Expressions and Equations
Introduction and Summary
This chapter is a bridge between pre-algebra and algebra. It starts with a review of the material learned at the end of pre-algebra, and leads right into working with variables. It explains how to simplify expressions and solve equations; in addition, it teaches the formal properties which allow us to solve equations.
Much of the information covered in the first section can be found in the last chapter of the Pre-Algebra SparkNote. This section discusses variables and how they are used. It outlines the steps for translating word statements into algebraic expressions and algebraic equations.
The second section explains what a term is and discusses how to simplify expressions by combining like terms. Simplifying expressions makes expressions a lot easier to think about and work with, and a most algebra problems require this skill.
The next section is a first look at solving equations. It discusses the meaning of a solution and it explains how to find a solution set from a replacement set. Using the skills learned in the first two sections, the reader will begin to solve equations.
The fourth section details a different method of solving equations-- solving by inverse operations. The section also focuses on solving equations by reversing multiple operations.
In the fifth section, we learn the properties that allow us to perform inverse operations--the properties of equality and the properties of operations and identities. These properties, which all real numbers have, are important to know both for solving equations and for future proofs involving equations. They are interesting to learn because they reveal the nature of the numbers we work with in all of mathematics.
Using these properties, the final section shows how to solve an equation that contains variables on both sides. Many of the equations we will encounter in algebra are of this type, and it is essential to know how to solve them.
Overall, this chapter is the first in-depth look at what we will be doing in the rest of algebra. It is impossible to understand the material covered in future chapters without a thorough understanding of this chapter. In fact, this chapter is essential to understanding all future mathematics. Equations play a huge role in Geometry, Algebra II, and Calculus, as well as in higher math, so it is crucial to know how to work with them and solve them.