The Language of Algebra

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638 Chapter 8 An Introduction to Algebra

Objectives

S E C T I O N 8.1

1Use variables to state properties of addition, multiplication, and division.

2Identify terms and coefﬁcients of terms.

3Translate word phrases to algebraic expressions.

4Evaluate algebraic expressions.

Coefﬁcient

The ﬁrst seven chapters of this textbook have been an in-depth study of arithmetic. It’s now time to begin the move toward algebra. Algebra is the language of mathematics. It can be used to solve many types of problems. In this chapter, you will learn more about thinking and writing in the language of algebra using its most important component—a variable.

The Language of Mathematics The word algebra comes from the title of the book Ihm Al-jabr wa’l muqabalah, written by an Arabian mathematician around A.D. 800.

1Use variables to state properties of addition, multiplication, and division.

One of the major differences between arithmetic and algebra is the use of variables. Recall that a variable is a letter (or symbol) that stands for a number. In this course, we have used variables on several occasions. For example, in Chapter 1, we let l stand for the length and w stand for the width in the formula for the area of a rectangle: A = lw. In Chapter 6, we let x represent the unknown number in percent problems.

The Language of Mathematics The word variable is based on the root word vary, which means change or changing. For example, the length and width of rectangles vary, and the unknown numbers in percent problems vary.

Many symbols used in arithmetic are also used in algebra. For example, a plus symbol is used to indicate addition, a minus symbol – is used to indicate subtraction, and an symbol means is equal to.

Since the letter x is often used in algebra and could be confused with the multiplication symbol , we usually write multiplication using a raised dot or parentheses. When multiplying a variable by a number, or a variable by another variable, we can omit the symbol for multiplication. For example,

2b means 2 b

xy means x y

8abc means 8 a b c

In the notation 2b, the number 2 is an example of a constant because it does not change value.

Many of the patterns that we have seen while working with whole numbers, integers, fractions, and decimals can be generalized and stated in symbols using variables. Here are some familiar properties of addition written in a very compact form, where the variables a and b represent any numbers.

• The Commutative Property of Addition

a b b a

Changing the order when adding does not affect the answer.

• The Associative Property of Addition

(a b) c a (b c)

Changing the grouping when adding does not affect the answer.

8.1 The Language of Algebra

639

• Addition Property of 0 (Identity Property of Addition)

a 0 a	and	0 a a	When 0 is added to any number, the
			result is the same number.

Here are several familiar properties of multiplication stated using variables.

• The Commutative Property of Multiplication

ab ba Changing the order when multiplying does not affect the answer.

• The Associative Property of Multiplication

(ab)c a(bc) Changing the grouping when multiplying does not affect the answer.

• Multiplication Property of 0

0	a 0	and	a	0 0 The product of 0 and any number is 0.
• Multiplication Property of 1
1	a a	and	a		1 a	The product of 1 and any number is that
1	a a	and	a		1 a	number.
						number.

Here are two familiar properties of division stated using a variable.

• Division Properties

a			a		Any number divided by 1
a	a	and	a	1 provided a 0	is the number itself.
1	a	and	a	1 provided a 0	Any number (except 0)
1			a		Any number (except 0)
					divided by itself is 1.

2 Identify terms and coefﬁcients of terms.

When we combine variables and numbers using arithmetic operations, the result is an algebraic expression.

Algebraic Expressions

Variables and/or numbers can be combined with the operations of addition, subtraction, multiplication, and division to create algebraic expressions.

The Language of Mathematics We often refer to algebraic expressions as simply expressions.

Here are some examples of algebraic expressions.

4a 7

10 y

15mn(2m)

This expression is a combination of the numbers 4 and 7, the variable a, and the operations of multiplication and addition.

This expression is a combination of the numbers 10 and 3, the variable y, and the operations of subtraction and division.

This expression is a combination of the numbers 15 and 2, the variables m and n, and the operation of multiplication.

