BC Math 9 Chapter 5

BC MATH 9 Pilot Edition Author

Laura Epp Numeracy Mentor Support SD43

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BC Math 9 - Pilot Edition Copyright © 2025, Edvantage Interactive

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About This Math 9 Chapter 5 is being piloted in classrooms during the 2025 school year. Edvantage Interactive is using this pilot to check clarity, pacing, examples, and assessment items so the final edition better supports student learning. We are grateful to the teachers who are trying new materials with their classes and taking time to share feedback. We also thank the students who are completing these lessons and pointing out where explanations can be clearer. Your participation helps us build a book that works in real classrooms.

For more information, please contact us.

Contents Chapter 1 Rational Numbers 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

What is a Rational Number?

Equivalent forms of Rational Numbers Comparing and Ordering Rational Numbers Operations With Decimals Adding and Subtracting Fractions Multiplying and Dividing fractions Order of Operations With Rational Numbers Applications Chapter Review

Chapter 2 Powers and Roots 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

What is a Power?

Exponent Laws Part 1 Exponent Laws Part 2 Combining Exponent Laws Square Roots Approximating Square Roots Scientific Notation (optional) Applications Part 1 Applications Part 2 Chapter Review

Chapter 3 Polynomials 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

What is a Polynomial?

Introduction to Algebra Tiles Adding Polynomials Subtracting Polynomials Multiplying Polynomials Dividing Polynomials Combined Polynomial Operations Applications of Polynomials Chapter Review

Chapter 4 Linear Relations 4.1 4.2 4.3 4.4 4.5 4.6

BC Math 9

Chapter 5 Linear Equations 5.1 5.2 5.3 5.4 5.5 5.6 5.7

What is a Linear Equation?

Solving 1 and 2 Step Equations With Integers Solving 2 Step Equations With Rational Numbers Solving Multi-step Equations With Brackets Solving Multi-step Equations With Variables on Both Sides

How to Solve a Word Problem Chapter Review

Chapter 6 Proportional Reasoning 6.1 6.2 6.3 6.4 6.5 6.6 6.7

What is Proportional Reasoning?

Converting Linear Metric Units Scale Diagrams Similar Triangles Similar Polygons Applications of Proportional Reasoning Chapter Review

Chapter 7 Statistics (Data Analysis) 7.1 7.2 7.3 7.4 7.5 7.6

What is Statistics?

Factors Affecting Data Collection Collecting and Organizing Data Graphing Data Interpreting Data (optional) Chapter Review

Chapter 8 Financial Literacy 8.1 8.2 8.3 8.4 8.5

What is Financial Literacy?

Banking Simple Interest Making a Budget Chapter Review

Cumulative Review of Math 9

What is a Linear Relation?

Graphing Linear Relations Part 1 Graphing Linear Relations Part 2 Interpreting Graphs of Linear Relations Applications of Linear Relations Chapter Review

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Introduction iii

Welcome to Edvantage Math: BC Math 9 Before diving into your course, we invite you to read this introduction. It outlines key ideas from modern cognitive science research; ideas that shaped how this textbook was designed to help you learn more effectively and remember for longer.

What Is Learning?

Let’s start with a research-based definition from Make It Stick: The Science of Successful Learning: “Learning is acquiring knowledge and skills and having them readily available from memory so you can make sense of future problems and opportunities.”

Brown, Roediger, & McDaniel (2014), p. 2 This means that learning isn’t just about reading or listening. It’s about getting ideas and skills into your long-term memory so that you can apply them when you face new challenges. That’s what this course, and this textbook, are designed to help you do.

How Your Brain Learns: A Simplified Model The diagram you’ll see below is a simplified version of the Cognitive Theory of Multimedia Learning (Mayer, 2005), which integrates decades of research on how people learn from words, images, and experiences.

The diagram highlights three key memory systems:

Reproduced with permission from Failsafe Strategies for Science and Literacy

•

Sensory memory: Information enters through your senses (especially eyes and ears).

•

Working memory: Where your brain holds and organizes new information.

•

Long-term memory: Where learning is stored permanently as it’s integrated into your prior knowledge.

Understanding how these systems interact is critical to you learning effectively.

Attention: The Gatekeeper of Learning You can’t learn what you don’t notice. Your sensory memory only processes what you give attention to. If you're distracted by your phone, for example, you won’t retain what your teacher is saying, no matter how good the explanation.

That’s why attention is the first essential skill for learning. This textbook is structured to help you focus on what matters most in each lesson.

Cognitive Load: Why Less Is More Your working memory can only hold about 4 to 7 pieces of new information at once and often fewer when the material is unfamiliar. If too much is thrown at you at once, your brain gets overwhelmed and can’t process it deeply enough to store in long-term memory.

That’s called cognitive overload, and it’s a major barrier to learning.

To address this, Edvantage Math: BC Math 9 is carefully designed to reduce unnecessary cognitive load so your brain can focus on understanding core ideas. Clear diagrams, structured problem-solving prompts, and scaffolded practice are just a few of the strategies we’ve built in to help.

iv Introduction

© Edvantage Interactive 2025

The Importance of Forgetting (and Remembering) Forgetting is normal. It happens to everyone. In fact, it begins as soon as we learn something. This is known as the Ebbinghaus Forgetting Curve and is illustrated in the graph below. But here’s the good news: forgetting can be slowed and even reversed by reviewing information at spaced intervals as shown in the graph.

This process is called retrieval practice, and it’s one of the most powerful learning strategies discovered in cognitive science. You’ll find many activities in this book designed to help you remember what you learn when you need it most.

This Book Supports How You Learn Partially Worked Solutions - Try Together Every sample problem is followed by a Try Together problem. This problem is a partially worked solution. This helps you focus on the key aspects of the problem without overloading your working memory with too many steps at once.

How is it Going?

At the end of each unit, you become the teacher. You'll mark a sample student’s quiz allowing you to retrieve, evaluate, and apply what you’ve learned. This kind of elaborative practice strengthens long-term memory and makes your learning more durable.

Remembering Ebbinghaus Periodically, you’ll encounter questions that revisit earlier topics even if they’re not part of your current unit. This deliberate spacing and retrieval strengthens memory and prepares you to use math concepts flexibly, not just repeat them on a test.

Final Thoughts If you’ve read this far, congratulations, you’ve already taken the first step in developing strategies that help you learn. This textbook is more than a source of information; it’s a tool to help you become a better learner.

Good luck on your journey through BC Math 9 and don’t forget to visit edvantagemath.com for additional resources and support.

© Edvantage Interactive 2025

Introduction v

References and Suggested Reading Ashman, G. (2022). Little Guide for Teachers: Cognitive Load Theory.

Brown, P. C., Roediger, H. L., & McDaniel, M. A. (2014). Make It Stick: The Science of Successful Learning.

Christodoulou, D. (2014). Seven Myths About Education.

Kirschner, P., Hendrick, C. (2020). How Learning Happens: Seminal Works in Educational Psychology and What They Mean in Practice.

Ebbinghaus, H. (1885/1913). Memory: A Contribution to Experimental Psychology.

Mayer, R. E. (2005). The Cambridge Handbook of Multimedia Learning.

