(s) and teaching 

example(s), review the solution(s), 

and then try again. (See Review 

Study Guide.) They can then complete 

the remaining parts, if they 

feel they need more practice. 

The following cooperative learning 

strategies might be used for 

arranging students to complete this 

Review of Key Concepts for 

Chapter 1: 

Pairs Drill: Students work in pairs 

on the review questions. Partners 

alternate questions, and then 

exchange and check each other’s 

answers and solutions. 

Pairs Check: A group of four students 

divides into two pairs. In 

each pair, one student does a review 

question, while the partner coaches. 

The partners then switch roles for 

the next question. The group of 

four reconvenes after the questions 

in each part of the Review have 

been completed. They then discuss 

questions that caused difficulty or 

had multiple possible solutions 

or answers. 


Review Study Guide 

Students can work through the 

Review and then check their 

answers against the answers at the 

back of the text. Refer to the table 

below to locate the section(s) of 

the text and specific teaching example(s) 

for any question(s) that are 

incorrect. Direct students to the 

specific teaching example(s) so that 

they can review the solution and 

then try the question again. In 

some cases, there will be students 

who are unsure of how to proceed 

with a question. Use the Review 

Study Guide to direct them to the 

appropriate teaching example(s). 

They can the model their solutions 

on the solution presented in the 

specific teaching example. 

Question 

Numbers 

1 

2 

3 

4 

5 

Review Study Guide 

Section 

(Teaching 

Example) 

1.2 (1, 2, 4) 

1.2 (3) 

1.2 (4, 5) 

1.2 (1, 2) 

1.2 (1, 2) 

Expectation 

by Code 

AG1.01 

AG1.01 

AG1.01 

AG1.01 

AG1.03 

AG1.01 

AG1.03 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

1.3 (1, 2, 3, 4) 

1.3 (1) 

1.3 (1) 

1.5 (1, 2) 

1.2, 1.3, 1.5 

1.5 (2) 

1.5 (3, 4) 

1.5 (2) 

1.7 

1.7 (1) 

1.7 (2) 

1.7 (3) 

1.7 (4) 

AG1.02 

AG1.02 

AG1.02 

AG1.03 

AG1.02 

AG1.01 

AG1.02 

AG1.02 

AG1.02 

AG1.02 

AG1.03 

AG1.03 

AG1.03 

AG1.03 

AG1.03 

AG1.03 

Review of Key Concepts 51

Common Errors 


when rearranging and solving 

equations. For example, for 

question 1, part g): 

3x 2y 8 

2y 3x 8 

2y 

3x 8 

 

2 2 

y 3 x 

4 

2 


to show every step in the solution 

rather than simply transpose terms 

across the equal sign, for example: 

3x 2y 8 

3x 3x 2y 3x 8 

2y 3x 8 

2 

 

y 

2 

3 x 8 

 

2 

y 3x 

8 

 

2 

2 

y 3 x 

 

2 8 2 

y 3 x 

4 

2 


when expanding and simplifying 

equations with negative terms. 

For example, for question 7, 

part b): 

3(x 1) (y 7) 2 

3x 1 y 7 2 

R x Students can add implied coefficients 

of 1 where appropriate, 

draw arrows to show how the distributive 

property is applied to 

expand, and keep the polynomial to 

be subtracted inside brackets until 

the “add-the-opposite rule” has 

been applied; for example: 

3(x 1) (y 7) 2 

3(x 1) [1(y 7)] 2 

3x 3 [1y 7] 2 

3x 3 [1y 7] 2 

3x 3 y 7 2 

• Students have difficulty creating 

systems of equations to represent 

and solve problems in questions 

14 to 18. 

R x The following suggestions may 

help students: 

• Use variables that reflect what 

they mean. 

• Create a table to organize the 

given information and the 

variables. 

• Draw diagrams to organize the 

given information and understand 

the problem. 

• Refer to the related teaching 

example in 1.7 Solving 

Problems Using Linear 

Systems in the student text. 


Technology Tips 


Fraction Function To use the 

Fraction function in question 2, 

parts a) and b) to convert the 

approximate decimal values of 

the intersection point coordinates 

to exact fractions, follow these 

steps immediately after the coordinates 

of the intersection point 

are displayed: 



Okui1 e 

2. To convert the Y coordinate to 


a 1 i 1 e 


Window Settings If students graph 

to find the point of intersection or 

to solve a system, the intersection 

point must be displayed in the window 

for the Intersect operation to 

work, for example, for questions 5, 

8, and 13 to 18. 

They can use a trial and error 

method to view the graphs of the 

system first, using the standard 

viewing window, and then adjust 

the Window settings until the point 

of intersection is displayed. Or they 

might calculate the intercepts for 

each equation in the system and use 

these to determine reasonable 

Window settings. 


Negative Versus Subtraction Key 

Remind students to use the correct 

key when entering equations and 

calculations. Using either the negative 

or subtraction key incorrectly 

will result in an ERROR message. 

Assessment 

Self-Assessment 

This Review of Key Concepts is an 

opportunity for students to assess 

themselves by completing selected 

questions and checking the answers 

against the answers in the back of the 

student text. They can then revisit 

any questions that they got wrong 

or had significant difficulty with. 

