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

In previous units, we learned that linear equations with one variable generally have one solution. However, linear equations with two variables have an infinite number of solutions. If we pair two linear equations together, we can solve for the pair of numbers that would solve both equations. This is called a system of linear equations. In this unit, we will learn how to solve systems of linear equations.

**Completing this unit should take you approximately 3 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: Solution of a System of Linear Equations

First, we'll look at how to determine whether a given ordered pair of numbers is a solution to a system of linear equations. We do this using the techniques you already learned in this course.

Watch this video to see examples of how we apply algebra techniques to testing solutions to systems of equations.

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

There are several methods for solving systems of linear equations. The first method we explore is using graphing to solve systems of linear equations.

Read this page, starting at the section on solving a system of linear equations by graphing. After you read, complete examples 5.3 through 5.7 and check your answers. You should use graph paper for these examples, since you will need to identify the intersection point between two lines.

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

Sometimes it is not practical to solve a system of equations by graphing. The substitution method is a non-graphical method for solving systems of equations, and it uses algebra techniques you already know.

For the substitution method, we solve one of the equations for one of the variables in terms of the other variable. Then, we substitute that expression into the second equation to solve for the variable. Then, once you have solved for one variable, you can solve for the other.

Watch this video to see examples of using the substitution method.

After you watch, complete this assessment and check your answers.

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

The elimination method is another non-graphical method for solving systems of linear equations. For this method, we use the addition property of equality. This states that if we add the same amount to both sides of an equation, the two sides of the equation will still be equal.

Watch these videos for examples of how to solve systems of equations with the elimination method using a few different methods.

After you watch, complete this assessment and check your answers.

### 6.5: Choosing a Strategy for Solving Systems of Linear Equations

It can be difficult to decide which of the three methods (graphing, substitution, or elimination) is best for solving a given system of linear equations.

Read this article and watch the videos. The beginning provides a nice summary of the main methods for solving systems of equations, and when you should use each one. The rest of the article offers a good overview of the different methods with a discussion of why each method was chosen for a given problem.

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

Linear systems of equations appear in many real-world applications. Systems of equations are often used to conduct comparisons, such as when comparing prices. It can be difficult to translate a world problem into equations and to determine the appropriate variables.

Watch this video for examples of common real-world word problems.

Read this article and watch the video. The article describes examples in which systems of equations can be used to solve real-world quantities. After you review, complete problems 1 to 4 and check your answers.

### 6.7: Graphing Systems of Linear Inequalities

We can use graphical methods to solve systems of linear inequalities, much like we graphed single inequalities earlier in this course.

Read this page until the section Solve Applications of Linear Inequalities. Pay attention to the steps needed to solve a system of linear inequalities.

After you read, complete examples 5.51 through 5.56 and check your answers.

### 6.8: Applications of Systems of Linear Inequalities

Linear inequalities appear in many real-world applications.

Read the section on solving applications of linear inequalities. Review the examples to see how the authors use systems of linear inequalities to solve problems involving cost of objects and budgets.

Linear programming uses a system of several linear inequalities to analyze a real-world situation. Linear programming takes a set of inequalities and determines the optimal or best solution for the given set of conditions. Read this article to learn about this application of linear inequalities.