8: Algebraic Thinking
- Page ID
- 155314
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- 8-1 Variables
- 8-2 Equals Relation and Equations
- 8-3 Functions
- 8-4 Equations in a Cartesian Coordinate System
- * Module B: Using Real Numbers in Equations–online
- Chapter 8 Review
- 8.1: The Language of Algebra
- In algebra, we use letters of the alphabet to represent variables. The letters most commonly used for variables are x,y,a,b, and c .
- 8.4: Solving Equations with Variables and Constants on Both Sides
- In all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we will learn to solve equations in which the variable terms, or constant terms, or both are on both sides of the equation.
- 8.9: Sequences and Their Notations
- One way to describe an ordered list of numbers is as a sequence. A sequence is a function whose domain is a subset of the counting numbers. Listing all of the terms for a sequence can be cumbersome. For example, finding the number of hits on the website at the end of the month would require listing out as many as 31 terms. A more efficient way to determine a specific term is by writing a formula to define the sequence.
- 8.10: Arithmetic Sequences
- In this section, we will consider specific kinds of sequences that will allow us to calculate depreciation. For example, companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.
- 8.11: Series and Their Notations
- The sum of the terms of a sequence is called a series. Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma, ∑, to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. In this section, we will learn how to use series to address annuity problems.
- 8.16: Systems of Equations - The Substitution Method
- Solving a system by graphing has its limitations. We rarely use graphing to solve systems. Instead, we use an algebraic approach. There are two approaches and the first approach is called substitution. We build the concepts of substitution through several examples and then conclude with a general four-step process to solve problems using this method.
- 8.17: System of Equations - The Addition Method
- The substitution method is often used for solving systems in various areas of algebra. However, substitution can get quite involved, especially if there are fractions because this only allows more room for error. Hence, we need an even more sophisticated way for solving systems in general. We call this method the addition method, also called the elimination method. We will build the concept in the following examples, then define a four-step process we can use to solve by elimination.
- 8.18: Applications with systems of equations
- We saw these types of examples in a previous chapter, but with one variable. In this section, we review the same types of applications, but solving in a more sophisticated way using systems of equations. Once we set up the system, we can solve using any method we choose. However, setting up the system may be the challenge, but as long as we follow the method we used before, we will be fine. We use tables to organize the parameters.