Upon completing the Review, 

students can also answer questions 

such as the following: 

• Did you work by yourself or with 

other students Why 

• What questions did you find 

easy difficult Why 

• How often did you have to check 

the related teaching example in 

the text to help you with a question 

For what questions 

Students should then make a list of 

questions that caused them difficulty, 

and identify the related sections and 

teaching examples. They can use 

this to focus their studying for a 

final test on the chapter’s content. 




Evaluation Resource Kit: 

Self-Check Chapter 1 Linear 

Systems 





Bank: 

Chapter 1 Linear Systems 

Review of Key Concepts 53

Chapter Test 

Materials 




• rulers 

Expectations 

The questions in this section are 

designed to assess student performance 

with respect to the following: 

Can the student model and solve 

problems involving the intersection 

of two straight lines [AGV.01] 

Specifically, can the student 


of two linear relations graphically, 

with and without the use of 

graphing calculators or graphing 

software, and interpret the intersection 

point in the context of a 

realistic situation [AG1.01] 



methods of substitution and 

elimination [AG1.02] 






graphs [AG1.03] 

Assessment Guide 

Use the following table to identify 

expectations with which students 

might be experiencing difficulty, and 

to locate related sections of the text 

and specific teaching examples, as 

required, in order to provide further 

instruction or review. 

Question 

Number 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Assessment Guide 

Expectation 

by Code 

AG1.01 

AG1.01 

AG1.01 

AG1.02 

AG1.02 

AG1.01 

AG1.02 

AG1.03 

AG1.03 

AG1.03 

AG1.03 

Section 

(Example) 

1.2 (1, 2) 

1.2 (1, 2) 

1.2 (4, 5) 

1.2 (1, 2) 

1.2 (1, 2) 

1.2, 1.3, 

1.5 

1.7 

1.7 ( 2) 

1.7 ( 1) 

1.7 ( 3) 

Using the Chapter Test 

Students could complete the Chapter 

Test and hand it in for final evaluation. 

Alternatively, students could use 

the Chapter Test for self-assessment, 

checking their answers with the 

answers at the back of the book. 

The alternate chapter test, found in 

MATHPOWER 10, Ontario Edition, 

Assessment and Evaluation Resource Kit, 

could then be the final test. Or, a 

final test could be created using 

the MATHPOWER 10, Ontario 


Bank. (See Related Resources.) 




Evaluation Resource Kit: 

Test Chapter 1 Linear Systems 



3. MATHPOWER 10, Ontario Edition, 

Computerized Assessment Bank: 

Chapter 1 Linear Systems 


Expectations 






in two variables by an algebraic 

method [AG1.02] 





algebraic method [AG1.03] 


Answer 

$200 000 for the house at one end 

of the street and $186 000 for the 

house at the other end. 

Using the Achievement 

Check 

(calculators) 

Students can be given different 

stages of assistance: 

• A student who needs Stage 1 

assistance cannot achieve level 4 

performance. 


assistance cannot achieve level 3 

or level 4. 


assistance is performing at level 1. 

For all stages of assistance, 

provide the following: 

This problem can be solved by 

finding the solution to a system of 

equations. One equation represents 

the total cost of the 15 houses. 

For Stage 1: The middle house can 

be represented by two different 

expressions depending on the end 

of the street from which you are 

working—equate these two expressions 

to create the other equation. 

For Stage 2: To find the other 

equation, let the value of the first 

house at one end of the street be x. 

The second house from this end 

costs x 3000. Continue on in this 

fashion to the middle house. Let 

the value of the first house at the 

other end of the street be y. The 

second house from this end costs 

y 5000. Continue on in this fashion 

to the middle house. Equate the 

two expressions for the middle 

house to create the other equation. 

For Stage 3: Use 

this diagram to 

represent the 

problem. Equate 

the two expressions 

for the middle 

house to create 

the other equation. 

The equation 

representing the 

total cost of the 15 

houses and can be 

found by finding 

the sum of the 

expressions for all 

the houses and 

equating it to $3 091 000. 

Rearrange both equations in the 

form Ax By C and then use 

elimination to find the solution. 

Common Errors 

A common error would be to use 

the middle house twice in the equation 

that represents the total cost of 

the 15 houses instead of choosing 

either x 21 000 or y 35 000. 

Assessment 






• Thinking/Inquiry/Problem 

Solving 








x 

M 

y 

Cost 

x 

x 3000 

x 6000 

x 9000 

x 12 000 

x 15 000 

x 18 000 

x 21 000 

y 35 000 

y 30 000 

y 25 000 

y 20 000 

y 15 000 

y 10 000 

y 5000 

y 

Chapter Test 55

Problem Solving: Use a Data Bank 

Materials 

• atlases, Canadian and world 

• encyclopedias and almanacs 

• international time zone map 

• Internet access, e.g., Statistics 

Canada Web site 

• reference books on the solar 

system and endangered species 

• road map of the United States 

Optional: 

• an interview with a librarian 

• university calendars 

Expectations 

Students will 

• select appropriate sources of data. 

• solve problems using the data. 


(variety of data banks) 

Provide each small group of students 

with a data bank. Allow them 

time to peruse the data bank, make 

a list of the sort of data that is available, 

and determine how the data 

bank is organized so that data can 

be retrieved. 

Groups can share what they have 

learned about their data banks with 

the class. 



or pairs. 

Page 56 

(international time zone map) 

Display an international time zone 

map for students to interpret. Have 

them locate their community on 

the map and express the current 

time in GMT (Greenwich Mean 

Time). For example, Ottawa is in 

time zone 5. This means it is 5 h 

behind GMT. If it is 13:12 in 

Ottawa, it is 13:12 5 or 18:12 

in Greenwich. 