640	Chapter 8 An Introduction to Algebra

Addition symbols separate expressions into parts called terms. For example, the expression x + 8 has two terms.

x		8
First term		Second term

Since subtraction can be written as addition of the opposite, the expression a2 3a 9 has three terms.

a2 3a 9	a2		( 3a)	( 9)
	First term		Second term	Third term

In general, a term is a product or quotient of numbers and/or variables. A single number or variable is also a term. Examples of terms are:

4, y, 6r, w3, 3.7x5,	3	,	15ab2
	n

Self Check 1

Identify the coefﬁcient of each

term in the expression: p3 12p2 3p 4

Now Try Problem 23

Caution! By the commutative property of multiplication, r6 6r and15b2a 15ab2. However, when writing terms, we usually write the numerical factor ﬁrst and the variable factors in alphabetical order.

The numerical factor of a term is called the coefﬁcient of the term. For instance, the term 6r has a coefﬁcient of 6 because 6r 6 r. The coefﬁcient of 15ab2 is 15 because 15ab2 15 ab2. More examples are shown below.

A term such as 4, that consists of a single number, is called a constant term.

Term	Coefﬁcient

8y2	8
0.9pq	0.9
3b	3
4	4
6x	61
x	1
t	1
27	27

This term could be written 34b .

Because 6x 16x 61 x

Because x 1x

Because t 1t

The coefﬁcient of a constant term is that constant.

The Language of Algebra Terms such as x and y have implied coefﬁcients of 1. Implied means suggested without being precisely expressed.

EXAMPLE 1 Identify the coefﬁcient of each term in the expression: 7x2 x 6

Strategy We will begin by writing the subtraction as addition of the opposite. Then we will determine the numerical factor of each term.

WHY Addition symbols separate expressions into terms.

Solution If we write 7x2 x 6 as 7x2 ( x) 6, we see that it has three terms: 7x2, x, and 6. The numerical factor of each term is its coefﬁcient.

The coefﬁcient of 7x2 is 7 because 7x2 means 7 x2.

The coefﬁcient of x is 1 because x means 1 x.

The coefﬁcient of the constant 6 is 6.

8.1 The Language of Algebra

641

It is important to be able to distinguish between the terms of an expression and the factors of a term.

	EXAMPLE 2	Is m used as a factor or a term in each expression?
a. m 6 b.		8m

Strategy We will begin by determining whether m is involved in an addition or a multiplication.

WHY Addition symbols separate expressions into terms. A factor is a number being multiplied.

Solution

a. Since m is added to 6, m is a term of m 6. b. Since m is multiplied by 8, m is a factor of 8m.

3 Translate Word Phrases to Algebraic Expressions.

The tables below show how key phrases can be translated into algebraic expressions.

Addition

the sum of a and 8	a 8

4 plus c	4 c

16 added to m	m 16

4 more than t	t 4

20 greater than F	F 20

T increased by r	T r

exceeds y by 35	y 35

Subtraction

the difference of 23 and P	23 P

550 minus h	550 h

18 less than w	w 18

7 decreased by j	7 j

M reduced by x	M x

12 subtracted from L	L 12

5 less ƒ	5 ƒ

Caution! Be careful when translating subtraction. Order is important. For example, when a translation involves the phrase less than, note how the terms are reversed.

18 less than w

w 18

Multiplication

the product of 4 and x		4x

20 times B	20B

twice r		2r

double the amount a		2a

triple the proﬁt P		3P

three-fourths of m		3	m
three-fourths of m	4		m

Division

the quotient of R and 19

s divided by d

the ratio of c to d

k split into 4 equal parts

Self Check 2

Is b used as a factor or a term in each expression?

a.27b

b.5a b

Now Try Problems 27 and 29

Caution! Be careful when translating division. As with subtraction, order is

d important. For example, s divided by d is not written s .

EXAMPLE 5

642	Chapter 8 An Introduction to Algebra

Self Check 3

Write each phrase as an algebraic expression:

a.80 less than the total t

b.23 of the time T

c.the difference of twice a and 15, squared

Now Try Problems 31, 37, and 41

EXAMPLE 3 Write each phrase as an algebraic expression:

a.one-half of the proﬁt P

b.5 less than the capacity c

c.the product of the weight w and 2,000, increased by 300 Strategy We will begin by identifying any key phrases.

WHY Key phrases can be translated to mathematical symbols.

Solution

a. Key phrase: One-half of	Translation: multiplication by	1
	1P.	2
The algebraic expression is:	1P.
	2
b. Key phrase: less than	Translation: subtraction

Self Check 4

COMMUTING TO WORK It takes Val m minutes to get to work

if she drives her car. If she takes the bus, her travel time exceeds this by 15 minutes. How long does it take her to get to

work by bus?

