Introduction

There are three common techniques for solving a system of equations: graphing, substitution, and elimination. These techniques are presented in the first three sections of this chapter. You will be introduced to the definition of a linear system, and learn how to check your solutions. The fourth section looks at special linear systems, where there is no solution because the lines are parallel or there are infinitely many solutions because the lines coincide. In Section 5.5, you will learn that solving a system by graphing can be used to solve an equation with variables on both sides. This is actually a technique that you will use extensively in future mathematics courses. The last lessons combine prior skills to allow you to graph systems of linear inequalities.

Lesson Objectives

After completing this lesson, you should be able to:

• estimate graphically the solutions to systems of two linear equations with two variables in real-world problems;
• write linear inequalities in two variables given a table of values, a graph, and a verbal description;
• graph the solution set of linear inequalities in two variables on the coordinate plane;
• graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist;
• estimate graphically the solutions to systems of two linear equations with two variables in real-world problems;
• graph the solution set of systems of two linear inequalities in two variables on the coordinate plane;
• solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides; and
• estimate graphically the solutions to systems of two linear equations with two variables in real-world problems.

Materials Needed

• Textbook: Big Ideas Math, Algebra 1
• Spiral notebook for notes
• Graphing calculator

How to Proceed

1. Read the Introduction and Objectives for this lesson. They will give you an idea of the major concepts you need to watch for as you read, as well as the kinds of skills you will be expected to have at the end of the lesson.
   
   
2. Survey pages 216-273 in your textbook. (Skim and/or look at the title, the introduction, the headings and subheadings, the graphics, and the important terms. The Chapter Review on pages 268-270 is a good place to see an overview of what you’ll be learning in Lesson Five.)
   
   
3. Read the assigned pages in your textbook for each section of this lesson. The assigned textbook readings are given at the beginning of each Section Summary.
   
   
4. Take notes on the assigned pages for each section. Pay attention to the list of key words in each section. The key words are important and should be remembered for future reference.
   
   
5. Read the Section Summaries on the course homepage. They will help you to understand what you read in the textbook. Read and study all examples given in the textbook.
   
   
6. Complete the Section Assignments neatly on paper to be scanned and saved as a PDF (see "Textbooks and Materials" in the Course Introduction). Show every step and box your answers as shown in the Sample Lesson Submission in the Resources section of this course. When you have completed all of the assignments, save them as a single PDF and upload your work using the Lesson Five: Assignment Upload.
   
   
7. Review your notes briefly every day until you complete the course. After you have finished this lesson, you may proceed to Lesson Six.

Before You Begin

Before you begin this chapter, look through the whole chapter to become acquainted with what will be covered. Remember that each chapter has two sections labeled "What Did You Learn?". This part of the book lists the core vocabulary and core concepts. Make sure that you have all of the core vocabulary and core concepts written in your notes and defined. You may also want to add some examples.

Now look at pages 217 and 218. Work through Maintaining Mathematical Proficiency and Mathematical Thinking. This will serve as a review of some skills and topics that you should know and will also need for this chapter. The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. The rise in the slope is the change in y (vertical change) and the run is the change in x (horizontal change). You have to "rise" before you can "run."

Also remember when you multiply or divide each side of the inequality by a negative number, you must reverse the inequality symbol.

Check your answers for Maintaining Mathematical Proficiency (page 217) in the Selected Answers section of your textbook, beginning on page A1.

Page 218 is a graphing calculator lesson on finding the point of intersection of a system.