As a class, read through the 

introduction on student text page 

56. As a student reads the problem 

aloud, have another student locate 

Honolulu and Oakland, California, 

on the time zone map. 

Discuss the solution to the problem 

as presented in the text and 

summarize the steps followed: 

Step 1: Make a plan: 

Divide the distance flown by 

the time taken, to calculate 

average speed. 

Step 2: Determine given and 

required information: 

The distance, 3875 km, and the 

local times at departure and 

arrival are given. The time taken 

must be calculated. 

Step 3: Calculate the time taken: 

Use a time zone map to determine 

the time zone conversion numbers: 

Honolulu is 10 and California 

is 8. 

Convert both local times to GMT: 

17:16 Honolulu time is 03.16 GMT. 

13:31 California time is 21:31 GMT. 

From 03:16 to 21:31 is 18 h 15 min 

or 18.25 h. 

Step 4: Calculate the average speed: 

3875 km 18.25 h 212 km/h 


(variety of data banks) 

Assign problems according to the 

availability of data banks and the complexity 

of the mathematics involved. 

Alternatively, students can work 

in groups to select two or three 

problems and make a plan for solving 

each. Plans should include the 

steps to be followed and the data 

bank(s) required. Plans could be 

submitted for approval and one 

problem selected and assigned. 



Page 57, problem 3 

Land area 

Step 1: Make a plan: 

Find the population and area of 

each province and then calculate 

the population density of each. 

Step 2: Use the Canadian Almanac 

to find the data. Organize it in a 

table: 

Province 

Nfld 

PEI 

NS 

NB 

Que 

ON 

Man 

Sask 

Alta 

BC 

Population 

1991 

(000) 

568 

130 

900 

724 

6 896 

10 085 

1 092 

989 

2 546 

3 282 

Land 

Area 

(000 

km 2 ) 

372 

6 

53 

72 

1 357 

891 

548 

571 

644 

930 

Density 

(pop. 

per 

km 2 ) 

1.5 

21.6 

17.0 

10.1 

5.1 

11.3 

2.0 

1.7 

4.0 

3.6 

Step 3: Select the least dense 

province to answer part a) and the 

densest province to answer part b). 

a) Each person in Newfoundland 

would receive the most land. 

b) Each person in PEI would 

receive the least land. 

Common Errors 

• Students are not aware of the 

types of information available in the 

different data banks. They also have 

difficulty retrieving the data. 

R x Begin the lesson with the 

Prerequisite Assignment on page 56 

of this teacher’s resource. 

Math Journal 

Explain why you chose the problem(s) 

you did in this section. 

Assessment 

Self-Assessment Checklist 

Students can use this checklist to 

assess how they solved the problem 

and communicated the solution: 

• Are the steps to the solution 

clearly laid out 

• Is the reasoning behind each step 

provided 

• Is the data source described 

• Is correct mathematical terminology 

used 

• Did you double-check all calculations 

and data 

• Did you check to ensure that you 

solved the problem 

• Did you check the reasonableness 

of your answer 

Journal 

Make a list of available data banks. 

For each, list the type of data available 

and describe how the data is 

organized for easy retrieval. 




Cross-Discipline 

Library Science 

(Internet access, university catalogues, 

interview with a librarian) 

Librarians specialize in data banks 

and information retrieval. Find out 

what special training a librarian 

receives in order to set up a library 

so that information is accessible and 

easy to retrieve and to help others 

access information in the library. 

Problem Solving: Use a Data Bank 57

Problem Solving: Model and Communicate Solutions 

Materials 


Optional: 


Expectations 

Students will 

• use a variety of mathematical 

models, including algebraic 

models, to solve problems. 

• communicate solutions to multistep 

problems in good mathematical 

form, giving reasons for 

the steps taken to reach the 

solution. 



1. How many edges, faces, and 

vertices are there on a cube 

[12 edges, 6 faces, 4 vertices] 

2. Use an equation to represent 

the relationship between x and y 

in this table of values. 

x 

y 

1 

8 

2 

11 

3 

14 

4 

17 

[y = 3x + 5] 

3. a) Explain how you would use a 

graphing calculator to find the 

equation in question 2. 

[Press o 1 and then enter the 

data into the Stat List Editor, the 

x-values into L1, and the y-values 

into L2. Then press o B4 to 

use the LinReg instruction to find 

the equation by pressing O 1 

GO2 G sBee 

e. 

b) Explain how you could use 

the graph generated by the 

Linear Regression instruction in 

part a) to determine 

a y-value for a given x-value. 

[Press O reand then use 

the Value operation by entering 

the given value for x and then 

pressing e.] 


Arrange students in small groups or 

pairs to work through the introductory 

material on pages 58 to 60 

with the class. Groups can then 

select or be assigned problems from 

Applications and Problem Solving. 

Note that “solution” is used to 

describe the process followed to 

arrive at the “answer.” 

Pages 58 to 60 

Ask a volunteer to read the first 

three paragraphs on student text 

page 58 aloud to the class. Discuss 

the value of communication in 

mathematics, both oral and written, 

in helping the communicator consolidate 

learning, organize thought 

processes, and determine possible 

errors or omissions. In other words, 

when people have to explain what 

they did, and why, to solve a mathematical 

problem, they must take the 

time to reflect on what they did so 

that they can communicate it to 

others who may be unfamiliar 

with the problem. 

