Download R.5 Algebraic Expressions

Section R.5
Algebraic Expressions 39
R.5 Algebraic Expressions
OBJECTIVES
1
Translate English Expressions
into the Language of
Mathematics
2
Evaluate Algebraic Expressions
3
Simplify Algebraic Expressions
by Combining Like Terms
4
Determine the Domain of a
Variable
In Words
We mentioned earlier that in algebra we use letters such as x, y, a, b, and c to represent
numbers. If a letter is used to represent any number from a given set of numbers, it is
called a variable. A constant is either a fixed number, such as 5 or 22, or a letter that
represents a fixed (possibly unspecified) number. For example, in Einstein’s Theory
of Relativity, E = mc2, E and m are variables that represent total energy and mass,
respectively, while c is a constant that represents the speed of light (299,792,458 meters
per second).
An algebraic expression is any combination of variables, constants, grouping symbols
such as parentheses ( ) or brackets [ ], and mathematical operations such as addition,
subtraction, multiplication, division, or exponents. The following are examples of algebraic expressions.
In algebra, letters of the alphabet are used to
represent numbers.
Work Smart
An algebraic expression is not the
same as an algebraic statement.
An expression might be x + 3 while
a statement might be x + 3 = 7.
Do you see the difference?
1
5v4 - v
4 - v
2y2 - 5y - 12z
3x + 4
Translate English Expressions into the Language of Mathematics
One of the neat features of mathematics is that the symbols we use allow us to express
English phrases briefly and consistently. There are certain words or phrases in English
that easily translate into math symbols. Table 2 lists various English words or phrases
and their corresponding math symbol.
Table 2 Math Symbols and the Words They Represent
#
Add (⫹)
Subtract (⫺)
Multiply ( )
Divide (/)
sum
difference
product
quotient
plus
minus
times
divided by
more than
subtracted from
of
per
exceeds by
less
twice
ratio
in excess of
less than
added to
decreased by
increased by
EXAMPLE 1 Writing English Phrases Using Math Symbols
Express each English phrase as an algebraic expression.
(a)
(b)
(c)
(d)
The sum of 3 and 8
The quotient of 50 and some number y
The number 11 subtracted from z
Twice the sum of a number x and 5
Solution
(a) Because we are talking about a sum, we know to use the + symbol, so
“The sum of 3 and 8” is represented mathematically as 3 + 8.
50
(b) “The quotient of 50 and some number y” is represented mathematically as .
y
(c) “The number 11 subtracted from z” is represented mathematically as z - 11.
(d) “Twice the sum of a number x and 5” is represented as an algebraic expression
as 21x + 52. We know the mathematical representation of the phrase is
21x + 52 rather than 2x + 5 because the phrase said “twice the sum,” which
means to multiply the sum of the two numbers by 2. The English phrase that
would result in 2x + 5 might be “the sum of twice a number and 5.” Do you
see the difference?
40
CHAPTER R
Real Numbers and Algebraic Expressions
Quick
1. A
is a letter used to represent any number from a given set of numbers.
2. A
is either a fixed number or a letter used to represent a fixed (possibly
unspecified) number.
In Problems 3–8, express each English phrase using an algebraic expression.
3. The sum of 3 and 11
4. The product of 6 and 7
5. The quotient of y and 4
6. The difference of 3 and z
7. Twice the difference of x and 3
8. Five plus the ratio of z and 2
2
Evaluate Algebraic Expressions
To evaluate an algebraic expression, substitute or replace each variable with its numerical value.
EXAMPLE 2 Evaluating an Algebraic Expression
Evaluate each expression for the given value of the variable.
-z2 + 4z
1
(a) 4x + 3 for x = 5
(b)
for z = 9
(c) ƒ 10y - 8 ƒ for y =
z + 1
2
Solution
(a) We substitute 5 for x in the expression 4x + 3.
4152 + 3 = 20 + 3 = 23
-z2 + 4z
.
z + 1
-1922 + 4 # 9
-81 + 36
-45
-9 # 5
-9
9
=
=
=
=
= #
9 + 1
10
10
5 2
2
2
(b) We substitute 9 for z in the expression
1
for y in the expression ƒ 10y - 8 ƒ .
2
1
- 8 ` = ƒ 5 - 8 ƒ = ƒ -3 ƒ = 3
` 10 #
2
(c) We substitute
Quick
In Problems 9–12, evaluate each expression for the given value of the variable.
9. -5x + 3 for x = 2
10. y2 - 6y + 1 for y = -4
w + 8
1
11.
for w = 4
12. ƒ 4x - 5 ƒ for x =
3w
2
EXAMPLE 3 Evaluating an Algebraic Expression
The algebraic expression 2x + 21x + 102 represents the perimeter of a rectangular
field whose width is 10 yards more than its length. See Figure 14. Evaluate the
algebraic expression for x = 4, 8, and 10.
