Ch 2 Sec 1: Slide #1

Columbus State Community College

Chapter 2 Section 1

Introduction to Variables

Ch 2 Sec 1: Slide #2

Introduction to Variables

1. Identify variables, constants, and expressions.

2. Evaluate variable expressions for given replacement values.

3. Write properties of operations using variables.

4. Use exponents with variables.

Ch 2 Sec 1: Slide #3

Expressions, Variables, and Constants

EXAMPLE 1 Writing an Expression and Identifying the Variable and Constant

(a) Maria increased her test average by 12 points.

a + 12

Write an expression for each rule. Identify the variable and the constant.

Variable Constant

(b) The price of a game dropped by $20

p – 20Variable Constant

Ch 2 Sec 1: Slide #4

Follow the rule. Subtract to find 90 – 20.

Evaluating an Expression

EXAMPLE 2 Evaluating an Expression

(a) Evaluate the expression when the original price is $90.

p – 20

Use this rule for finding the price of a game: The price of a game dropped by $20. The expression is p – 20.

90 – 20

70

Replace p with 90.

The new price will be $70.

Ch 2 Sec 1: Slide #5

Follow the rule. Subtract to find 78 – 20.

Evaluating an Expression

EXAMPLE 2 Evaluating an Expression

(b) Evaluate the expression when the original price is $78.

p – 20

Use this rule for finding the price of a game: The price of a game dropped by $20. The expression is p – 20.

78 – 20

58

Replace p with 78.

The new price will be $58.

Ch 2 Sec 1: Slide #6

3x –4m –d n3x –4m –1d 1n

Numerical Coefficients

The number part in a multiplication expression is called the numerical coefficient, or just the coefficient.

The numerical coefficients are 3, –4, –1, and 1 respectively.

Ch 2 Sec 1: Slide #7

CAUTION

CAUTIONIf an expression involves adding, subtracting, or dividing, then you do have to write +, –, or ÷. It is only multiplication that is understood without writing an operation symbol.

5 + x 5 – x 5 ÷ x 5x

Add x Subtract x Divide by x Multiply by x

Ch 2 Sec 1: Slide #8

The Perimeter of a “STOP” Sign

s

The shape of a common “STOP” sign is called an “Octagon.” An octagon has 8 equal sides as shown in the diagram below.

s

s

ss

ss

s

STOP 8 s

To find the distance around an object, called the perimeter, simply add the outside edges together. The expression (rule) can be written in shorthand form as shown below.

Ch 2 Sec 1: Slide #9

Evaluating an Expression with Multiplication

15 in. 15 in.

15 in.

15 in.15 in.

15 in.15 in.

15 in.

STOP

8 s

EXAMPLE 3 Evaluating an Expression with Multiplication

The expression (rule) for finding the perimeter of an octagon is 8s. Evaluate the expression when the length of one side of the “STOP” sign is 15 inches. See the diagram below.

8 • 15 inches

120 inches

The total distance around this “STOP” sign (perimeter) is 120 inches.

Multiply.

Replace s with 15 inches.

Ch 2 Sec 1: Slide #10

Evaluating an Expression with Several Steps

EXAMPLE 4 Evaluating an Expression with Several Steps

30d + 50

A car rental company charges a flat fee of $50 plus $30 per day to rent a certain car. The expression (rule) for finding the amount to charge a customer is shown below. Evaluate the given expression for a person who rents this car for 6 days.

Replace d with 6, the number of days.

30 ( 6 ) + 50 Follow the order of operations. Multiply first.

180 + 50 Add.

230 The cost of renting the car for 6 days is $230.

Ch 2 Sec 1: Slide #11

A Rectangular Garden

24 feet

Suppose you wanted to put a fence around a rectangular-shaped flower garden. The length of the garden is 24 feet and the width is 16 feet. How much fencing material would you need to finish the job?