Mirabelli, S., Sandner, L. (2023) Failsafe Strategies for Science and Literacy

vi Introduction

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5 Linear Equations This chapter is your guide to understanding and solving linear equations. You will explore a range of strategies to find solutions, both symbolically and pictorially. Hands-on and visual activities are integrated throughout to provide a connection between theory and applications. While most equations will have one clear solution, some might surprise you with no solution at all, or many different answers. You will build foundational skills and confidence to help you solve problems in math and beyond.

Big Ideas of the Chapter: •

Solving equations algebraically and pictorially.

•

Preservation of equality.

•

Verifying the solution of an equation (checking your answer).

•

Rewrite expressions in terms of the variables involved. For example: speed is distance divided by time can be written as distance is equal to speed multiplied by time.

•

The distributive property is a(b + c) = ac + bc

•

There are a variety of strategies that can be used to simplify solving (for example: multiplying by LCM to eliminate fractions and cross multiplying).

•

Linear equations of the following forms have x as a variable, a, b, c, d as real numbers, where a is not zero:

•

Linear equations can be used to solve problems in context.

The graph on the right shows that the amount of fuel in the gas tank depends on the distance driven by the vehicle.

What do you notice, what do you wonder?

©Edvantage Interactive 2025

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5.1 What is a Linear Equation?

Warm Up Competency Focus - apply multiple strategies to solving problems, in both abstract and contextual form.

1. Solve the Math Mobile.

What number does the circle represent? The triangle? How do you know?

2. The Magic of Math Ask your friend what their favorite number from 1 to 9 is. Multiply it by 3 and remember that number.

Ask your friend to multiply 37 037 times the number from step 1 What happens? Try it with a different number. Explain the pattern?

Linear equations are mathematical equations that represent a straight line when graphed. The simplest example is y = x. When x = 0, y = 0. When x = 1, y = 1. y is the same value as x.

Why do we study linear equations? To help us understand the world around us, so we need to understand how things change. The simplest kind of change is a constant change. A linear equation helps us to describe and represent this kind of change, a change that happens at a constant rate. For example, the table on the next page is based on an hourly salary of 20 dollars an hour.

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S = 20t Hours

Salary

1

$20 $40

2 100

$2000 $4000

200

The linear equation for this table is S = 20t. We can use this linear equation to: • Make predictions like: How much do you earn after an 8 hour shift?

• Solve problems like: How many hours do I need to work to buy headphones that cost $150?

Linear equations have a degree of 1, meaning that the largest exponent for the variables , but you will NOT see terms x and y is 1. This means you will see terms like y, 3y, x, 2x and with , x2, x3, y2 in linear equations.

Try This ...

Circle the linear equations: Hint: look at the exponents of x and y.

Preservation of Equality

You may have heard a math teacher at school say this: “Whatever you do to one side of an equation, you must do to the other side.”

This is the foundation of solving equations, and allows us to preserve or keep the equation balanced. The image below illustrates this statement using a balance instead of an equation.

Add a marble on the right

No longer balanced - right side is heavier ©Edvantage Interactive 2025

Add a marble on each side

Still balanced

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Verify Your Solution

When solving any equation, it is helpful to check your solution. Double checking is a useful practice in the real world too. You may have heard of “measure twice, cut once” in carpentry.

Pilots also use a variety of double-check procedures to ensure safety and mitigate risks before, during, and after flights. Checking your solution will help you catch and fix your mistakes. It can also help avoid finding a solution that is impossible or unreasonable, a solution that seems right until you put it in context and find out you have divided by zero, gone back in time, or taken the square root of a negative number.

Sample Problem — Checking Your Solution Is x = 2 a solution for 2x –1 = 3?

What to Think About 1. Check if x = 2 by substituting or replacing x with 2.

How to Do It

2x – 1 = 3 2(2) – 1 = 3

2. Simplify.

4–1=3 3=3

3. Is this true or false statement? If it is true, it is a solution. If it is false, it is not a solution.

True x = 2 is a solution

Try Together— Checking Your Solution Joe and Silas are solving the equation 2x + 3 = 5x – 6. Joe got a solution x = 4 and Silas got x = 3. Who is correct?

Joe’s Solution

Silas’ Solution

1. Check if x = 4.

1. Check if x = 3

2. Replace x with 4.

2. Replace x with 3

2( ____ ) + 3 = 5( ___ ) – 6

2(___) + 3 = 5(___) – 6

____ + 3= ____ – 6

___ + 3 = ___ – 6

___ = ___

___ = ___

This is true / not true.

This is true / not true

Is Joe correct?

Is Silas correct?

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Your Turn — Checking Your Solution Which of the following equations have x = 1 as a solution x+2=3

How Many Solutions?

x–2=1

2x = -2

-x – 4 = -5

0.75x = 0.75

A linear equation can have one solution, no solution, or infinitely many solutions.

After you simplify both sides, compare the variables and constants: • If you get x = a number there is one solution.

• If you get a false statement involving 2 numbers like 4 = 7, there is no solution.

• If you get a true statement involving 2 numbers like –3 = –3 then there are infinite solutions.

Sample Problem — How Many Solutions?

For the equation 2(x + 3) = 2x + 6, check to see if x = 0, 1, –10 and ½ are solutions. Don’t forget to use BEDMAS. x=0

x=1

x = –10

x=½

2 (0 +3) = 2(0) + 6

2 (1 +3) = 2(1) + 6

2 (–10 +3) = 2(–10) + 6

2 (½ +3) = 2(½) + 6

2(3) = 6

2(4) = 8

2(-7) = -20 + 6

2(½ +

6=6

8=8

-14 = -14

7=7

True

True

True

True

∴ x is a solution

∴ x is a solution

∴ x is a solution

∴ x is a solution

)=1+6

All four numbers are solutions. In fact, all real numbers will be solutions to this equation. There are an infinite number of solutions. In lesson 5.4, you will learn how to prove that these two expressions are equal.

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Try Together— How Many Solutions?

When we solve linear equations we often get one solution. Is there always one solution? Consider x+1 = x+1 and x+1=x+2 when x = 0, x = 2 and x = 3. x+1=x+1

x+1=x+2

When x = 0 ___ + 1 = ___ + 2

___ + 1 = ___ + 1

___ = ___

___ = ___

This is true / false

This is true / false

When x = 2 ___ + 1 = ___ + 1

___+1 = ____+2

___ = ____

___ = ____

This is true / false

This is true / false

When x = 3 ___ + 1 =___ + 1

___ + 1 =___ + 1

___ = ____

___ = ____

This is true / false

This is true / false

What do you notice? Try a few other numbers mentally. Why is this happening?

Your Turn — How Many Solutions?

Look at the following equations and determine whether they have no solution or infinite solutions. How do you know?

3x – 1 = 3x – 1

-x + 4 = -x + 7

x = x – 1.5

Important Note: Sometimes when we solve problems there is a solution, but sometimes there are no solutions, or there are many, many possible solutions.

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Extend and Explore

Look at the graphs of the 2 straight lines y = x + 1 and y = x + 2.

What do you notice about the two lines?

Do they ever cross each other?

How does that relate to solving x + 1 = x + 2?