At the bottom of student text 

page 58, a problem about a large 

cube is presented, along with three 

possible mathematical models that 

could be used to solve the problem. 

After discussing the cube problem 

with the students, have them turn 

to page 59 and, as a class, discuss 

each method. 


Method 1: Build and Interpret a 

Physical Model 

Challenge students to explain, in 

their own words, the reasoning or 

logic behind this solution. Elicit the 

role of the physical model in this 

solution. (It helps one visualize the 

problem and therefore understand 

the problem. The actual model is 

not used to arrive at an answer. The 

answer is reached through logic.) 

Method 2: Model Algebraically 

Elicit the steps used in this solution: 

Step 1: Data are collected from the 

diagrams and recorded in a table. 

Step 2: A relationship is determined 

and an equation or formula created 

to represent the relationship. 

Step 3: The formula is used to find 

the answer to the problem. 

Method 3: Model Graphically 

Elicit the steps used in this solution: 

Step 1: Data are collected from the 

diagrams and recorded in a table. 

Step 2: The data are graphed to 

display the relationship. 

Step 3: The answer to the problem 

is interpolated from the graph. 

Before assigning the Applications 

and Problem Solving problems, 

brainstorm characteristics of a good 

solution and record them on the 

board for students to refer to when 

self-assessing their own solutions. 

For example: 

Characteristics of a Good 

Mathematical Solution 

1. The problem is clearly stated, 

or restated if necessary. 

2. The steps taken to solve the problem 

are clear, with the reasoning 

behind each step provided. 

3. The materials used to solve 

the problem are described. 

4. The mathematical model used 

to solve the problem is described. 

5. Correct mathematical terminology 

is used. 

6. All relevant diagrams, tables, 

and calculations are included. 

7. The answer to the problem is 

clearly stated. 



Remind students to refer to 

Characteristics of a Good 

Mathematical Solution to help them 

communicate their solutions. 

For problems 1, 2, and 3, 

students might create a table of 

values and then either use algebraic 

modelling (by creating an equation 

or formula) or graph and interpolate. 

For problem 4, students could 

use a diagram or a physical model 

or both. 

For problem 5, students might 

organize or model the data in a 

table of values, look for a pattern, 

and then use the pattern to determine 

which numbers cannot be 

written as a difference of two 

squares (2, 6, 10, 14, 18, …). 

Alternatively, they might create a 

formula for the positive integers 

from 1 to 40 that cannot be written 

as a difference of two squares 

(4n 1). 

Problem Solving: Model and Communicate Solutions 59



Square pattern 

If the pattern in the first five diagrams 

continues, how many shaded 

squares will there be in the 100th 

diagram 

Step 1: Collect data from the diagrams 

and organize it in a table: 

Diagram 

Number, n 

1 

2 

3 

4 

5 

Number of 

Shaded 

Squares, s 

1 

4 

7 

10 

13 

Step 2: Decide on a model: 

Create an equation to represent the 

relationship between the diagram 

number and the number of shaded 

squares. 

Step 3: Enter the data into the 

StatList Editor of a graphing 

calculator and then use the Linear 

Regression instruction to determine 

the equation: 

s 3n 2 

Step 4: Use the equation to predict 

the number of shaded squares in 

the 100th diagram (n 100): 

s 3n 2 

3(100) 2 

298 

There will be 298 shaded squares in 

the 100th diagram. 

Common Errors 

• Students have difficulty determining 

an equation to represent a 

relationship in a table of values. 

R x Students can begin by listing 

several possible relationships for 

one pair of data values in the table 

and then try the relationships on 

other pairs of data values. For 

example, for the table of values 

shown in Sample Solution: 

Possible Relationships 

2 and 4 2 2 4 

2 2 4 

2 3 2 4 

2 4 4 4 

Try the relationships on another pair: 

3 and 7 3 2 5 NO 

3 2 6 NO 

3 3 2 7 YES 

Check on another pair: 

1 and 1 1 3 2 1 YES 

The relationship is n 3 2 s, 

or s 3n 1. 

Assessment 

Self-Assessment Checklist 

Students can create their own 

checklists for evaluating a written 

solution to a mathematical problem. 

They can then use the checklist 

to assess their written solutions. 





Problem Solving: Using the Strategies 

Materials 

• atlases and time zone map 

Optional: 


Edition, Assessment and Evaluation 

Resource Kit, Assessment Master XX 

(Problem Solving Checklist) 

Expectations 

Students will use a variety of strategies 

to solve problems. 


Arrange students in small groups or 

pairs. Students can select problems 

or problems can be assigned. 



Measurement 

G 

9 

13 

5 

5 C 

4 

5 B 

4 

9 5 1 A 1 1 3 

4 3 E 

4 F 

D 2 2 

4 

H 13 

B is 4 4, C is 5 5, so A is 1 1. 

B is 4 4, A is 1 1, so E is 3 3. 

A is 1 1, E is 3 3, so D is 2 2. 

D is 2 2, A is 1 1, C is 5 5, 

so F is 4 4. 

F is 4 4, C is 5 5, so G is 9 9. 

G is 9 9, F is 4 4, so H is 13 13. 

A B C D E F G H 

1 16 25 4 9 16 81 169 

321 cm 2 


Missing sums 

Find A, B, C, D, E, and F: 

Row 1: A B A B 36 

A B 18 

Column 4: B C C A 32 

A B = 18 18 2C 32 

C 7 

Row 2: A C A C 34 

A C 17 

C 7 so A 7 17 

A 10 

If A 10 and A B 18, 

then B 8. 