Now Try Problem 67

Self Check 5

SCHOLARSHIPS Part of a $900 donation to a college went to the scholarship fund, the rest to the building fund. Choose a variable to represent the amount donated to one of the funds.Then write an expression that represents the amount donated to the other fund.

Sometimes thinking in terms of speciﬁc numbers makes translating easier. Suppose the capacity was 100. Then 5 less than 100 would be 100 5. If the capacity is c, then we need to make c 5 less. The algebraic expression is: c 5.

Caution! 5 c is the translation of the statement 5 is less than the capacity c and not 5 less than the capacity c.

c. Key phrase: product of	Translation: multiplication
Key phrase: increased by	Translation: addition

In the given wording, the comma after 2,000 means w is ﬁrst multiplied by 2,000; then 300 is added to that product. The algebraic expression is: 2,000w 300.

To solve application problems, we let a variable stand for an unknown quantity.

EXAMPLE 4	Swimming A

pool is to be sectioned into 8 equally
wide swimming lanes.Write an algebraic
expression that represents the width of
each lane.		x

Strategy We will begin by letting x the width of the swimming pool in feet. Then we will identify any key phrases.

WHY The width of the pool is unknown.

Solution The key phrase, sectioned into 8 equally wide lanes, indicates division. Therefore, the width of each lane is x8 feet.

Painting A 10-inch-long paintbrush has two parts: a handle and bristles. Choose a variable to represent the length of one of the parts. Then write an expression to represent the length of the other part.

Strategy There are two approaches.We can let h the length of the handle or we can let b the length of the bristles.

WHY Both the length of the handle and the length of the bristles are unknown.

Now Try Problem 12

Solution Refer to the ﬁrst drawing. If we let h the length of the handle (in inches), then the length of the bristles is 10 h.

Now refer to the second drawing. If we let b the length of the bristles (in inches), then the length of the handle is 10 b.

8.1 The Language of Algebra

643

h	10 – h

10 in.

10 – b	b

10 in.

	EXAMPLE 6	Enrollments Second

semester enrollment in a nursing program was 32
more than twice that of the ﬁrst semester. Let x
represent the enrollment for one of the semesters.
Write an expression that represents the
enrollment for the other semester.
Strategy There		are two	unknowns: the	Somos/Veer
enrollment ﬁrst		semester and	the enrollment

second semester. We will begin by letting x the enrollment for the ﬁrst semester.
WHY Because the second-semester enrollment is related to the ﬁrst-semester
enrollment.
Solution
	Key phrase: more than		Translation: addition
	Key phrase: twice that		Translation: multiplication by 2

The second semester enrollment was 2x 32.

4 Evaluate Algebraic Expressions.

To evaluate an algebraic expression, we substitute given numbers for each variable and perform the necessary calculations in the proper order.

Self Check 6

ELECTIONS In an election, the incumbent received 55 fewer votes than three times the challenger’s votes. Let x represent the number of votes received by one candidate. Write an expression that represents the number of votes received by the other.

Now Try Problem 91

EXAMPLE 7 Evaluate each expression for x 3 and y 4:

a. y3 y2	b. y x c. 0 5xy 7 0	d.	y 0
			x ( 1)

Strategy We will replace each x and y in the expression with the given value of the variable, and evaluate the expression using the order of operation rules.

WHY To evaluate an expression means to ﬁnd its numerical value, once we know the value of its variable(s).

Solution	Substitute 4 for each y. We must write
a. y3 y2 ( 4)3 ( 4)2	4 within parentheses so that it is the
	base of each exponential expression.	5 14
	base of each exponential expression.	5 14
64 16	Evaluate each exponential expression.	64
64 16	Evaluate each exponential expression.	16
		16
48	Do the addition.	48
48	Do the addition.	48

Self Check 7

Evaluate each expression for

a2 and b 5:

a.0 a3 b2 0 b. a 2ab

a 2 c. b 3

Now Try Problems 73 and 85

644	Chapter 8 An Introduction to Algebra

Caution! When replacing a variable with its numerical value, we must often write the replacement number within parentheses to convey the proper meaning.

b.y x ( 4) 3

4 3

Substitute 4 for y and 3 for x. Don’t forget to write the sign in front of ( 4).