Remembering Ebbinghaus – Simplifying Expressions 1. Which of these expressions is equal to 3x + 2? a.

b.

c.

2. Create 2 expressions that simplify to –x – 1.

3. a. Multiplying by one third is the same as dividing by ____. b. Dividing by one half is the same as multiplying by ___. c. Multiplying by ¾ is the same as dividing by ___. d. Dividing by 10/11 is the same as multiplying by ___.

4. Use the table of values for the linear equation y = x + 1 to plot 3 points on the grid provided. Draw a line through the points.

©Edvantage Interactive 2025

x

y

-1 0

0 1

1

2

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5.1 Practice 1. Identifying Linear Equations - Circle the linear equations: a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

2. Linear Equations - Fill in the blank a. You can check your solution to a linear equation by _______________ your answer into the equation, to see if it makes a true or false statement. b. Equations with a degree of one can have no solution, ______ solution, or infinite solutions. c. The degree of a linear equation is always ______. d. The graph of a linear equation looks like a ____________________. e. As we solve an equation, we want to make sure we don’t change the equation. This is why we talk about the preservation of ________________.

3. Working with Linear Equations - Solve a. b.

4. Simplify the expressions a.

b.

c.

d.

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5. Preserve the Equality Circle the equation on the right that is equivalent to the one on the left.

Equation

Which is Equivalent?

a. b. c. d. e.

6. Verify Your Solution Check to see if x=1 is a solution to the following equations. Write yes or no. Show your work. a.

b.

d.

c.

7. How Many Solutions?

How many solutions are there for each equation? How do you know?

a.

b.

c.

v

d.

v

e.

f. Create an equation that has many solutions.

g. Create an equation that has no solutions.

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5.2 Solving 1- and 2-Step Equations With Integers Warm Up Competency Focus - Apply strategies to solve problems in contextualized situations.

Jamie bought 4 bags of dill pickle chips and a chocolate bar for $13.50. The chocolate bar was $1.50. How much did each bag of chips cost?

How did you figure it out?

Equation strips are a visual representation of solving equations. The two strips represent each side of the equal sign. In the image below the top strip represents an unknown value and 4.

The bottom strip represents 15. As an equation this is written ? + 4 = 15.

Equation Strips

?

4 15

Visually you can determine that to make both strips equal you would make the unknown value 11. In the next example, what is the value of ‘?’? ?

?

20

50 To figure out the answer as an equation, we can write the following: Top Strip = Bottom Strip ? + ? +20 = 50 2? + 20 = 50 2? = 30 ? = 15

Try This ...

1. Write an equation to match the pictures below. What are the missing values? How do you know? a. b. c.

5 13

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6 30

6 42

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2. Create an equation strip and share it with a partner or the class. Choose a number for the bottom strip, then divide the top strip into sections that include an unknown value.

Your equation strip

Your partner’s equation strip

Unknown value = ?

Unknown value = ?

Isolating the Variable

“To isolate” means making a person feel alone or apart from others. In math, it means having the variable , x, by itself on one side of the equation and all other terms on the opposite side.

Let’s take the equation x + 1 = 2 as an example where we isolate x. x + 1 = 2 x+1–1=2–1 x = 1

Solving 1-Step Linear Equations

We want to isolate x, so we will do the opposite of adding 1.

We subtract 1 from both sides.

We simplify and x has been isolated.

A 1-Step linear equation is an equation where you can find the value of the variable by doing just one operation, like adding, subtracting, multiplying, or dividing once.

When we use an equation, remember preservation of the equality. What you do to one side of the equation, you must do to the other. Your goal is to isolate the variable, x. Your main tool to isolate a variable is to use opposite operations. For example: • If the variable is being multiplied by 2, the DIVIDE both sides by 2.

• If the variable is being divided by 2, the MULTIPLY both sides by 2.

• If we are adding 4 to the variable, then SUBTRACT 4 from both sides.

• If we are subtracting 4 from the variable, then ADD 4 to both sides.

Sample Problem — Solving 1-Step Linear Equations (Add and Subtract) x+7=5

What to Think About 1. We want to isolate x. The opposite of adding 7 is subtracting 7.

2. Subtract 7 from both sides 3. Simplify 4. Check answer. Is the statement true or false 5. It is true so the solution is correct. ©Edvantage Interactive 2025

How to Do It x+7=5 x+7–7=5–7 x = –2 –2 + 7 = 5 5 = 5 Answer is true Chapter 5 Linear Equations (Pilot)

331

Sample Problem — Solving 1-Step Linear Equations (Opposite is Divide) 3x = 33

What to Think About

How to Do It

1. We want to isolate x. We are multiplying by 3.

2. Do the opposite of multiplying by 3. Divide both sides by 3.

3. Simplify.

x = 11

4. Check answer.

3 (11) = 33 33 = 33 (True)

Try Together— Solving 1-Step Linear Equations (Opposite is Multiply)

1. We want to isolate x. We are dividing x by 2.

2. Do the opposite of dividing by 2. Multiply both sides by 2.

____

3. Simplify.

x = ____

= 12 ( ___ )

= 12

4. Check answer.

_____ = _____ (Is this true? _____)

Your Turn — 1-Step Linear Equations 1. For which equation is the solution x = –2? (Hint: replace x with –2) a.

b.

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c.

d.

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Your Turn — 1-Step Linear Equations - continued 2. Solve for x a.

b.

c.

d.

e.

f.

g.

h.

3a. Create a one step linear equation whose solution is x = 8. (Hint: pick one operation and one number like adding 4, or multiplying by 3).

3b. Share your equation with a partner. Write out their equation and check to see if 8 is a solution.

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Solving 2-Step Linear Equation

A 2-Step linear equation is an equation where you need to do two operations to solve for the variable. 2x + 1 is an example of an expression with 2 operations, multiplying and adding.

What does the expression 2x + 1 mean? Start with a number, x. First double it, then add 1.

If we want to get back to where we started, we need to undo those two operations.

We need to do the opposite of doubling and adding. But in what order? It can help to think of shoes and socks. We start with bare feet and put on socks, then shoes. If we want to get back to bare feet, which should we remove first? Socks or shoes? The shoes, because they were the last thing we put on. In the expression 2x + 1, to isolate x we need to subtract 1 first (opposite to adding 1) and then divide by 2 (opposite of multiplying by 2).

Sample Problem — Solving 2-Step Linear Equations (Opposite is Divide) Solve for x.

What to Think About

How to Do It

1. Subtract 5 from both sides.

2. Simplify.

3. Divide both sides by 3.

4. Simplify.

Try Together— Solving 2-Step Linear Equations (Opposite is Multiple) Solve for x.

What to Think About

How to Do It

1. _________ to both sides.

2. Simplify.

3. _________ to both sides by 5.

4. Simplify.

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Your Turn — Solving 2-Step Linear Equations 1. Solve for x. a.

b.

c.

d.

e.

f.

g.

h.

2a. Create a 2-Step linear equation whose solution is x = 10 and solve it.

2b. Switch with a partner. Write their equation here and check to see if x = 10 is a solution.

Remembering Ebbinghaus – Simplify using Order of Operations 1. a.

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b.

c.