Column 2: B C B E 40 

C 7, B 8 8 7 8 E 40 

E 17 

Column 3: A A B F 41 

A 10, B 8 20 8 F 41 

F 13 

Column 1: A A B D 37 

A 10, B 8 20 8 D 37 

D 9 

If A 10, B 8, C 7, D 9, 

E 17, and F 13, then: 

Row 3 is B B B C 31. 

Row 4 is D E F A 49. 

Assessment 

Problem Solving Checklist 

Use Assessment Master XX, 

Problem Solving Checklist, in the 

Assessment and Evaluation Resource 

Kit to assess students’ problemsolving 

abilities while observing 

them engaged in problem solving, 

and/or when assessing a written 

submission or presentation of a 

solution to a problem. 




Problem Solving: Using the Strategies 61

CHAPTER 1 

Student Text Answers 

ANSWERS 

Getting Started p. 2 

1. 10; The check digit will be 0. 

2. a) 9 b) 9 c) 2 d) 8 

3. a) No, the check digit should be 6. 

b) Yes, the check digit is correct. 

c) Yes, the check digit is correct. 

4. Answers may vary. 123 456 717; 223 456 740 

5. a) 10 m b) 0 

c) The check digit is equal to 10 m if m 0 

and 0 if m 0. 

Review of Prerequisite Skills p. 3 

1. a) x 2 b) 2x 8 c) 3y 5 

d) 5a 3 e) 6x 14 f) 5z 8 

g) 7t 41 h) 2x 9 

2. a) 6x b) 2c c) x 

d) 3n e) x 2y f) 3p r 

3. a) 8 b) 2 c) 6 d) 5 

4. a) 7 b) 3 c) 2 

d) 12 e) 1 2 f) 3 2 

g) 4 h) 5 i) 5 2 

j) 4 k) 1 l) 2 

5. a) x 11 3y b) x 5y 8 

c) x 2y 4 d) x 5 3y 

 

2 

6. a) y 3 2x b) y x 2 

c) y 1 2x 

d) y 3x 4 

 

4 

2 

10. a) (3, 1) b) (5, 2) c) (1, 6) 

d) (4, 8) e) (4, 5) f) (2, 1) 

11. a) 9x 4y 1 b) 13m 2 6m 19 

c) a 3b 10 d) e 2 

12. a) x 8y 10 b) t 2 5t 11 

c) 9a 3b 1 d) 12e 1 

Section 1.1 pp. 4–5 

1 Ordered Pairs and One Equation 

1. a) (1, 13), (24, 10) b) (2, 4), (12, 0) 

c) (2, 3) d) (0.5, 2.5) 

2. a) 3, 9, 10, 2 b) 2, 9, 11, 2 

c) 1, 5, 13, 10 d) 5, 3, 4, 7 

2 Ordered Pairs and Two Equations 

1. a) (1, 2) b) (3, 1) c) (2, 3) 

d) (6, 8) e) (2, 5) f) (4, 7) 

2. a) (4, 3) b) (6, 3) c) (1, 0) 

d) Answers may vary. (0, 0) 

3 Problem Solving 

1. a) 55 b) 3 days c) $55 

2. a) The equations represent the same graph. 

b) Answers may vary. (1, 2), (2, 3) 

3. The equations represent parallel and distinct lines. 

The lines never intersect. 


Practice 

1. a) (5, 4) b) (1, 2) 

c) (3, 5) d) (2, 3) 

2. a) (2, 3) b) (2, 0) 

c) (2, 3) d) (3, 2) 

3. a) (3, 1) b) (1, 6) 

c) (4, 1) 

d) infinitely many solutions e) (6, 0) 

f) (3, 4) g) no solution 

h) (2, 1) i) (2, 1) 

j) (3, 2) k) (4, 1) 

l) no solution m) (5, 1) 

n) infinitely many solutions o) (1, 2) 

p) (2, 2) 

4. a) (0.5, 2) b) (2, 1.5) 

c) (1, 0.5) d) (1.5, 2.5) 

5. a) (1.5, 0.8) b) (6.7, 1.7) 

c) (3.9, 0.3) d) (2.7, 0.3) 

e) (2.3, 3) f) (2.6, 5.1) 

6. a) one solution b) no solution 

c) infinitely many solutions d) one solution 

e) no solution f) no solution 

7. Austria: 9, Germany: 16 

8. a) (20, 500) b) 20 months 

c) Champion 

9. (6, 3) 

Chapter 1, Student Text Answers A 1–1

10. (2, 4), (1, 2), (8, 2) 

11. (3, 1), (5, 1 ), (4, 0) 

3 

12. parallelogram 

13. Answers may vary. 

a) x y 5 b) 2x 2y 8 

c) x 2y 4 


a) x y 5, x y 1 

b) x y 1, 2x 2y 2 

15. The system has infinitely many solutions: all points 

on the line x 2y 6 0. 

