### Unit 6: Systems of Linear Equations and Inequalities

You have seen in Units 2 and 5 that linear equations in one variable usually have 1 solution and linear equations in two variables have infinitely many solutions. What would happen if two linear equations in two variables had to be solved together? That would mean that a pair of numbers would have to satisfy both equations at the same time. This pair of numbers would be the solution of a system of linear equations. Some of the mixture problems from Unit 3 could be solved by setting up a system of linear equations, because they involve two given relationships between two variables. In this unit, you will learn three different methods of solving systems of linear equations and use them to solve a variety of world problems. You will also find solutions of systems of linear inequalities in two variables and to apply them to real-life situations. For example, if you want to determine the price for and amount of two types of candy for a party, you have the constraints of the total amount of candy needed (GREATER than a given amount) and the amount of money you can spend (LESS than the amount you have). The quantities of two types of candy you buy have to satisfy both constraints. In this unit, you will learn how to identify all the possibilities for similar problems.

**Completing this unit should take you approximately 14 hours.**

Upon successful completion of this unit, you will be able to:

- determine the number of solutions of a given system of linear equations;
- classify systems of linear equations according to the number of solutions;
- solve systems of linear equations using graphing, substitution, or elimination;
- locate on a coordinate plane all solutions of a given system of inequalities;
- create systems of equations and use them to solve world problems; and
- use systems of inequalities to model word problems, and interpret their solutions in the context of the problem.

### 6.1: Solving Systems of Linear Equations

### 6.1.1: Solution of a System of Linear Equations

Read this page. You can skim through the beginning of the article, as it reviews the concepts you already know. Focus on the definition of a system of equations and its solution as well as the discussion of how many solutions a system of equations can have.

Watch this video and take notes. This video demonstrates how to find out whether an ordered pair of numbers is a solution of a given system of equations.

This article contains practice problems for the concepts you have just learned. Work through Example A, and complete practice problems 9-12. Once you have completed the practice problems, check your answers against the answer key.

### 6.1.2: Solving Systems of Linear Equations by Graphing

Watch this video and take notes. This video demonstrates the method of solving systems of linear equations in two variables by graphing.

Complete this exercise. Graph the equations in the system by using intercepts or slope-intercept method, whichever is more appropriate. Then, determine the coordinates of the point of intersection.

- This article contains more practice problems for solving systems of equations by graphing. Work through Example B and the example in the Guided Practice section, then complete practice problems 13-22. Watch the "Solving Linear Systems by Graphing" video embedded in the text if you need help. Once you have completed the practice problems, check your answers against the answer key.

### 6.1.3: Solving Systems of Linear Equations Using the Substitution Method

Watch this video and take notes. In this video, a system of equation is solved using substitution.

Read the introduction and section titled "Example (a system with a unique solution)." This reading provides a step-by-step method of solving any system of linear equations in two variables by substitution. However, as you will see in the rest of the subunit 6.1, some systems are better suited for solving by substitution than others.

Complete this exercise. Solve the given system of equations by substitution, then fill in your values of x and y in the appropriate tabs on the right side of the page, and select "Check Answer."

### 6.1.4: Solving Systems of Linear Equations Using the Elimination Method

Complete this exercise. It gives you an opportunity to practice the examples similar to the one you saw in the Khan Academy: video. Fill in your values for x and y in the appropriate places.

Read this page. This reading explains the concepts behind the elimination method and provides different examples of systems that are convenient to solve by elimination.

Watch these videos and take notes. In the first video, you will see an example of a system that is ideal for solving by elimination. The system discussed in the second video can also be solved by elimination, but an extra step is required before one of the variables can be eliminated. In the third example, both equations need to be replaced by equivalent in order to eliminate one of the variables.

Complete this exercise. Solve the given system of equations by elimination.

### 6.1.5: Strategy for Solving Systems of Linear Equations: Choosing a Method

You have seen three different methods for solving systems of linear equations in two variables. In this subunit, you will focus on choosing the most efficient method of solving a particular system.

Read this article and watch the videos embedded in the text. Focus on the summary of the three methods in the "Guidance" section, which highlights when each method is most appropriate. After reading, complete practice problems 1-6. Try to choose the most efficient method for solving each system. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise. It provides more practice solving various systems of equations. While Khan Academy suggests a method for solving each system, you might want to think whether this method is most appropriate and possibly choose another one.

### 6.1.6: Classifying Systems by the Number of Solutions

Recall that systems of linear equations in two variables can have one, none, or infinitely many solutions. In this subunit, you will explore the last two cases in greater detail.

Read this page. Also, make sure to watch the "Special Types of Linear Systems" video embedded in the text. Then, complete practice problems 7-24. Keep in mind that you do not have to actually solve a system to classify it as inconsistent or dependent; it is enough to show that slopes of both lines are the same and the y- intercepts are different (in case of inconsistent system) or that the equations in the system are equivalent (in case of dependent system). Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise. Use your skills acquired in the previous assignment to determine the number of solutions of each system.

### 6.2: Solving Word Problems by Using Systems of Equations

Read this page. This reading reviews the step-by-step process for solving word problems: finding information in the text of the problem, translating verbal statements into mathematical statements, and solving the resultant system of equations. After reading and working through the example, click on "new problem" at the bottom of the page, and solve seven problems. Alternatively, you may generate a worksheet of seven problems by clicking on the button at the bottom of the page titled "Click Here for a Randomly Generated Worksheet and Answers." The solutions will be provided at the end of the worksheet.

Read the entire text, and watch the videos embedded in the text. (You may skip Example B, as you have already seen it in subunit 6.1.6.) This reading discusses a variety of real-world situations which can be described by systems of linear equations. Note that some of these systems can be inconsistent or dependent, and you will learn to interpret what this means in the context of the given situation. After reading, complete practice problems 1-10. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise. It contains more word problems that can be solved by setting up and solving a system of equations.

### 6.3: Systems of Linear Inequalities

### 6.3.1: Graphing Systems of Linear Inequalities

Read the article until the section "Linear Programming - Real-World Systems of Linear Inequalities." Scroll down to the section "Practice Problems," and watch "Systems of Linear Inequalities" video embedded in the text. Then, complete practice problems 6-16. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise. Drag the points to move each line to correct position and select appropriate shading and line option (solid or dashed) to represent each inequality.

### 6.3.2: Applications of Systems of Linear Inequalities

Linear programming is a process of analyzing a real-world situation by using a system of several linear inequalities. Read this article and watch the videos embedded in the text. Then, complete practice problems 1-11. Once you have completed the practice problems, check your answers against the answer key.

Scroll down to the section "System of Equations and Inequalities; Counting Methods: Review," and complete the odd-numbered problems for 3-41, but do not complete problem 37. This set of practice problems will allow you to assess your mastery of the concepts in Unit 6. Once you have completed the practice problems, check your answers against the answer key.