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5.2 Practice 1. 1-Step Equations – Solve for x. a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k. 0.9x = –99

l. x – 0.75 = –0.25

2. Using 1-Step Equations – For which equations is the solution x = –1? (Hint: replace x with –1) a.

b.

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c.

d.

e.

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3. 2-Step Equations – Solve for x. a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k.

l.

m.

n.

o.

4. Create a 2-Step Equation – Create two different 2-Step linear equations that have a solution of x = –0.5.

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5.3 Solving 2-Step Equations with Rational Numbers Warm Up Competency Focus - apply strategies to solve problems in contextualized situations Choose one of the two questions below. Explain your answer.

1. Would you rather buy a jacket that is 30% off no sales tax OR 20% off with a 5% sales tax?

2. Would you rather buy a gaming console that is 5% off with a $25 rebate after purchase OR 10% off and no tax?

Rational Numbers and Equations

Recall rational numbers are all the numbers that can be written as a fraction where the numerator and denominator are integers and the denominator is not zero. They include integers, fractions, and terminating or repeating decimals. For example, −3, , 0.75 and –0.3 are all rational numbers.

In the last section you learned how to solve 2-Step equations. You can still use those strategies when faced with equations that have rational numbers as coefficients, constants, and solutions. In this section we will introduce new strategies for solving equations involving rational numbers (fractions and decimals). These new strategies offer choice and flexibility, boost understanding and often save time. We will also work with formulas, and use our skills to efficiently find solutions to problems in context, like converting from Fahrenheit to Celsius.

Sample Problem — Solving 2-Step Equations with Rational Numbers Solve for x using one of the strategies covered in the previous section.

What to Think About

How to Do It

We want to isolate x.

Add

to both sides.

Simplify (add fractions).

Simplify (reduce a fraction).

Multiply both sides by 2.

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Try Together— Solving 2-Step Equations with Rational Numbers Solve for x.

What to Think About

How to Do It

Isolate x.

Subtract ______ from both sides.

Simplify.

Multiply both sides by ______.

Simplify.

Your Turn — Solving 2-Step Equations with Rational Numbers Solve for x.

2.

1.

3.

4. For which equations is x = one half a solution? a.

b.

c.

d.

e.

f.

5. Create a 2-Step linear equation whose solution is x= one quarter and solve it.

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Alternate Strategy for Solving 2-Step Equations with Rational Numbers

Solving equations with fractions or decimals can be more challenging. Multiplying every term by the least common multiple (LCM) of the denominators or a suitable multiple of 10 for decimals helps clear these decimals or fractions from the equation. This process results in an equivalent equation with whole numbers, making the steps to solve a 2‑step equation simpler and errors less likely.

Sample Problem — Solving 2-Step Equations with Rational Numbers: Fractions Solve for x.

What to Think About

How to Do It

Find the LCM of the denominators 4 and 2.

It is 4.

To multiply both sides by 4 (the LCM), we will need to multiply each term by 4.

Simplify.

Add 4 to both sides.

Divide both sides by 3.

Sample Problem — Solving 2-Step Equations with Rational Numbers: Decimals Solve for x.

What to Think About

How to Do It

What could we multiply everything by to

0.25x + 0.75 = 1.5

eliminate the decimals?

Multiply every term by 100.

100(0.25x) + 100(0.75) = 100(1.5)

Simplify.

25x + 75 = 150

Subtract 75.

25x = 75

Divide by 25.

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x=3

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Try Together— Solving 2-Step Equations with Rational Numbers Solve for x.

What to Think About

How to Do It

Multiply everything by ____.

Simplify.

–4x + ____ = 2

Subtract ____ from both sides.

–4x = 1

Divide both sides by ____.

x = ____

Your Turn —Solving 2-Step Equations with Rational Numbers Solve for x.

1.

2.

3. 0.6x – 1.2 = 3.6

4. –0.3x + 8.5 = 11.5

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Formulas

Formulas are powerful tools in math and science that allow us to quickly find missing information and solve problems. For example on the highways in British Columbia the speed limit is 50 km/hr, unless otherwise posted. The 50 km/hr (speed) represents 50 km (distance) in 1 hour (time). As a formula this is written as: speed =

=

To rewrite this formula in terms of time, we multiply both sides by time (t): st = d And then divide both sides by speed (s):

Try This ...

1. Rewrite the speed formula in terms of distance (d).

2. Rewrite the formula Interest = principal(p) * rate(r) * time(t) two different ways.

3. To calculate final velocity, you can use the formula vf = vi + at where vi is initial velocity, a is acceleration and t is time.

Rewrite this formula in terms of time.

Extend and Explore

Why is the sum of 3 consecutive integers divisible by 3? Show that the sum of 3 consecutive integers is divisible by 3.

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Remembering Ebbinghaus – Lowest Common Multiple 1. What is the LCM for the following? a. 10 and 20

b. 2, 4 and 6

c. 3, 6 and 9

d. 20 and 30

e. 2, 3 and 7

f. 3, 6 and 10

5.3 Practice 1. 2-Step Equations with Rational Numbers – Solve for x

a.

e.

i.

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b.

c.

d.

f.

g.

h.

j.

k.

l.

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5.3 Practice – continued 2. Find a Solution – Show that

is a solution for a and c. Show that 2.5 is a solution for b and d.

a.

b.

c.

d.

3. Using Formulas – Rewrite these statements into formulas in terms of the variables indicated. a. The area of a rectangle is the length times the width. (width)

b. The perimeter of a rectangle is the sum of the lengths of the four sides. (width)

c. The circumference is 2πr, where r is the radius. (radius)

d. Area of a triangle is half the base times the height. (base)

Extend and Explore

Prove that the sum of two odd numbers is even.

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5.1 – 5.3 Check Your Understanding 1. Which equations are equivalent to –x + 8 = –16. a.

b.

c.

d.

e.

f.

g.

h.

i.

2. The perimeter of a regular hexagon is 162cm (perimeter is the distance around the outside and a regular hexagon is a 6 sided polygon with equal sides). Write an equation to find the length of a side. Then use it to find the length of one side.

3. Make an equation that you will solve where you will add 2 to both sides and then divide by 3 to isolate the variable x.

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4. Create two equations whose solution is –5 and solve them.

5. Draw a picture to represent 3x – 3 = 6. How could you use this picture to find x?

How is it Going?

1. Below is a student’s quiz with 9 questions. Some answers are correct and some are not. Give a check mark if the question is solved correctly. If it is not, circle the error and correct the error in the space provided on the right of each question.

a.

b.

c.

d.

e.

f.

g.

h.

i. –7 – x = 0 –x = 7 x = –7

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5.4 Solving Multi-step Equations with Brackets Warm Up Competency Focus - develop, demonstrate and apply mathematical understanding through play, inquiry and problem solving.

1.

Think of a number.

2.

Double the number

3.

Add 10

4.

Divide by 2 (take half of it)

5.

Subtract the number you started with in step 1.

6.

What number did you get at the end? Try it with a different number.

7.

What do you notice? What do you wonder?

Think More: Why does it work? Can you make a number trick of your own?

The Distributive Property

Recall the distributive property is a(b + c) = ab + ac Now let’s prove the distributive property if a = 3, b = 4 and c= 5 through inductive reasoning.

