# Chapter 5: Systems of Equations and Inequalities

## Introduction

Here you'll learn all about *systems* in algebra. You will learn about systems of equations and how to solve them graphically and algebraically. You will also learn about systems of inequalities and how to solve them graphically and use them to solve real-world problems involving maximizing and minimizing.

## Chapter Outline

- 5.1. Graphical Solutions to Systems of Equations
- 5.2. Substitution Method for Systems of Equations
- 5.3. Elimination Method for Systems of Equations
- 5.4. Applications of Systems of Equations
- 5.5. Graphs of Linear Inequalities
- 5.6. Graphical Solutions to Systems of Inequalities
- 5.7. Applications of Systems of Inequalities

### Chapter Summary

## Summary

You learned that a *system* of equations or inequalities means more than one equation or inequality.

To solve a system of equations you can graph the system and look for the point of intersection or use one of two algebraic methods (substitution or elimination). Sometimes a system of equations has no solution because it the two lines are parallel. Other times the system has an infinite number of solutions because the lines coincide.

A linear inequality appears as a region in the Cartesian plane. To solve a system of linear inequalities, graph both and look for where their solution regions overlap. This region is often called the feasible region. Systems of linear inequalities can help you solve problems where you have multiple constraints on different variables and you are trying to figure out how to maximize or minimize something (like profit or cost). The maximum or minimum values will occur at one of the vertices of the feasible region according to the Vertex Theorem. These types of problems are sometimes referred to as linear programming problems.