Module 7 Introduction

Module 7: Systems of Linear Equations

Module 7 Introduction

Engaging in a hobby has many advantages. Recreational pursuits are enjoyable and can reduce stress. In addition, becoming active in a hobby can prolong life and keep disease at bay. Studies have shown that hobbies that engage the brain can delay the onset of Alzheimer’s disease and dementia in senior citizens. Brain-friendly hobbies include puzzles, card games, genealogy, and even blogging! You don’t have to wait until you have grandchildren to enjoy the benefits of pursuing a hobby, however. By engaging in your favourite pastimes, you can not only improve your health but also your manual dexterity, your focus and concentration, and your creativity.

In this module you will learn about systems of linear equations in the context of recreational pursuits such as photography, volunteerism, and speed stacking. You will build on the knowledge that you have gained in previous modules related to linear functions and their properties. You will learn how to model a situation using a linear system. You will also learn both graphing and algebraic strategies for determining and verifying solutions to these systems.

Besides health benefits, recreational pursuits can also have social benefits. Fishing, dancing, staging a production, and other pursuits can help to connect you with other people who possess common interests and passions. You can exchange ideas and share methods.

Whatever your hobbies are, sharing what you do with someone else and perhaps teaching the basics can be even more rewarding than doing hobbies in isolation. In this module you will continue to work with your peers as you learn and practice using strategies to solve linear systems. Together with partners, you will model situations with linear systems. You will also consider the advantages and disadvantages of different solution strategies. By doing so, you can increase your own understanding of the concepts presented and can help others to better understand them as well.

In this module you will investigate the following questions:

• What are systems of equations, and how can they be solved by graphical and algebraic methods?

• How can you use systems of linear equations to model and solve problems?

• Why can a system of linear equations have different numbers of solutions?

To investigate these questions, you will focus on the following lessons and consider these lesson questions.

Lesson	Title	Lesson Questions
1	Solving Linear Systems by Graphing	What is meant by the solution of a system of equations? How is solving a mathematical problem different from verifying a solution to the problem?
2	Modelling Linear Systems	How do you decide what parts of a problem should be represented by variables? How do you organize the information you need to create a linear system?
3	Solving Linear Systems by Substitution	How do you decide which variable to isolate? Does it matter? Why would you change a mathematical equation into an equivalent expression?
4	Solving Linear Systems by Elimination	In using the elimination method, how do you know whether equations need to be multiplied, added, or subtracted? How does adding or subtracting the equations of a linear system affect the solution to the system?
5	Number of Solutions of a Linear System	Why can linear systems have different numbers of solutions? How do the equations of a linear system indicate the number of solutions of that system?
6	Solving Problems with Linear Systems	Why is it necessary to learn several methods for solving a problem? How can you recognize which strategy is most appropriate for solving a given system of equations?

 

You will explore the concepts of this lesson in fun and engaging ways. In addition to working through related examples in your textbook, you will participate in group discussions and investigations. You will have opportunities to develop approaches to problem solving by sharing ideas with your peers. The lessons in this module include several applets as well as videos that will provide relevant contexts for the math that you will learn. As usual, you will engage in inquiry-based activities that will spark your interest and lead you to insights into mathematics. You will continue to solve reality-based problems, which will help you to see how these concepts are applied in day-to-day living.

Some of the most interesting work you will do in this module is found in the Connections section. You will undertake a unit project related to vacation planning. As well, you will have the opportunity to extend the concepts in the lesson by looking at additional material that relates to the strategies and ideas underlying systems of equations and their graphs and solutions. These include exploring 3-D graphs, online calculators, and linear inequalities.

 

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