Left Side

Right Side

a(b+c)

ab +ac

3(4+5)

is the same as

3(4) + 3(5)

3(9)

is the same as

12 + 15

27

is the same as

27

The values are the same.

Try This ...

1. Rewrite the following expressions using the distributive property. a.

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b.

c.

d.

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Sample Problem — Using the Distributive Property to Solve an Equation Solve for x.

What to Think About

How to Do It

1. Use distributive property.

2. Add 3 to both sides.

3. Multiply both sides by 2.

Try Together— Using the Distributive Property to Solve an Equation Solve for x.

How to Do It

What to Think About

0.1(10x – 10) = 11

Use the distributive property.

_________________ = 11

Add ______ to both sides.

x = 12

Your Turn — Using the Distributive Property to Solve an Equation Solve for x.

1.

2.

3.

4.

5.

6.

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Alternate Strategy for Solving Equations with Brackets

Sometimes an equation with brackets can be solved more quickly by dividing first instead of multiplying out the brackets. For example, to solve 2(x +3) = 14: 2(x +3) = 14

Divide both sides by 2.

x+3=7

Then subtract 3.

x=4 This method works because dividing by the number in front of the brackets immediately cancels that factor, leaving a simpler equation.

Use this strategy when the number outside the brackets divides evenly into the number on the other side of the equation. Check the relationship between those two numbers first. If the coefficient in front of the brackets is a factor of the constant on the other side, dividing first will usually be the quickest way to the answer.

Try This ...

1. Use the alternate strategy to solve for x. a. –3(x – 4) = 9

b. 5(2x – 1) = 35

c.

d.

Remembering Ebbinghaus - Combining Like Terms 1. Simplify the following expressions a. 2x-4x+7x-9x

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b. -x+5-x+8-x-13

c. 2(x-2)+3(x-3)+4(2-x)

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5.4 Practice 1.Distributive Property - Rewrite these expressions a.

b.

c.

d.

a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k.

l.

2. Solve for x.

Extend and Explore

You have 12 identical marbles and a balance scale. One of the marbles is heavier. How many weighings will it take to find the heavier marble? Can you do it in less than 4? What would change if it is not known whether the marble is heavier or lighter?

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5.5 Solving Multi-step Equations with Variables on Both Sides Warm Up Competency Focus - represent mathematical ideas in concrete, pictorial and symbolic forms The image below shows 5 bags of marbles on the left with 4 extra marbles and 4 bags on the right with 10 extra marbles. If each bag has the same amount of marbles, how many are in each bag if the scale is balanced? Show your thinking.

Solving Equations with Variables on Both Sides

When an equation has variables on both sides, the goal is to gather all the variable terms together so that only one variable remains on one side. Think of it like moving everything of the same kind to the same team. You can use the same opposite-operation strategies you learned for two-step equations: adding to undo subtraction, subtracting to undo addition, multiplying to undo division, or dividing to undo multiplication. The key is to combine like terms so you can simplify and work with fewer pieces.

Sample Problem — Multi-step Equations with Variables on Both Sides Solve for x.

What to Think About

How to Do It 8x − 12 = 6x + 26

Subtract 6x from both sides.

8x − 6x − 12 = 6x − 6x + 26

Add 2x to both sides.

2x−12 + 12 = 26 + 12

Divide by 2.

2x = 38 x = 19

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Try Together— Multi-step Equations with Variables on Both Sides Solve for x

What to Think About

How to Do It 3x – 5 = 2x + 6

Move all x terms to one ____.

3x – ____ – 5 = 2x – ____ + 6 x – 5 + ____ = 6 __ ____

Add ____ to both sides.

x = ____

Your Turn —Multi-step Equations with Variables on Both Sides Solve for x.

1. x + 10 = 15 + 6x

Solving Equations with Variables on Both Sides and Brackets

2. 8 – 4x = 5 – 3x

3. 1 – 14x = x – 9

When an equation contains brackets on one or both sides, the first step is to remove those brackets so you can work with the terms directly. Recall that this is called expanding or using the distributive property. After expansion, you will have an equation with variables on both sides. Then the goal is to gather the x terms together and then isolate the variable by using the opposite operations you already know. Careful expansion makes every other step easier and ensures that the equation stays balanced as you solve it.

For example, take the equation 2(x – 3) = x – 7. This can be expanded and solved by: 2x – 6 = x – 7

x = ­–1 And the answer can be checked by changing x to –1

2(–1 –3) = –1 –7 2(–4) = –8

–8 = –8 Answer is true or correct.

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Try This ...

1. Solve the equations using the distributive property. a. 5(2x – 3) + 7 = 7x +7

b. 5(x + 7) = 2x – 4

c. 1.5(x + 4) = 3.5 – x

2. Solve and verify:

Solving Variables on Both Sides with Fractions and Decimals

When an equation includes fractions or decimals, the first challenge is that these numbers can make each step longer and more complicated. A way to simplify the work is to clear the fractions before you begin solving. To do this, find a number that is a common multiple of all the denominators and multiply every term of the equation by that number. This keeps the equation balanced while turning every fraction into a whole number. In the sample below, multiplying everything by 20, the lowest common multiple of 5 and 4, removes all fractions in a single step.

After that, solving the equation is just like the two-step equations you already know. By clearing fractions or decimals first, you make the rest of the solution faster, clearer, and less prone to mistakes.

LCM of 5 and 4 is 20.

Multiply by 20 and simplify.

Subtract 5x and subtract 60.

Divide by 3.

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Similarly, with decimals, we can multiply by a power of 10 to eliminate the decimal numbers. In the sample below we multiply everything by 10. This is an optional method. Of the 2 methods shown here, which do you prefer? Would it depend on whether you had access to a calculator?

Method 1 Method 2 0.4x – 3 = –0.4 Multiply everything by 10 to eliminate decimals 4x – 30 = –4 Add 30 4x = 26 Divide by 4 x = 6.5

0.4x – 3 = –0.4 0.4x = 2.6

Add 3 Divide by 0.4

x = 6.5

Try This ...

1. Solve for x. a.

b.

c.

e.

f.

g. 0.5(2x – 3.4) = 1.2x + 0.8 h. –1.3(0.4x + 2.5) = –0.6x – 3.1

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d.

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Solving 2 Variables on Both Sides and Cross Multiplying

When two fractions are equal, you can solve for the unknown by cross multiplying. This means you multiply the numerator of one fraction by the denominator of the other and set the two products equal. If and b and d are not zero, you can rewrite the equation as ad = bc.

Cross multiplying quickly removes the fractions and gives an equation with whole numbers that is easier to solve. This method is especially helpful when working with ratios, proportions, or percent problems.

Sample Problem — Cross Multiplying Solve for x.

What to Think About

How to Do It

Cross multiply.

Expand and simplify.

Divide both sides by 10.

Sample Problem — Cross Multiply with x in the Denominator Solve for x.

What to Think About

How to Do It

Note that x cannot be zero.

3 is the same as 3/1.

Cross Multiply.

Divide by 3. Check to make sure x is not zero.

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Try Together— Cross Multiplying Solve for x

What to Think About

How to Do It

Cross multiply.

Distributive property.

Simplify.

