### Unit 9: Systems of Equations and Inequalities

Systems of equations are handy when exploring the relationship between two or more things. For example, the profit a business makes is dependent on both the cost to produce goods or services and the money it brings in from selling those goods or services. We will introduce methods for solving systems and interpreting the solution when you find one. We will also introduce techniques for graphing systems of inequalties that include combinations of linear and quadratic equations. You will use the skills you have learned from working with linear and quadratic functions and from graphing and solving linear inequalities to master the topics in this unit.

**Completing this unit should take you approximately 3 hours.**

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

- solve systems of linear equations in two and three variables using graphs, substitution, and addition;
- categorize solutions of systems of linear equations;
- solve systems of nonlinear equations using substitution and elimination;
- graph single and systems of nonlinear inequalities; and
- construct an augmented matrix from a system of linear equations.

### 9.1: Systems of Linear Equations in Two Variables

This section will introduce the basic characteristics of a system of linear equations.

Now that you know some methods for solving linear systems, it is vital to understand the solution. Linear systems have either one solution, no solutions, or infinitely many solutions, which we can determine by analyzing the solution.

We continue our study of systems of linear equations by learning different methods for solving them. You will learn to use graphs, substitution, and the addition method to solve linear systems.

Systems are great for modeling the relationship between behaviors we can observe. In this section, you will explore the profits of a fictional company. You will combine your knowledge of graphing linear functions, solving systems of linear equations, and analyzing the results to understand when a company will break even.

### 9.2: Systems of Linear Equations in Three Variables

We will further our study of systems of linear equations in this section by adding a variable. A linear function with three variables is similar to one with two, but the algebra used to solve them can get complex quickly. In this section, you will verify whether an ordered triple is a solution to a linear system with three variables and learn how to use elimination to find a solution.

In this section, you will learn how to analyze the solution to a system with three variables.

- Finally, you will learn how to set up a system as a matrix.

### 9.3: Systems of Non-Linear Equations

Now, we will explore graphical and algebraic methods for finding the solution to a system of equations containing combinations of different equations, including linear, quadratic, and circles.

Finally, we will explore non-linear inequalities. You will graph a quadratic inequality and a system of non-linear inequalities. Graphing non-linear systems of inequalities is similar to graphing inequalities, but you will need to take additional steps to determine the region where there are solutions to both inequalities.