Simplify: ( 4) 4.

c. 05xy 7 0 0	5(3)( 4) 7 0	Substitute 3 for x and 4 for y.
0	60 7 0	Do the multiplication: 5(3)( 4) 60.
0	67 0	Do the subtraction: 60 7 60 ( 7) 67.

		67		Find the absolute value of 67.
d.	y 0		4 0	Substitute 3 for x and 4 for y.

	x ( 1)		3 ( 1)
			4	In the denominator, do the subtraction:
			4	3 ( 1) 3 1 4.
		1		Do the division.
		1		Do the division.

ANSWERS TO SELF CHECKS
1.	1, 12, 3, 4 2.	a. factor b. term 3. a. t 80	b.	32T c. (2a 15)2
4.	(m 15) minutes	5. s amount donated to scholarship fund (in dollars);
900 s amount donated to building fund (in dollars)			6.	x number of votes

received by the challenger; 3x 55 number of votes received by the incumbent

7. a. 17 b. 18 c. 0

	S E C T I O N			8.1 STUDY SET
		VOCABULARY


	Fill in the blanks.
1.				are letters (or symbols) that stand for
			numbers.
2.			The word		comes from the title of a book
			written by an Arabian mathematician around
			A.D. 800.

3.Variables and/or numbers can be combined with the operations of arithmetic to create algebraic

4. A is a product or quotient of numbers and/or

variables. Examples are: 8x, 2t , and cd 3.


6.	A term, such as 27, that consists of a single number is
	called a			term.
7.	The		of the term 10x is 10.
8.	To	4x 3 for x 5, we substitute 5 for x

and perform the necessary calculations in order.

CONCEPTS

9.CUTLERY The knife shown below is 12 inches long. Write an expression that represents the length of the blade.

h in.

5. Addition symbols separate algebraic expressions into parts called .

10.SAVINGS ACCOUNTS A student inherited $5,000 and deposits x dollars in American Savings. Write an expression that represents the amount of money left to deposit in a City Mutual account.

	$5,000
American Savings	City Mutual
$x	$?

11.a. MIXING SOLUTIONS Solution 1 is poured into solution 2. Write an expression that represents the number of ounces in the mixture.

Solution 1 20 ounces

Solution 2 x ounces

b.SNACKS Cashews were mixed with p pounds of peanuts to make 100 pounds of a mixture. Write an expression that represents the number of pounds of cashews that were used.

PEANUTS

CASHEWS

p pounds ? pounds

MIX

100 pounds

12.BUILDING MATERIALS

a.Let b the length of the beam shown below (in feet). Write an expression that represents the length of the pipe.

b.Let p the length of the pipe (in feet). Write an expression that represents the length of the beam.

15 ft

8.1 The Language of Algebra

645

NOTATION

Complete each solution. Evaluate each expression for a 5, x 2, and y 4.

13.9a a2 9() (5)2

9(5)

14.x 6y () 6()

Write each expression without using a multiplication symbol or parentheses.

15.	4 x	16.	P r t
17.	2(w)	18.	(x)(y)

GUIDED PRACTICE

Use the following variables to write each property of addition and multiplication. See Objective 1.

19.a. Write the commutative property of addition using the variables x and y.

b.Write the associative property of addition using the variables r, s, and t.

20.a. Write the commutative property of multiplication using the variables m and n.

b.Write the associative property of multiplication using the variables x, y, and z.

21.Write the multiplication property of zero using the variable s.

22.Write the multiplication property of 1 using the variable b.

Answer the following questions about terms and coefﬁcients.

See Example 1.

23.Consider the expression 3x3 11x2 x 9.

a.How many terms does the expression have?

b.What is the coefﬁcient of each term?

24.Consider the expression 4a2 6a 1.

a.How many terms does the expression have?

b.What is the coefﬁcient of each term?

25.Complete the following table.

646Chapter 8 An Introduction to Algebra

26.Complete the following table.

	Term	4a	2r	c	3lw	d	x
					4	9
	Coefﬁcient

Determine whether the variable c is used as a factor or as a
term. See Example 2.
27.	c 32		28.	24c 6
29.	5c		30.	a b c

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable. See Example 3.