Divide by 8.

Your Turn —Cross Multiplying 1. Solve for x. a.

b.

c.

, x cannot be ____ .

2. Find x and the perimeter of this triangle. Hint: It is an isosceles triangle with two equal sides.

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Remembering Ebbinghaus - Matching Words to Expressions Match the correct term on the right to the statements on the left. _____ 1. A number plus 5

a. 3x

_____ 2. Triple a number

b.

_____ 3. 4 less than a number

c.

_____ 4. Half of a number

d. x – 4

_____ 5. The total of x and 10

e. 2(x +2)

_____ 6. Twice the sum of x and 2

f. x + 10

_____ 7. Double x and then add 2

g. x + 5

_____ 8. One third of x

h. 2x + 2

5.5 Practice 1. Solving Equations with Variables on Both Sides - What does this picture represent? Write the equation to solve it.

2. Solve and check a.

b.

c.

d.

e.

f.

g.

h.

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i.

j.

k.

l.

m.

n.

o.

p.

q.

r.

s.

t.

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4. Cross Multiply – Solve these equations

a.

b.

5. Solve these equations. State the value that is not permissible for x. a.

, x not equal to ____ ?

b.

x not equal to ____ ?

6. Find x and the perimeter of this triangle.

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5.6 How to Solve a Word Problem Warm Up Competency Focus - connect math concepts to each other and other areas A man has to get a fox, chicken and a sack of corn across the river in a boat. The boat can carry him and one other item.

If the fox and chicken are left alone, the fox will eat the chicken. If the chicken and the corn are left alone, the chicken will eat the corn. How can he do it?

Working with Words and Expressions

The four operations, addition, subtraction, multiplication and division can represent a wide variety of words. For example finding the “total” represents addition. In the table below, write as many words as you can to represent the operations described in each cell.

Words for adding

Words for subtracting

Words for multiplying

Words for dividing

Expressions can represent words with mathematical language using numbers, variables, and operations in a compact way. For example, to represent how much money is made from a job with an hourly salary of 17.85 dollars an hour, start with letting x be the number of hours worked.

Multiplying 17.85 by x would give me the total money earned. This can be written as 17.85x

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Try This ...

Find the expressions on the right that correspond to the words on the left.

1. _____ Nevena has x pencils and buys 5 more pencils. How many pencils does she have?

a. 4x + 6

2. _____ Tom’s allowance is twice the number of chores he completes in one week. x = number of chores

b. 15x

3. _____ Total cost of a box of books that sell for 5 dollars each. x is the number of books

c. 3x + 10

4. _____ Weekly earnings if you work x hours at 15 dollars per hour

d. x + 5

5. _____ Temperature after x hours, if it drops 1.5 degrees per hour, and starts at 24 degrees.

e. 2x

6. _____ Cost of a cell phone plan which costs 10 dollars plus 3 dollars per gigabyte of data used. x = number of gigabytes

f. 3x2

7. _____ Perimeter of a rectangle whose length is 3 more than its width. x = width

g. 24 – 1.5x

8. _____ Area of rectangle whose length is 3 times its width x = width

h. 5x

Steps to Solving Word Problems

A variable is a letter that stands for an unknown value. An equation is a statement that two expressions are equal.

Step 1 – What are you trying to find?

Read the question twice.

• Look at the last sentence.

• What is the question being asked? Let x be the answer.

Step 2 – Write an equation with x Make an equation that connects the key words in the problem.

• For example, Area = length × width, or • Profit = revenue – cost.

Step 3 – Solve the equation or make x alone Use strategies learned in this chapter to isolate x.

• Preserve equality. What you do to one side, you must do to the other.

• Simplify.

Step 4 – State the answer • Answer the question with words and use the correct units.

Step 5 – Check your solution • Substitute your value of x back into the equation. Is it a true statement?

• Does this answer make sense? Is it reasonable?

Tip to use all the time: Whenever possible draw a picture, make a chart, and highlight important numbers and words in the question.

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Sample Problem — Applications Sam has 3 times as much money saved as Taylor. Together they have 100 dollars. How much money does Taylor have saved?

What to Think About

How to Do It

Step 1 – What are you trying to find?

Let x be the amount of money Taylor has saved.

Step 2 – Write an equation with x.

Sam has 3x dollars, Taylor has x dollars, Together they have 100 dollars.

The equation is 4x=100.

Step 3 – Solve the equation or make x alone.

Step 4 – State the answer.

Step 5 – Check your solution.

Taylor has saved 25 dollars.

If Taylor has 25 dollars, Sam has 3(25) or 75 dollars. Together, they have 25+75 = 100 dollars.

Try Together— Applications On a math test, Ivy got two correct answers for every wrong answer. There were 30 questions on the test. How many did she get wrong?

What to Think About

Step 1 – What are you trying to find?

Step 2 – Write an equation with x.

How to Do It Let x be the number of _________________________.

Ivy got ____ answers wrong and ____ answers right. The total number of questions is 30.

The equation is __________________ .

Step 3 – Solve the equation or make x alone.

x + ____ = ____ 3x = ____ ____ = 10

Step 4 – State the answer.

Step 5 – Check your solution.

Ivy got ____ questions __________ on the math test.

Questions _________ + questions ________ = total number of questions or 10 +____ = 30

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Your Turn — Applications 1. Jordan scored points in a basketball game by making 2 and 3 point shots. He made 5 more 2 point shots than 3 point shots and scored 45 points in total. How many of each shot did he make?

2. A number is 5 times another number. The sum of the two numbers is 36. Hint: let x be the smaller of the two numbers.

3. The sum of two consecutive numbers is 31. Consecutive means “in a row”. For example: 4 and 5 are consecutive numbers. 3 and 4, 7 and 8 are also consecutive. Hint: Let x be the smaller number. The bigger number will be one more than that because they are consecutive)

Remembering Ebbinghaus - Graphing Linear Equations Match the lines in the graphs below to the list of equations.

1.

2.

4

3.

4

4

3

3

3

2

2

2

1

1

1

0

0

0

−1

−1

−1

−2

−2

−2

−3

−3

−4

−4

−4

−3

−2

−1

0

1

2

3

4

a

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−3 −4

−3

−2

−1

0

1

2

3

−4

4

b

−4

−3

−2

−1

0

1

2

3

4

c

y=x

y=x

y = –2x

y=x+2

y = 2x

y = 2x + 2

y=x–2

y = ½x

y = 2x – 2

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5.6 Practice 1. Match the terms on the left with the statements on the right a. _____ Jonathan has x books and buys 3 more books. How many books does he have

i. 4x + 10

b. _____ Jewel’s allowance is three the number of chores they complete in one week. x = number of chores.

ii. 21.5x

c. _____ Total value of x nickels.

iii. 2x + 25

d. _____ Weekly earnings if you work x hours at 21.50 dollars per hour.

iv. x + 3

e. _____ Temperature after x hours, if it drops 1.25 degrees per hour, and starts at 23 degrees.

v. 3x

f. _____ Cost of a cell phone plan which costs 25 dollars plus 2 dollars per gigabyte of data used. x = number of gigabytes.

vi. 2x2

g. _____ Perimeter of a rectangle whose length is 5 more than its width. x = width

vii. 23 ­– 1.25x

h. _____ Area of rectangle whose length is twice its width x = width.

viii. 5x

2. Beside each word indicate the operation implied: +, –, ×, ÷ a. total

b. decrease

c. sum

d. loss

e. quotient

f. double

g. difference

h. split evenly

i. increase

j. triple

k. product

l. plus

m. minus

n. less than

o. shared equality

p. reduce

q. gain

r. fewer

s. times

t. more than

3. Which steps are missing in solving a word problem.

1.

