After completing this unit you should be able to:
Algebra is a branch of mathematics that uses mathematical statements to describe relationships between things that vary over time. These variables include things like the relationship between supply of an object and its price. When we use a mathematical statement to describe a relationship, we often use letters to represent the quantity that varies, since it is not a fixed amount. These letters and symbols are referred to as variables. (See the Appendix One for a brief review of constants and variables.)
The mathematical statements that describe relationships are expressed using algebraic terms, expressions, or equations (mathematical statements containing letters or symbols to represent numbers). Before we use algebra to find information about these kinds of relationships, it is important to first cover some basic terminology. In this unit we will first define terms, expressions, and equations. In the remaining units in this book we will review how to work with algebraic expressions, solve equations, and how to construct algebraic equations that describe a relationship. We will also introduce the notation used in algebra as we move through this unit.
The basic unit of an algebraic expression is a term. In general, a term is either a number or a product of a number and one or more variables. Below is the term 3ax.
The numerical part of the term, or the number factor of the term, is what we refer to as the numerical coefficient. This numerical coefficient will take on the sign of the operation in front of it. The term above contains a numerical coefficient, which includes the arithmetic sign, and a variable or variables. In this case the numerical coefficient is 3 and the variables in the term are a and x. Terms such as xz may not appear to have a numerical coefficient, but they do. The numerical coefficient is 1, which is assumed.
An expression is a meaningful collection of numbers, variables, and signs, positive or negative, of operations that must make mathematical and logical sense. Expressions:
An example of an expression is:
In an expression, the signs of operation separate it into terms. The sign also becomes part of the term that it follows. The expression above contains two terms, the first term is 3ax and the second term is +11wx2y. The addition sign separates the two terms. For example, in the expression given above the plus sign (+) separates the 3ax from 11wx2y and is also part of the second term. Terms that do not have a sign listed in front of them are understood to be positive.
Below are several examples that are not expressions.
|x + y||This statement tells us "x plus multiplied by y". This does not make mathematical or logical sense. This collection of symbols is nonsense.|
|y = 2x 1||This statement is not an expression because expressions are not allowed to contain the equal sign.|
NOTE: The operation of multiplication can be represented by using a x, , or by placing items to be multiplied in parentheses, brackets or braces, or in the case of variables, just written next to one another. The statements a xb, a b, (a)( b), and ab are equivalent. In this booklet we will use the latter three representations.
An equation is a mathematical statement that two expressions are equal. The following three statements are equations:
|4 + 5 = 9||x 35 = 56k2 + 3||x + 3 = 15|
The first equation, 4 + 5 = 9, contains only numbers; the other two, however, also contain variables. All three contain two expressions separated by an equal sign:
When an equation contains variables you will often have to
solve for one of those variables. Using equations to solve for
a variable will be discussed later in this booklet.