Algebra, usually introduced during the middle or early high school years, is often students’ first encounter with reasoning abstractly and symbolically. This branch of mathematics entails a sophisticated set of rules applied to a variety of situations. To get started, students need to become familiar with the basic rules and will use these as building blocks as their course progresses.
The Concept of a Variable
At the heart of algebra lies the use of alphabetic letters to represent numbers. These letters are known as variables, and they stand for numbers that are as yet unknown. For example, suppose you’re told that some number plus one equals five. Algebraically, you could write this as x + 1 = 5, or n + 1 = 5 or b + 1 = 5 -- variables can be represented by any letter, although some, such as x and y, are more commonly encountered than others.
Terms and Factors
Students of algebra must quickly become familiar with the concept of a “term.” Terms can consist of a variable, a single number or the combination of numbers and variables multiplied together. For instance, in x + 1 = 5, “x”, “1” and “5” are all considered terms. Likewise, 4y is a term: here, four is being multiplied by the variable y, though the multiplication sign isn’t typically written. In a multiplication such as this, the term is said to be a product of two factors -- in this case, the term “4y” is a product of the factors “4” and “y.”
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Symmetry of Equations
In algebra, equations -- mathematical sentences showing equality -- possess symmetry. That is, the terms on one side of the equals sign can be flipped with the terms on the other side of the equal sign. This is perhaps best demonstrated via an example: for instance, x + 1 = 5 is equivalent to 5 = x + 1.
Commutative and Associative Properties
There are assorted number properties you’ll encounter during algebra, but to start off, it is most useful to know the commutative and associative properties. The commutative property posits that the order of terms may be reversed when dealing with the operations of addition or multiplication. For an arithmetic example of this, consider that 4_5 is equivalent to 5_4; for an algebraic example, p + 3 is the same as 3 + p. The associative property deals with how terms – usually three -- are grouped within parentheses, and it can be applied to addition, subtraction and multiplication. It is best demonstrated through examples: 1 + (3 – 2) produces the same result as (1 + 3) – 2; likewise, 6(2x) is equivalent to (6*2)x.
Dealing With Negatives
You’ll often encounter negative numbers in algebra. You may sometimes find it helpful to think of subtraction as addition of a negative number. For instance, x – 4 is the same as x + (-4). When multiplying or dividing two negative terms, the result will always be positive: -7 * -7 = 49, and -7 * -x = 7x. When multiplying or dividing a negative term and a positive term, the result will be negative: -9/3 = -3, just as -9r/3 = -3r.