16 feet

24 feet

16 feet

24 feet + 16 feet + 24 feet + 16 feet = 80 feet of fencing

Ch 2 Sec 1: Slide #12

Rectangles

l

In general, the expression (rule) for finding the amount of fencing needed to surround a rectangular garden can be found as follows.

w

l

w

l + w + l + w = amount of fencing= 2l + 2w

Ch 2 Sec 1: Slide #13

There is no operation between the 2 and the l and there is no operation between the 2 and the w, so it is understood to be multiplication.

Evaluating an Expression with Two Variables

EXAMPLE 5 Evaluating an Expression with Two Variables

2 l + 2 w

The expression (rule) for finding the perimeter of a rectangle is 2l + 2w. Evaluate the expression of a rectangular table that has a length, l, of 5 feet and a width, w, of 2 feet.

2 ( 5 feet ) + 2 ( 2 feet )

10 feet + 4 feet

Replace l with 5 feet and w with 2 feet.

Add.

14 feet The perimeter of this table is 14 feet.

(a)

Ch 2 Sec 1: Slide #14

Evaluating an Expression with Two Variables

EXAMPLE 5 Evaluating an Expression with Two Variables

a – b

Complete the table below to show how to evaluate each expression.(b)

a • bValue of a Value of b

5 – 7 is –25 7

–4 – 9 is –13 –4 9

6 – –3 is 9 6 –3

–2 – –8 is 6 –2 –8

5 • 7 is 35

–4 • 9 is –36

6 • –3 is –18

–2 • –8 is 16

1.

2.

3.

4.

Expression ( Rule )

Ch 2 Sec 1: Slide #15

Writing Properties of Operations Using Variables

EXAMPLE 6 Writing Properties of Operations Using Variables

Use the letter n to represent any number.

Use the variable n to state this property: When any number is divided by 1, the quotient is the number.

n1

= n

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Understanding Exponents Used with Variables

EXAMPLE 7 Understanding Exponents Used with Variables

(a) m5

Rewrite each expression without exponents.

can be written as m • m • m • m • m

(b) 8 x y4 can be written as 8 • x • y • y • y • y

Coefficient is 8. y4

The exponent applies only to y.

(c) –5 b3 c2 can be written as –5 • b • b • b • c • c

Coefficient is –5. b3 c2

m is used as a factor 5 times.

Ch 2 Sec 1: Slide #17

Evaluating Expressions with Exponents

EXAMPLE 8 Evaluating Expressions with Exponents

(a) g2 when g is –4

Evaluate each expression.

g2 means g • g

16

–4 • –4

Replace each g with –4.

Multiply –4 times –4.

So g2 becomes ( –4 )2, which is ( –4 ) ( –4 ), or 16.

Ch 2 Sec 1: Slide #18

Multiply two factors at a time.Replace m with –3, and replace n with –2.

Evaluating Expressions with Exponents

EXAMPLE 8 Evaluating Expressions with Exponents

(b) m3 n2 when m is –3 and n is –2

Evaluate each expression.

m3 n2 means m • m • m • n • n

9 • –3 • –2 • –2

–3 • –3 • –3 • –2 • –2

–27 • –2 • –2

54 • –2

–108

So m3 n2 becomes ( –3 )3 ( –2 )2, which is

( –3 ) ( –3 ) ( –3 ) ( –2 ) ( –2 ), or –108.

Ch 2 Sec 1: Slide #19

Multiply two factors at a time.

–2 w v2 means –2 • w • v • vSo –2 w v2 becomes –2 ( –4 ) ( 3 )2, which is

( –2 ) ( –4 ) ( 3 ) (3 ), or 72.

Replace w with –4 and replace v with 3.

Evaluating Expressions with Exponents

EXAMPLE 8 Evaluating Expressions with Exponents

(c) –2 w v2 when w is –4 and v is 3

Evaluate each expression.

8 • 3 • 3

–2 • –4 • 3 • 3

24 • 3

72

Ch 2 Sec 1: Slide #20

Introduction to Variables

Chapter 2 Section 1 – Completed

Written by John T. Wallace