What are you trying to find? Let x be the thing you are trying to find. Usually the last sentence has this information.

2.

Write an equation with x using the information from the question.

3.

________________

4.

State the answer

5.

______________

4. 1. One number is 7 times another number. The difference between the two numbers is 30. Find the numbers.

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5. The length of a rectangle is 3 times its width. The perimeter is 48cm. What is the width of the rectangle?

6. Two consecutive integers add to 35. Write an equation to represent this situation and then solve it.

7. A taxi costs $4.00 plus $2.28 per km. a. How much does a 10 km ride cost?

b. Create an equation for C = cost of the taxi, if x is the number of km.

c. Use the equation to find out the cost of a 100km trip.

d. Use the equation to find out how far you can go for 100 dollars

8. Michelle and Laura were walking around Perimeter Trail in Mundy Park, Coquitlam, BC. The trail is 4km long. Estimate the area inside the trail if it is roughly the shape of a rectangle. The ratio of the width to the length is approximately 3:2

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9. An annual event called the Tribal Journeys or Canoe Journeys is held during the summer months. One community hosts and paddlers from other communities make the canoe journey for 5 days of celebrations. A group from the Haida nation traveled by canoe to attend a celebration. Another group traveled 5 times as far as the Haida paddlers. The total distance traveled by the two groups was 600 km. How far did the people from Haida nation travel by canoe?

Extend and Explore

An even number can be represented by 2x, where x is an integer. An odd number can be represented by 2x+1, where x is an integer. Is the sum of 2 odd numbers always even? Show your thinking.

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5 Chapter Review 1. Unscramble the words below and write one of these symbols next to each word.

a. mus

b. crudtop

c. friendefec

d. nitoquet

e. laque

f. lotta

g. heards

h. slos

i. dreacese

j. mites

k. scareeni

2. Solve the following

a.

b.

c.

3. Change or create an equation of each type listed below and solve it.

a. 2 Step Integer

b. 2 Step Fractions

c. 2 Step Decimals

d. 2 Step Brackets

e. 2 Step Variables on Both Sides

f. Combination

4. Rearrange these steps in the correct order. a. 3x=9 b. 5x-2x=2x-2x+9 c. 5x+3=2x+12 d. x=3 e. 5x+3-3=2x+12-3 ©Edvantage Interactive 2025

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5. Solve these linear equations

a.

e.

b.

c.

f.

g.

h.

j.

k.

l.

n.

o.

d.

i.

m.

p.

q.

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r.

t.

s.

u.

6. Solve 3(x + 4) = 12 in two ways.

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7. The following equations have no solution OR infinitely many solutions. Simplify. Write no solution, or the symbol for infinity below each one. a. 2x + 9 = x + x + 4 + 5

b. 3x + 1 = 4x + 2 – x + 1

c. 5(x – 2) = x + 4(x - 3) + 2

e. 0.5(x + 6) + 2(0.25x + 2)=2x + 7 – x

d. 4(x – 3) + 13 – 2x = 2(x + 1)

8. Rewrite the formulas in terms of the variable indicated in the chart below. a. P = 2(l + w), Solve for w, width.

b. V =

d. F=ma, Solve for acceleration , a.

e. I=Prt, Solve for time, t.

Solve for h, height.

c. s =

, Solve for time, t.

f. V=lwh , Solve for height, h.

9. Two consecutive integers add to 23. Write an equation to represent this situation and then solve it.

10. Rewrite the formula of the area of a trapezoid in terms of the height, h.

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11. Find x and the perimeter of the isosceles triangle.

12. Amir bought a shirt and jeans for 72 dollars. If the cost of the shirt is 12 dollars more than half the price of the jeans, what does each item cost?

13. In a horse race Heartache is running at 15m/s and True Love is running at 17m/s. If Heartache had a 10 meter head start, how long will it take in seconds for True Love and Heartache to be tied?

14. A group of university students bought a winning lottery ticket valued at one million dollars. There are 4 students in the group. If the amount is shared evenly what does each of them receive? If they had agreed that one of them should receive twice the amount the other 3 do, how would they split the money?

15. Jen and Soyoung solved this equation. 3(x + 2) = x + 2(x + 3). Jen got x = 8 and Soyoung got x = –1. Check the two solutions and explain the result.

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16. The sum of two consecutive even numbers is 154. What are the two numbers? (Hint: an even number can be represented as 2x, where x is an integer).

17. Three consecutive numbers add to 36. Write an equation to represent this and solve it. Find the 3 numbers.

Extend and Explore 1. On an analog clock, between 1 pm and 2 pm, when does the minute hand overlap the hour hand?

2. Calculate the following a.

b.

c.

2d. What do you notice? How can you explain this? Hint: Let x = the single digit number

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How Did It Go? Part 1: Reflecting on Chapter 5 - Check all that apply to you ☐ The content of this chapter is challenging. ☐ I am starting to understand linear equations. ☐ I can solve simple equations. ☐ I find word problems challenging. ☐ I need more time to practice before the test. ☐ I understand and can use the distributive property. ☐ I feel more confident solving equations than last year. ☐ I can solve 2-step equations. ☐ I understand how to preserve equality in an equation. ☐ I have a few questions before the test. ☐ I am confident about the content of this chapter. ☐ I can help others solve linear equations. ☐ I can use many different strategies to solve linear equations. ☐ I feel confident applying skills to word problems. ☐ I am ready to write the test.

Part 2: Reflecting on solving equations

In this chapter you learned a variety of different ways to solve linear equations. A set of skills you will continue to use in your life in a variety of different ways. Below are three ways to solve the same equation. Which way do you prefer and why?

a.

c.

b.

Here are two more ways to solve the same equation. Which method do you prefer and why?

3.1(x – 7) = 6.2

Distributive property

3.1x – 21.7 = 6.2

Add 21.7

3.1x = 27.9

Divide by 3.1

x = –5

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3.1(x – 7) = 6.2

Divide by 3.1 Add 7

x–7=2 x=9

Chapter 5 Linear Equations (Pilot)

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Vocabulary List for Linear Equations B

Balance – Keeping both sides of an equation equal; the idea that what is done to one side must be done to the other.

C

Check – Substituting the solution back into the original equation to confirm its correctness.

Coefficient – The number that multiplies a variable in an expression (for example, 4 in 4x).

Constant – A fixed number that does not change, in contrast to a variable. The 3 in 5x+3.

Consecutive – Describes items (such as numbers or like terms) that follow one after another in sequence. Like 4, 5, 6 or 12, 13, 14, 15.

D

Distributive Property – The property that states a(b + c) = ab + ac, meaning you multiply a number by each term inside the

parentheses.

