Unit 1: Variables and Variable Expressions
In algebra, we use letters to represent numbers in equations. We call these letters variables, because the numbers they represent vary. For example, let’s say your salary is $10 per hour. If you worked two hours, you would be paid $10 × 2, or $20. If you worked five hours you would be paid $10 × 5, or $50. To generalize this we can say that if you work h hours, you will be paid $10 × h.
In this unit, we learn that the properties of numbers also apply to the letters we use in these formulas: you treat variables the same way you treat numbers in calculations. We discuss how to recognize like terms, which you can add and subtract as if they were numbers.
Completing this unit should take you approximately 4 hours.
1.1: Variables, Constants, and Coefficients
The first step in understanding algebra is knowing the definitions of variables, constants, and coefficients. These are all important parts of an algebraic equation.
1.2: Replacing Variables with Their Values
We often know the value of a variable for a given problem. For the example given at the start of the chapter, we said that if you make $10 per hour and you work h hours, your pay would be $10 × h. Let's say you worked 40 hours in a given week. For that week, we say h = 40. So, your pay would be $10 × 40 = $400.
Being able to substitute a value in for a variable appropriately is an important skill in algebra. You also need to understand how to write variable expressions when multiplication is involved.
1.3: Order of Operations Review
Mathematicians use the convention order of operations to ensure we always evaluate expressions with many operations in the same way. This ensures we complete complicated calculations the same way every time.
1.4: Commutative Property of Addition and Multiplication
In addition to knowing the order of operations, we need to understand some properties of arithmetic before we can begin evaluating more complicated algebraic expressions.
The commutative property of addition and multiplication states that the order of the input numbers does not matter. For example, 3 + 2 = 5 and 2 + 3 = 5. Likewise, 3 × 2 = 6 and 2 × 3 = 6.
1.5: Associative Property of Addition and Multiplication
The associative property states that we can add or multiply several numbers in any order. We use this property frequently, such as when we switch terms around to simplify algebraic expressions. For example, (1 + 1) + 3 = 5 and 1+ (1 + 3) = 5.
1.6: Distributive Property of Multiplication over Addition/Subtraction
The distributive property is another property we use extensively to simplify algebraic expressions. It allows us to "distribute" a multiplicative factor over an addition or subtraction. For example, for the expression 4 × (3 + 2), we can distribute the multiple of four across the addition in the parentheses to get (4 x 3) + (4 × 2) = 12 + 8 = 20.
1.7: Definition and Examples of Like Terms
Like terms have the same variables using the same exponents, but may have different numerical coefficients. You need to recognize like terms to add and subtract them. For example, 3x and 8x are like terms because they have the same variables and exponents. The terms 3x and 8x2 are not like terms because, although the variables are the same, the exponents on the variables are different.
1.8: Simplifying Expressions by Combining Like Terms
In general algebraic expressions, we open the parentheses to combine like terms. Due to the commutative and associative properties, we can move like terms around to combine them. This makes it easier to solve complicated-looking expressions.