17. a) (12.5, 9); (48, 24); (16, 18) 

Modelling Math p. 14 

a) (t, d) (50, 1000) b) 50 

c) less than 50 d) greater than 50 

Career Connection p. 15 

1. south: 5000, north: 125 000 


Practice 

1. a) x 8 3y b) x 4y 13 

c) x 7y 7 d) x 2y 1 

2. a) y 11 6x b) y 5x 9 

c) y x 2 d) y 3x 4 

3. a) (2, 2) b) (1, 1) 

c) (2, 1) d) (2, 3) 

e) (3, 0) f) (3, 2) 

g) (4, 5) h) (5, 0) 

i) (2, 3) j) (2, 2) 

k) (1, 1) l) (3, 4) 

m) (1, 0) n) (1, 3) 

o) no solution p) (3, 1) 

q) infinitely many solutions 

r) (1, 5) s) no solution 

t) (1, 1) u) (1, 1) 

4. a) 1 2 , 1 7 

 

b) 11 , 1 

11 

c) 3, 6 5 

d) 1, 1 3 

1 

 

e) 1, 2 7 

f) 

4 3 , 1 3 

2 

g) 3 , 1 8 

5 5 

h) 3 4 , 1 2 

i) 

1 7 , 4 5 


5. a) (24, 18) b) (3, 2) 

c) 3 2 , 2 

d) 5 3 , 1 6 

6. a) Fairweather Mountain is 3970 m higher than 

Ishpatina Ridge. Fairweather Mountain is 188 m 

less than seven time higher than Ishpatina Ridge. 

b) Fairweather Mountain: 4663 m, 

Ishpatina Ridge: 693 m 

7. a) The angles are complementary. Six degrees less 

than y is three times x. 

b) x 21°, y 69° 

8. a) The total number of tickets sold is 550. The 

total revenue from tickets is $9184. 

b) adult tickets: 323, student tickets: 227 

9. (1, 2), (9, 14), (3, 2) 

10. a) (5, 4) b) (4, 5) 

c) (1, 5) d) 1 2 , 1 2 

11. A 3, B 2 

12. a) (1, 4, 2) b) (2, 1, 3) 

13. m 1 

14. n 1 2 


a) (h, C) (4, 270) 

b) Quality is cheaper for less than 4 h. ABC is cheaper 

for more than 4 h. 

c) 10 h of work 


1 Equivalent Forms 

1. Answers may vary. (0, 6), (1, 5), (2, 4) 

2. a) 2x 2y 12 b) yes 

3. a) 3x 3y 18 b) yes 

4. Yes, they all have the same solution. 


a) 2x 2y 4, x y 2, 2x 2y 4 

b) 2x 2y 8, x y 4, 2x 2y 8 

c) 2x y 7, 4x 2y 14, 4x 2y 14 

d) 2y 8x 6, 3y 12x 9, 4y 16x 12 

2 Equivalent Systems 

1. (5, 2) 

2. (5, 2) 

3. a) 2x 2y 6, x y 7 b) (5, 2) 

4. They all have the same solution. 

5. Answers may vary. x y 3, x y 1 

A 1–2 

Chapter 1, Student Text Answers

3 Adding Equations 

1. (2, 1) 

2. a) 2x y b) 5 c) 2x y 5 

3. They all pass through (2, 1). 

4. They are equivalent systems. They have the 

same solution. 

5. They are equivalent systems. They have the 

same solution. 


Practice 

1. a) (5, 2) b) (3, 5) 

c) (1, 7) d) (1, 2) 

2. a) (2, 6) b) (1, 3) 

c) (4, 1) d) (3, 2) 

e) (2, 1) f) (5, 3) 

3. a) (1, 1) b) (2, 1) 

c) (6, 3) d) (2, 0) 

4. a) 4, 17 b) 20, 7 

5. a) (1, 2) b) (2, 2) 

c) (3, 1) d) no solution 

e) (1, 0) f) infinitely many solutions 

g) (4, 2) h) (3, 2) 

i) (2, 3) 

6. a) (9, 4) b) (3, 8) 

c) (2, 1) d) 1 3 , 1 

e) 2, 1 2 

f) 

5 9 , 1 9 

g) infinitely many solutions 

h) 4 5 , 3 5 

i) no solution 

7. a) (1, 3) b) (0.2, 0.1) 

c) (4, 3) d) (3, 4) 

e) (0.5, 0.3) f) (0.4, 1.1) 

8. a) (6, 10) b) (3, 4) 

c) (6, 4) d) (3, 3) 

e) (6, 8) f) (1, 1) 



a) substitution b) elimination 

c) substitution d) elimination 

e) elimination f) elimination 

10. a) There are 10 provinces. Three times the number 

of names with First Nations origins is equal to twice 

the number of names with other origins. 

b) 4 

11. ham: $5, roast beef: $6 

12. a) (1, 3) b) (1, 6) 

c) (2, 3) d) (2, 1) 

13. a) x a, y b b) x 3a, y b 

14. (3, 2), (2, 4), (0, 2) 

15. a 2, b 3 

16. (4, 6) 

17. a) 10 b) 6 

18. a) 2 b) 3 

19. (2, 5) 

20. Answers may vary. 2x 3y 3, x 2y 16 


a) 2x 3y 19, 2x 3y 11 

b) 3x 2y 2, 4x 5y 19 

c) 2x 3y 0, 3x 6y 1 

Technology Extension pp. 34–35 

1 Solving Systems Using a Graphing 

Calculator Program 

1. b) Each of the following systems has AE BD 0. 

In the system ax by c, kax kby kc, one 

equation is a multiple of the other. Thus, there are 

infinitely many solutions. CE BF ckb bkc 0. 