E

Equation – A mathematical statement representing the equality of two expressions (for example, 2x + 3 = 11).

Evaluate – To calculate the numerical value of an expression.

Equality – The relationship between two expressions that have the same value.

Equivalent equation – An equation that, while it may look different, has the same solution set as the original equation. 2x = 6 has the

same solution as 3x = 9.

Expression – A combination of numbers, variables, and operations that represents a value (for example, 3x + 5).

G

Greater than – A comparison symbol (>) showing that one value is larger than another. 4 > 3

I

Inverse – A number or operation that reverses the effect of another; for example, the inverse of addition is subtraction.

Inequality – A mathematical statement that compares two quantities using symbols such as <, >, ≤, or ≥.

L

Linear equation – An equation that graphically represents a straight line and has a degree of one. The exponents of the 2 variables are

both 1. Example: y = 9x – 2

Less than – A comparison symbol (<) indicating that one value is smaller than another. 3 < 4 Like terms – Terms in an expression that contain the same variable(s) raised to the same power and can be combined.

N

No solution – Describes an equation that has no value for the variable which will make the equation true.

P

Prove – To provide logical reasoning or mathematical evidence that confirms a statement is true.

R

Reciprocal – For any nonzero number a, its reciprocal is

. Examples: 2 and ½ , -1 and -1, ¾ and

.

S

Simplify – To reduce an expression to its simplest form by combining like terms and/or performing operations.

Solve – To find the value(s) of the variable(s) that make an equation true.

Solution – The value(s) obtained when an equation is solved.

V

Verify – To check and confirm that a proposed solution satisfies the original equation.

374 Chapter 5 Linear Equations (Pilot)

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Failsafe Vocabulary Check The Frayer Model is a graphic organizer that can help you learn and understand vocabulary in a structured way.

The example below shows how to fill out the Frayer Model. In the blank Frayer Model pick one of the three terms and fill it using the example as a guide. ☐ Preservation of Equality ☐ Distributive Property ☐ Simplify a linear expression

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Chapter 5 Linear Equations (Pilot)

375

Chapter 5 Answer Key 5.1 1. a, b, d, f, i 2. a. Substituting b. one c. one d. straight line e. equality 3. a. Circle is 3, hexagon is 4.5 b. Circle is 11, hexagon is 15 4. a. −4x + 2 13 7 b. x − 2 3 c. 20x − 21 d. 0

5. a. 2x + 2 = 10 x b. − 2 = 6 2 c. −6x + 4 = −10

2. a, c, e 3. a. 4 b. 4 c. 2 d. 2 e. 5 f. −3 g. 1

h. 16 i. −100 3 j. − 2 1 k.

3 l. 99 m. −12 n. 80

o. 8000 4. Sample Answers: 2x + 2 = 1

d. x + 2 = 10

e. −0.45x + 0.5 = 5 6. a. Yes b. no c. yes d. yes 7. a. Infinite

5.3 1. a. 7 5 7 c. −9

b. −

d. −6

b. none

e. −6

c. one

g. 20

d. one e. infinite f. Left and right side will be identical g. general form: ax + b = ax + c, where b ̸= c and a ̸= zero

5.2

f. −18 h. 13 17 i.

4 j. 2 k. −30

l. −175 2. a. Sample Answer:

1. a. −7

1 + 10 = 9 3 −1 + 10 = 9

b. 3

−3

c. 31 d. 24 e. −5

f. −99 g. 9

h. 1 i. −40

j. −36

k. −110 l. 0.5

376 Chapter 5 Linear Equations (Pilot)

−0.5x + 0.75 = 1

9=9

A 3. a. w = l P + 2l b. =w 2 C c. r = 2π 2A d. b = h Extend and Explore: Hint: let one of the numbers be 2a + 1.

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Check Your Understanding 5.1–5.3

2. a. −1 b. 3 4 c.

3 1 d.

2 e. 10

1. a, b, d, e, g, i 2. 27 cm 3. Sample Answer: 3x − 2 = 1 add 2 to both sides 3x = 3 divide both sides by 3 x=1

f. 20 g. 100 h. 2

4. Sample Answers: 5x = −25 or 0.5x + 2.5 = 0 5.

i. −5

j. 0.04 5 k. − 3 l. 0 2 m.

5 n. 30

9

x x x1 1 1

111111 =

5 5 b. c. −100 d. correct How is it going? a.

3 2 1 g. correct h. 4 i. correct correct f.

2

5.4 1. a. −2x + 10 2 b. x + 4 3 5 c. −10x + 2 d. 4x − 6 2. a. 1 b. −1 c. 9

d. 5 e. 6 11 f.

2 g. 10 h. 4 i. 4.2 j. 2.6

e.

o. 12 15 p. − 14 108 q.

7 r. 5 s. 7 t. 10

3. a. 4 3 b.

2 4. a. 21, x ̸= −1 b. 8, x ̸= 1

5. x = 8, perimeter = 24

5.6 1. a. x + 3 b. 3x c. 5x d. 21.5x e. 23 − 1.25x f. 2x + 25

g. 4x + 10

k. 0.1 1 l. − 3

2. a. +

5.5

b. −

1. Sample Answer: 5x + 2 = x + 6 remove one x tile and 2 one tiles from both sides 4x = 4 each x tile can be matched with a one tile x=1

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h. 2x2

c. +

d. − e. ÷ f. ×

g. −

Chapter 5 Linear Equations (Pilot)

377

h. ÷

11. x + 4, P = 44

5. a. 6

i. +

12. Jeans: $40

b. −9 c. 4

j. ×

1 2 14 e.

3 f. −90

k. ×

d. −

l. +

m. − n. −

Shirt: $32

13. 5 seconds 14. 3 get $200,000, one gets $400,000 15. All solutions work 16. 76 and 78 17. 11, 12, 13

g. 99

o. ÷

h. 90 7 i.

2 j. 1

p. − q. + r. −

s. × t. +

3. Step 3: Solve the equation. Step 5: Check your solution.

4. 5 and 35 5. 6 cm 6. 17 and 18 7. a. $26.8 b. C = 2.28x + 4 c. $232 d. 42.1 km 8. Area ≈ 1.1 km2 9. 100 km

k. −3 51 l.

20 13 m.

11 n. 3 o. 27 p. 5 q. −1 r. 21 s. 2 39 t.

29 u. 12 6.

3x + 12 = 12

OR

x + 4 = 4 (divide both sides by 3)

3x = 0

Chapter Review 1. a. Sum + b. Product ×

c. Difference − d. Quotient ÷ e. Equal = f. Total + g. Shared ÷ h. Loss − i. Decrease − j. Times × k. Increase + 2. a. Two circles b. Two circles c. Four rectangles 3. Answers will vary 4.

5x + 3 = 2x + 12 5x + 3 − 3 = 2x + 12 − 3

⇒

x=0

x=0 7. a. ∞

b. no solution c. ∞ d. no solution e. ∞ 8. a. w =

P −l 2

V πr2 d c. t = s F d. a = m I e. t = Pr V f. h = lw 9. 11 and 12 10. h = 2A a+b b. h =

5x − 2x = 2x − 2x + 9 3x = 9 x=3

378 Chapter 5 Linear Equations (Pilot)

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