5.5.1 Variables and Expressions - OpenTextBookStore

5.5.1 Variables and Expressions

Learning Objective(s) 1 Evaluate expressions with one variable for given values for the variable. 2 Evaluate expressions with two variables for given values for the variables.

Introduction

Algebra involves the solution of problems using variables, expressions, and equations. This topic focuses on variables and expressions and you will learn about the types of expressions used in algebra.

Variables and Expressions

One thing that separates algebra from arithmetic is the variable. A variable is a letter or symbol used to represent a quantity that can change. Any letter can be used, but x and y are common. You may have seen variables used in formulas, like the area of a rectangle. To find the area of a rectangle, you multiply length times width, written using

the two variables l and w.

l ? w

Here, the variable l represents the length of the rectangle. The variable w represents the width of the rectangle.

You may be familiar with the formula for the area of a triangle. It is 1 b ? h . 2

Here, the variable b represents the base of the triangle, and the variable h represents

the height of the triangle. The 1 in this formula is a constant. A constant, unlike a 2

variable, is a quantity that does not change. A constant is often a number.

An expression is a mathematical phrase made up of a sequence of mathematical symbols. Those symbols can be numbers, variables, or operations (+, ?, ?, ?). Examples of expressions are l ? w and 1 b ? h .

2

Example

Problem Identify the constant and variable in the expression 24 ? x.

Answer

24 is the constant. x is the variable.

Since 24 cannot change its value, it is a constant. The variable is x, because it could be 0, or 2, or many other numbers.

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Substitution and Evaluation

Objective 1, 2

In arithmetic, you often evaluated, or simplified, expressions involving numbers.

3 ? 25 + 4

25 ? 5

142 - 12 4

3- 1 44

2.45 + 13

In algebra, you will evaluate many expressions that contain variables.

a + 10

48 ? c

100 ? x

l ? w

1b?h 2

To evaluate an expression means to find its value. If there are variables in the expression, you will be asked to evaluate the expression for a specified value for the variable.

The first step in evaluating an expression is to substitute the given value of a variable into the expression. Then you can finish evaluating the expression using arithmetic.

Example

Problem Evaluate 24 ? x when x = 3.

24 ? x 24 ? 3

Substitute 3 for the x in the expression.

24 ? 3 = 21

Subtract to complete the evaluation.

Answer

21

When you have two variables, you substitute each given value for each variable.

Example

Problem Evaluate l ? w when l = 3 and w = 8.

l ? w 3 ? 8

Substitute 3 for l in the expression and 8 for w.

3 ? 8 = 24

Multiply.

Answer

24

When you multiply a variable by a constant number, you don't need to write the multiplication sign or use parentheses. For example, 3a is the same as 3 ? a.

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Notice that the sign ? is used to represent multiplication. This is especially common with algebraic expressions because the multiplication sign ? looks a lot like the letter x, especially when hand written. Because of this, it's best to use parentheses or the ? sign to indicate multiplication of numbers.

Example

Problem Evaluate 4x ? 4 when x = 10.

4x ? 4 4(10) ? 4

Substitute 10 for x in the expression.

40 ? 4 36

Remember that you must multiply before you do the subtraction.

Answer

36

Since the variables are allowed to vary, there are times when you want to evaluate the same expression for different values for the variable.

Example

Problem

John is planning a rectangular garden that is 2 feet wide. He hasn't decided how long to make it, but he's considering 4 feet, 5 feet, and 6 feet. He wants to put a short fence around the garden. Using x to represent the length of the rectangular garden, he will need x + x + 2 + 2, or 2x + 4, feet of fencing.

Answer

How much fencing will he need for each possible garden length? Evaluate the expression when x = 4, x = 5, and x = 6 to find out.

2x + 4 2(4) + 4

8 + 4 12

For x = 4, substitute 4 for x in the expression. Evaluate by multiplying and adding.

2x + 4 2(5) + 4 10 + 4

14

For x = 5, substitute 5 for x. Evaluate by multiplying and adding.

2x + 4 2(6) + 4 12 + 4

16

For x = 6, substitute 6 for x and evaluate.

John needs 12 feet of fencing when x = 4, 14 feet when x = 5, and 16 feet when x = 6.

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Evaluating expressions for many different values for the variable is one of the powers of algebra. Computer programs are written to evaluate the same expression (usually a very complicated expression) for millions of different values for the variable(s).

Self Check A Evaluate 8x ? 1 when x = 2. Summary Variables are an important part of algebra. Expressions made from variables, constants, and operations can represent a numerical value. You can evaluate an expression when you are provided with one or more values for the variables: substitute each variable's value for the variable, then perform any necessary arithmetic. 5.5.1 Self Check Solutions Self Check A Evaluate 8x ? 1 when x = 2. Substituting 2 for x gives 8(2) ? 1. First multiply to get 16 ? 1, then subtract to get 15.

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5.5.2 Associative, Commutative, and Distributive Properties

Learning Objective(s) 1 Identify and use the commutative properties for addition and multiplication. 2 Identify and use the associative properties for addition and multiplication. 3 Identify and use the distributive property.

Introduction

There are many times in algebra when you need to simplify an expression. The properties of real numbers provide tools to help you take a complicated expression and simplify it.

The associative, commutative, and distributive properties of algebra are the properties most often used to simplify algebraic expressions. You will want to have a good understanding of these properties to make the problems in algebra easier to work.

The Commutative Properties of Addition and Multiplication

Objective 1

You may encounter daily routines in which the order of tasks can be switched without changing the outcome. For example, think of pouring a cup of coffee in the morning. You would end up with the same tasty cup of coffee whether you added the ingredients in either of the following ways:

? Pour 12 ounces of coffee into mug, then add splash of milk. ? Add a splash of milk to mug, then add 12 ounces of coffee.

The order that you add ingredients does not matter. In the same way, it does not matter whether you put on your left shoe or right shoe first before heading out to work. As long as you are wearing both shoes when you leave your house, you are on the right track!

In mathematics, we say that these situations are commutative--the outcome will be the same (the coffee is prepared to your liking; you leave the house with both shoes on) no matter the order in which the tasks are done.

Likewise, the commutative property of addition states that when two numbers are being added, their order can be changed without affecting the sum. For example, 30 + 25 has the same sum as 25 + 30.

30 + 25 = 55 25 + 30 = 55

Multiplication behaves in a similar way. The commutative property of multiplication states that when two numbers are being multiplied, their order can be changed without affecting the product. For example, 7 ? 12 has the same product as 12 ? 7.

7 ? 12 = 84 12 ? 7 = 84

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