In the system ax by c, ax by d, the lines 

are parallel and distinct when c ≠ d. There is no 

solution, and CE BF cb bd b(c d) 0, 

since c d. 

2. a) (1, 2) b) (7, 10) 

c) infinitely many solutions 

d) no solution 

2 Solving Systems Using 

Preprogrammed Calculators 

1. a) (2, 4) b) (2, 3) c) 

1 2 , 1 

2. a) infinitely many solutions 

b) no solution 


1 Expressions in Two Variables 

1. a) x y b) x y 

c) 5y x d) 6x 2y 

2. a) x y b) x y 

3. a) x 7y b) x 15y 

4. a) x y b) 10x 

c) 5y d) 10x 5y 

5. a) x y b) 0.07x 

c) 0.06y d) 0.07x 0.06y 


1.2 Solving Linear Systems Graphically 

Rubric for the Achievement Check on page 15 of the student text 

Level 1 

Level 2 

Level 3 

Level 4 

The student 

Knowledge/ 

Understanding 

interprets the 

graph incorrectly, 

or interprets the 

graph correctly 

with substantial 

assistance 

interprets the graph 

with some errors, or 


correctly with some 

assistance 


correctly with no 

assistance, 

showing some 

understanding of 

optimal ranges 


accurately and 

completely with no 

assistance, and 

understands the 

optimal ranges for 

each plan 

The student 

Application 

applies the concept 


points incorrectly 



points with some 

errors or some 

omissions 



points correctly 

with no assistance 

to find the critical 

points for each plan 



points with no 

assistance to find 

the optimal range 

for each plan and 

to determine the 

meaning of each 

intersection point 

The student 

Communication 

provides limited 

or incomplete 

descriptions for the 

use of each plan 

describes a 

situation for the 

use of some of 

the plans 

clearly describes a 

situation for the 

best use of each 

plan with some 

conditions 

clearly and 

concisely describes 

each plan’s best 

situation, outlining 

any conditions for 

changing plans 

Copyright © 2000 McGraw-Hill Ryerson Limited Chapter 1, Achievement Check Rubrics R 1–1

1.5 Solving Linear Systems by Elimination 


Level 1 

Level 2 

Level 3 

Level 4 

The student 

Knowledge/ 

Understanding 

finds the vertices 

with substantial 

assistance; estimates 

the area from 

counting squares 

on the graph 

uses graphing, 

substitution, or 

elimination to find 

the vertices with 

some accuracy; 

attempts calculation 

of the area 

uses substitution or 

elimination to find 

the correct vertices; 

calculates the 

correct area 

uses elimination to 

find the correct 

vertices; calculates 

the area efficiently 

The student 

Thinking/ 

Inquiry/ 


needs assistance to 

approach the 

problem 

uses graphing to 

estimate the 

vertices; may use an 

algebraic method 

with some 

accuracy; attempts 

to use the formula 

for the area 

determines that the 

vertices are the 

intersections of the 

lines; uses lengths 

of line segments to 

find area 

uses the 

intersections of the 

lines as vertices; 

uses horizontal and 

vertical lengths to 

find area 

The student 

Application 

shows limited 

ability to halve 

the area 

halves the area 

incorrectly, for 

example; halves 

both the height 

and the base of 

the triangle 

halves the height or 

the base correctly 

to find a solution 

halves the area 

correctly and gives 

multiple solutions 

The student 

Communication 

uses limited 

justification and/or 

mathematically 

incorrect 

terminology 

uses some 

justification for 

the solution; may 

include a graphical 

solution 

clearly justifies 

solution, and gives 

correct graphical 

and algebraic 

explanations 


multiple solutions 

with some 

complexity in 

explanation 


Chapter 1 Test 


Level 1 

Level 2 

Level 3 

Level 4 

The student 

Knowledge/ 

Understanding 

incorrectly 

attempts an 

algebraic method to 

solve the system of 

equations 

attempts an 

algebraic method to 


equations with 

some accuracy 

uses an algebraic 

method to correctly 


equations 

uses the most 

efficient algebraic 

method to solve the 

system of equations 

accurately 

The student 

Thinking/ 

Inquiry/ 


needs Stage 3 

assistance to model 

the problem 

needs Stage 2 


the problem 

needs Stage 1 


the problem 

uses systems of 

equations to model 

the problem with 

no assistance 

The student 

Application 

shows limited 

ability to apply 

systems of 

equations 

shows some ability 

to apply systems of 

equations 

correctly applies 

systems of 

equations; may 

recognize some 

patterns to simplify 

solution 

correctly applies 

systems of 

equations; 