Figure 14
x ⫹ 10
x
Section R.5
Algebraic Expressions 41
Solution
We evaluate the algebraic expression 2x + 21x + 102 for each value of x.
x = 4:
x = 8:
x = 10:
2142 + 214 + 102 = 8 + 21142 = 8 + 28 = 36 yards
2182 + 218 + 102 = 16 + 21182 = 16 + 36 = 52 yards
21102 + 2110 + 102 = 20 + 21202 = 20 + 40 = 60 yards
Quick
13. The algebraic expression 107x represents the number of Japanese Yen that you
could purchase for x dollars. Evaluate the algebraic expression for x = 100, 1000,
and 10,000 dollars. (SOURCE: Yahoo! Finance)
5
14. The algebraic expression 1x - 322 represents the equivalent temperature in
9
degrees Celsius for x degrees Fahrenheit. Determine the equivalent temperature
for x = 32, 86, and 212 degrees Fahrenheit.
3
Simplify Algebraic Expressions by Combining Like Terms
Often, we can simplify complicated algebraic expressions into simpler algebraic
expressions. One technique for doing this is through the elimination of parentheses
and combining like terms. A term is a number or the product of a number and one or
more variables raised to a power. In algebraic expressions, terms are separated by
addition signs. For example, consider Table 3, where various algebraic expressions are
given and the terms of the expression identified.
Table 3
Algebraic Expression
Terms
5x + 4
5x, 4
2
7x - 8x + 3
7x 2, -8x, 3
3x 2 + 7y 2
3x 2, 7y 2
Notice in the algebraic expression 7x2 - 8x + 3, we first rewrite it with only addition
signs as in 7x2 + 1-8x2 + 3 to identify the terms.
Terms that have the same variable(s) and the same exponent(s) on the variable(s)
are called like terms. For example 3x and 8x are like terms because both terms have the
variable x raised to the 1st power. Also, -4x3y and 10x3y are like terms because both
have the variable x raised to the 3rd power along with y raised to the 1st power. The numerical factor of the variable expression is called the coefficient. For example, the coefficient of 3x is 3; the coefficient of -4x3y is -4. For terms that have no coefficient,
such as xy, the implied coefficient is one. The implied coefficient for -z would be
negative one because -z = -1 # z.
We combine like terms by using the Distributive Property “in reverse.”
EXAMPLE 4 Combining Like Terms
Simplify each algebraic expression by combining like terms.
(a) 5x + 3x
(b) z + 7z - 5
Solution
(a) 5x + 3x = 15 + 32x = 8x
(b) We should note that z = 1 # z because of the multiplicative identity. So
z + 7z - 5 = 1z + 7z - 5
= 11 + 72z - 5
= 8z - 5
Real Numbers and Algebraic Expressions
Quick
15. A
is a number or the product of a number and one or more variables
raised to a power.
In Problems 16–19, simplify each expression by combining like terms.
16. 4x - 9x
17. -2x2 + 13x2
18. -5x - 3x + 6 - 3
19. 6x - 10x - 4y + 12y
Many times we must rearrange the terms in an algebraic expression using the commutative properties of real numbers to simplify an algebraic expression.
EXAMPLE 5 Combining Like Terms Using the Associative
and Distributive Properties
Simplify each algebraic expression.
(a) 13y2 + 8 - 4y2 + 3
(b) 12z + 5 + 5z - 8z - 2
Solution
(a) We rearrange the terms so that we can use the Distributive Property.
13y2 + 8 - 4y2 + 3 = 13y2 - 4y2 + 8 + 3
Use the Distributive Property “in reverse” = 113 - 42y2 + 8 + 3
Combine like terms: = 9y2 + 11
(b) Again, we rearrange terms before using the Distributive Property.
12z + 5 + 5z - 8z - 2 = 12z + 5z - 8z + 5 - 2
Use the Distributive Property “in reverse” = 112 + 5 - 82z + 5 - 2
Combine like terms: = 9z + 3
Quick
In Problems 20–22, simplify each expression by combining like terms.
20. 10y - 3 + 5y + 2
21. 0.5x2 + 1.3 + 1.8x2 - 0.4
22. 4z + 6 - 8z - 3 - 2z
We also may need to first remove parentheses using the Distributive Property
before we can collect like terms. This is consistent with the Rule for Order of Operations as expressions must be simplified by multiplying before adding or subtracting.
EXAMPLE 6 Combining Like Terms Using the Distributive Property
Simplify each algebraic expression.
(b) x + 4 - 21x + 32
(a) 51x - 32 - 2x
Solution
(a) We first must use the Distributive Property to remove the parentheses.
⁄
51x - 32 - 2x = 5x - 15 - 2x
= 5x - 2x - 15
Combine like terms: = 3x - 15
Rearrange terms:
(b) Distribute -2 to remove the parentheses.