31.The sum of the length l and 15

32.The difference of a number and 10

33.The product of a number and 50

34.Three-fourths of the population p

35.The ratio of the amount won w and lost l

36.The tax t added to c

37.P increased by two-thirds of p

38.21 less than the total height h

39.The square of k, minus 2,005

40.s subtracted from S

41.1 less than twice the attendance a

42.J reduced by 500

43.1,000 split n equal ways

44.Exceeds the cost c by 25,000

45.90 more than twice the current price p

46.64 divided by the cube of y

47.3 times the total of 35, h, and 300

48.Decrease x by 17

49.680 fewer than the entire population p

50.Triple the number of expected participants

51.The product of d and 4, decreased by 15

52.The quotient of y and 6, cubed

53.Twice the sum of 200 and t

54.The square of the quantity 14 less than x

55.The absolute value of the difference of a and 2

56.The absolute value of a, decreased by 2

57.One-tenth of the distance d

58.Double the difference of x and 18

Translate each algebraic expression into words. (Answers may vary.) See Example 3.

59.34 r

60.3 d

61.t 50

62.c 19

63.xyz

64.10ab

65.2m 5

66.2s 8

Answer with an algebraic expression. See Example 4.

67.MODELING A model’s skirt is x inches long. The designer then lets the hem down 2 inches. What is the length of the altered skirt?

68.PRODUCTION LINES A soft drink manufacturer produced c cans of cola during the morning shift. Write an expression for how many six-packs of cola can be assembled from the morning shift’s production.

69.PANTS The tag on a new pair of 36-inch-long jeans warns that after washing, they will shrink x inches in length. What is the length of the jeans after they are washed?

70.ROAD TRIPS A caravan of b cars, each carrying 5 people, traveled to the state capital for a political rally. How many people were in the caravan?

Evaluate each expression, for x 3, y 2, and z 4.

See Example 7.

71.	y		72.	z
73.	z 3x		74.	y 5x
75.	3y2 6y 4		76.	z2 z 12
77.	(3 x)y		78.	(4 z)y
79.	(x y)2 0z y 0		80.	[(z 1)(z 1)]2
81.		2x y3	82.		2z2 x
		y 2z			2x y2

Evaluate each expression. See Example 7.

83.b2 4ac for a 1, b 5, and c 2

84.(x a)2 (y b)2 for x 2, y 1, a 5, and b 3

85.a2 2ab b2 for a 5 and b 1

a x

86. y b for x 2, y 1, a 5, and b 2

87. 2 [2a (n 1)d] for n 10, a 4.2, and d 6.6

	a(1 rn)
88.		for a 5, r 2, and n 3

	1 r
89.	(27c2 4d2)3 for c			1 and d 1
				3	2
90.	b2 16a2 1		for a 1 and b 10
	2
					4

APPLICATIONS

91.VEHICLE WEIGHTS A Hummer H2 weighs 340 pounds less than twice a Honda Element.

a.Let x represent the weight of one of the vehicles. Write an expression for the weight of the other vehicle.

b.If the weight of the Element is 3,370 pounds, what is the weight of the Hummer?

92.SOD FARMS The expression 20,000 3s gives the number of square feet of sod that are left in a ﬁeld after s strips have been removed. Suppose a city orders 7,000 strips of sod. Evaluate the expression and explain the result.

Strips of sod, cut and ready to be loaded on a truck for delivery

93.COMPUTER COMPANIES IBM was founded 80 years before Apple Computer. Dell Computer Corporation was founded 9 years after Apple.

a.Let x represent the age (in years) of one of the companies. Write expressions to represent the ages (in years) of the other two companies.

b.On April 1, 2008, Apple Computer Company was 32 years old. How old were the other two computer companies then?

8.1 The Language of Algebra

647

94.THRILL RIDES The distance in feet that an object will fall in t seconds is given by the expression 16t2.

Find the distance that riders on “Drop Zone” will fall during the times listed in the table.

Time Distance

(seconds) (feet)

WRITING

95.What is a variable? Give an example of how variables are used.

96.What is an algebraic expression? Give some examples.

97.Explain why 2 less than x does not translate to 2 x.

98.In this section, we substituted a number for a variable. List some other uses of the word substitute that you encounter in everyday life.

REVIEW

99. Find the LCD for 125 and 151 .

3 3 5

100. Simplify: 3 5 5 11

101.Evaluate: a23b

102.Find the result when 78 is multiplied by its reciprocal.

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