recognizes patterns 

for a more efficient 

solution 

The student 

Communication 

shows limited 

justification and/or 

incorrect 

mathematical 

terminology 

shows some 

justification and 

may include an 

attempted algebraic 

explanation 


solution and gives a 

correct algebraic 

explanation 

clearly justifies an 

efficient solution 

and gives a correct 

algebraic 

explanation 


2 Equations in Two Variables 

1. a) x y 8 b) x y 5 

c) y 3x 1 d) y 2x 1 

2. a) l w 40 b) 2b 3t 61 

3 Systems of Equations 

1. a) x y 7, x y 3 b) y 2x, y x 4 

2. a) x y 256 b) 5x 2y 767 

3. a) p r 295, p r 11 

b) l w 6, l w 46 

c) c 2d, c d 17 

d) b f 331, 10b 15f 3915 

e) x y 180, x 4 3y 


Practice 

1. a) $140 b) $15 

c) $210 d) 0.04x 

2. a) 30 kg b) 200 L 

c) 0.3x litres d) 0.09m kilograms 

3. a) 240 km b) 40x kilometres 

y 

c) 12 h d) hours 9 0 


4. 147, 108 

5. $2000 at 6%, $6000 at 4% 

6. 200 km at 100 km/h, 270 km at 90 km/h 

7. 16 km/h, 4 km/h 

8. $6000 at 4%, $9000 at 5% 

9. x 34, y 10 

10. 10 mL of the 5% solution, 40 mL of the 10% 

solution 

11. 25 mL 

12. x 32, y 20 

13. 495 km/h, 55 km/h 

14. 2.7 m by 1.2 m 

15. 30 min 

16. 2.5 

17. 400 km 

18. x q r 

, y q r 

 

2 2 

19. a) 1.8 h b) 135 km 

20. 8, 14, 31 

21. No, since a b c. 


23. a) 24 m by 2 m b) not possible 

c) 24 m by 2 m d) not possible 

Career Connection p. 46 

1. 100 g of 18-karat gold, 50 g of 9-karat gold 


a) Cost: C 2n 2000; Revenue: C 10n 

b) 250 c) 2250 

Rich Problem pp. 48–49 

1 Graphing and Interpreting Data 

2. a) 95% b) 5% 

3. 4 million years ago 

2 Communication 

3. a) (14, 50) 

b) Fourteen million years ago, the populations 

were equal. 

4. No, the graphs only show percents, not absolute 

numbers. 

5. a) 5; 5 


1. y 5x 20; y 5x 120 

2. (14, 50) 

Review of Key Concepts pp. 50–53 

1. a) (4, 1) b) (4, 3) 

c) (3, 2) d) no solution 

e) (1, 3) f) infinitely many solutions 

g) (2, 1) h) 1 2 , 5 

2. a) (1.9, 2.2) b) (0.1, 0.7) 

3. a) infinitely many solutions b) no solution 

c) one solution d) no solution 

4. Sahara Desert: 9 million square kilometres; 

Australian Desert: 4 million square kilometres 

5. a) d represents the total cost or revenue; 

p represents the number of paddles. 

b) (62.5, 1125) c) greater than 62 

6. a) (2, 2) b) (1, 1) c) (4, 2) 

d) infinitely many solutions e) (1, 3) 

f) no solution g) (1, 5) 

7. a) (3, 2) b) (1, 1) 

h) 1, 1 3 

8. Mount Pleasant: 16, Centreville: 15 

9. a) (1, 2) b) (2, 1) c) (3, 2) 

d) (1, 0) e) no solution 

f) infinitely many solutions g) (2, 3) 

h) (4, 1) 

A 1–4 

Chapter 1, Student Text Answers

10. Methods may vary. 

a) substitution: (4, 5) 

b) elimination: (1, 1) 

c) substitution: 2, 1 2 

d) elimination: (1, 2) 

11. (2, 3) 

12. a) (3, 4) b) (0.6, 0.5) 

13. one night: $150, one meal: $15 

14. 36 cars and 9 vans 

15. $5000 Canada Savings Bond, $10 000 Provincial 

Government Bond 

16. 75 kg of 24% nitrogen, 25 kg of 12% nitrogen 

17. 40 km/h; 280 km/h 

18. 210 km 

Chapter Test pp. 54–55 

1. a) (4, 3) b) (2, 3) 

c) (1, 0) d) (1, 2) 

2. a) (0.7, 3.7) b) (2.4, 1.1) 

3. a) The lines intersect at exactly one point. 

b) The lines are parallel and distinct. 

c) The lines are coincident. 

4. a) (2, 2) b) 3, 1 2 

5. a) (1, 1) b) (2, 1) 

6. a) (3, 2) b) (2, 2) 

c) infinitely many solutions 

d) no solution e) (6, 4) 

f) 2 3 , 1 3 

g) 4 7 , 2 7 

a) extinct: a species that no longer exists; extirpated: 

a species no longer existing in the wild, but 

existing elsewhere; endangered: a species facing 

imminent extirpation or extinction; threatened: a 

species likely to become endangered if limiting 

factors are not reversed; vulnerable: a species of 

special concern because of characteristics that 

make it particularly sensitive to human activities 

or natural events 

b) all living things, including plants and animals 

10. Alberta 


Problem Solving p. 60 

1. 298 

2. 50 

3. 92 units 

4. infinitely many; they pass through the centre of the 

rectangle 

5. 1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 

23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40 


1. 253 14 

2. a) 20 cm 2 b) 5 cm 2 

3. 05:00 Wednesday 

4. 321 cm 2 

5. 16 

6. 12 

7. a) 94 b) 50 

8. 31, 49 

h) (4, 1) i) (5, 6) 

j) (3, 3) 

7. Mackenzie River: 4241 km, Yukon River: 3185 km 

8. 240 g of 30% fruit, 360 g of 15% fruit 

9. term deposit: $4000, municipal bond: $9000 

10. 50 km/h, 550 km/h 



1. a) 8 h 29 min b) 179 km/h 

2. Jupiter, Saturn 

3. a) Newfoundland 

b) Prince Edward Island 

4. (information taken from the web site of the 

Canadian Museum of Nature: http:// 

www.nature.ca/english/eladback.htm)

Achievement Check - McGraw-Hill Ryerson

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