⁄
⁄
CHAPTER R
⁄
42
x + 4 - 21x + 32 = x + 4 - 2x - 6
Rearrange terms: = x - 2x + 4 - 6
Combine like terms: = -x - 2
Section R.5 Algebraic Expressions 43
EXAMPLE 7 Combining Like Terms Using the Distributive Property
Simplify each algebraic expression.
(a) 51x - 32 - 17x - 42
(b)
1
8x - 3
14x + 32 +
2
5
Solution
(a) When there is a subtraction sign in front of parentheses, we treat the
subtraction sign as -1.
51x - 32 - 17x - 42 = 51x - 32 - 1 # 17x - 42
Use the Distributive Property
to remove the parentheses: = 5x - 15 - 7x + 4
Rearrange terms: = 5x - 7x - 15 + 4
Combine like terms: = - 2x - 11
8x - 3
a
1 #
1
=
a and rewrite
= (8x - 3).
b
b
5
5
1
8x - 3
1
1
14x + 32 +
= 14x + 32 + 18x - 32
2
5
2
5
(b) We use the fact that
Use the Distributive Property
to remove parentheses:
Simplify:
Rearrange terms; rewrite fractions
with least common denominator:
Combine like terms:
1#
1
1
1
4x + # 3 + # 8x - # 3
2
2
5
5
3
8
3
= 2x +
+ x 2
5
5
=
10
8
15
6
x + x +
5
5
10
10
18
9
=
x +
5
10
=
Quick
In Problems 23–28, simplify each expression by combining like terms.
23. 31x - 2) + x
24. 51y + 32 - 10y - 4
25. 31z + 42 - 213z + 12
26. - 41x - 22 - 12x + 42
1
15x + 5
5x - 1
5x + 9
27. 16x + 42 28.
+
2
5
3
2
4
Determine the Domain of a Variable
When working with an algebraic expression, the variable may only be allowed to take
on values from a certain set of numbers. For example, because division by zero is not
defined, any value of the variable that causes division by zero must be excluded from
1
the set of numbers that the variable can take on. So, in the expression , the variable x
x
cannot take on the value 0, because this would cause division by 0, which is not defined.
DEFINITION
The set of values that a variable may assume is called the domain of the variable.
EXAMPLE 8 Determining the Domain of a Variable
Determine which of the following numbers are in the domain of the variable x for the
4
.
expression
x + 3
(a) x = 3
(b) x = 0
(c) x = - 3
44
CHAPTER R Real Numbers and Algebraic Expressions
Solution
We need to determine whether the value of the variable causes division by 0. That
is, we need to determine if the value of the variable causes x + 3 to equal 0. If it does,
we exclude it from the domain.
(a) When x = 3, we have that x + 3 = 3 + 3 = 6, so 3 is in the domain of
the variable.
(b) When x = 0, we have that x + 3 = 0 + 3 = 3, so 0 is in the domain of the
variable.
(c) When x = - 3, we have that x + 3 = - 3 + 3 = 0, so -3 is NOT in the
domain of the variable.
Quick
29. What is the domain of a variable?
Work Smart: Study Skills
Selected problems in the exercise
sets are identified by a symbol.
For extra help, view the worked
solutions to these problems on the
book’s CD Lecture Series.
In Problems 30–32, determine which of the following numbers are in the domain of
the variable.
(a) x = 2
(b) x = 0
(c) x = 4
(d) x = - 3
x
2
x + 3
31.
30.
32. 2
x
+
3
x - 4
x + x - 6
R.5 EXERCISES
1–32. are the Quick
s that follow each EXAMPLE
In Problems 45–60, evaluate each expression for the given value of
the variable. See Objective 2.
Building Skills
45. 4x + 3 for x = 2
In Problems 33–44, express each English phrase using an algebraic
expression. See Objective 1.
46. -5x + 1 for x = 3
33. The sum of 5 and a number x
34. The difference of 10 and a number y
35. The product of 4 and a number z
36. The ratio of a number x and 5
47. x2 + 5x - 3 for x = - 2
48. y2 - 4y + 5 for y = 3
49. -z2 + 4 for z = 5
50. -2z2 + z + 3 for z = - 4
51.
2w
for w = 3
w2 + 2w + 1
52.
4z + 3
for z = 3
z2 - 4
53.
v2 + 2v + 1
for v = 5
v2 + 3v + 2
42. Three times a number z increased by the quotient of
z and 8
54.
2x2 + 5x + 2
for x = 3
x2 + 5x + 6
43. The quotient of some number y and 3 increased by
the product of 6 and some number x
55. ƒ 5x - 4 ƒ for x = - 5
37. A number y decreased by 7
38. A number z increased by 30
39. Twice the sum of a number t and 4
40. The sum of a number x and 5 divided by 10
41. Three less than five times a number x
44. Twice some number x decreased by the ratio of a
number y and 3
56. ƒ x2 - 6x + 1 ƒ for x = 2
57. ƒ 15y + 10 ƒ for y = -
3
5