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Language of Algebra
Language of Algebra
Using letters to represent numbers is a fundamental characteristic of
algebra.
In this chapter, letters will represent the numbers of arithmetic. As you become
familiar with the symbols and terminology of algebra, operations on arithmetic
numbers will be reviewed.
SOME SYMBOLS FOR OPERATIONS
OBJECTIVES
Upon completing this section you should be able to:
1. Identify the operation symbols of arithmetic.
2. Perform these operations on whole numbers, fractions, and decimals.
Remember that when adding or subtracting fractions, you must find a common
denominator.
When adding or subtracting decimals, align the decimal points.
In multiplying fractions multiply numerators to obtain the numerator of the
product and multiply denominators to obtain the denominator of the product.
The symbol + is used to indicate a sum. a + b (read as "a plus b") is used to
indicate the sum of a and b.
The symbol - is used to indicate the operation of subtraction, a - b (read as
"a minus b") indicates the difference of the numbers a and b.
The operation of multiplication (the product of two numbers) can be represented
in several ways. You should become familiar with each of these.
1. The symbol X is often used to represent multiplication, a X b (read as "a
times b") represents the product of the two numbers a and b.
The symbol X is not usually used with letters in algebra because it can
be confused with the letter X.
2. a * b (read as "a times b") is also used to represent the product of the
numbers a and b.
The symbol * is not usually used with numbers because it could be confused with
a decimal point.
3. If two letters or a number and a letter are written together without an
operation sign, multiplication is indicated.
The process of writing two letters or a number and a letter together without
an operation sign is called juxtaposition. This method of indicating
multiplication cannot be used for two numbers, since 52 does not mean "5 times
2."
4. When two sets of parentheses have no operation sign between them,
multiplication is indicated, (a)(6) means "a times b." Also, a letter or number
preceding a parenthesis with no sign in between indicates multiplication. a{b)
means "a times 6" and 7(x) means "7 times x."
4.2 * 5.1 would be a confusing way to indicate the product of 4.2 and 5.1.
Notice that the sum or product of two numbers does not change if we reverse the
order in which they are written. For example,
3 + 5 = 8; also 5 + 3 = 8
4 * 6 = 24 and 6 * 4 = 24. This characteristic is called the commutative
property.
Does this property hold for subtraction?
(Consider 7 - 2 and 2 - 7.)
(4.2)(5.1) is a much better way to indicate the product of these two numbers
than the ways shown earlier.
Recall that division is the inverse operation of multiplication. Therefore,
Complete the following:
Is division commutative?
This information will be helpful to remember as you work the problems in the
following exercise set.
Remember to reduce fractions.
Of these several methods for indicating a product, the last two are most
commonly used.
The quotient of two numbers, a and b, can be written in two ways,and
- both indicate "the number a is divided by
the number b."
The fractional representation for division is most commonly used in algebra.
Recall from arithmetic that a fraction whose numerator is greater than the
denominator is called an improper fraction.
Therefore,
is an improper fraction.
Recall also that an improper fraction can be written as a mixed number.
Likewise, a mixed number may be written as an improper fraction.
Improper fraction form is generally a more convenient form to work with. If
division or multiplication is to be performed, then improper fraction form is
almost a necessity.
Some problems may involve only one
operation, while others may require two or more.
It is important to know the symbols for the various operations. It is also
important to know what operation or series of operations to use in solving a
specific We meet these problems every
day.
Solution The operation is addition. We would indicate the total cost as
$1.89 + $.65 + $.59.
Then adding, we obtain
$1.89 + $.65 + $.59 = $3.13.
The operation of subtraction often occurs in problems that involve making
change, balancing checkbooks, and discounting prices.
In preparation for later work get used to using parentheses to indicate
multiplication.
Solution The operation in this example is subtraction. We need to subtract
the cost of the order from $5.00.
$5.00 - $3.13 = $1.87
Solution This is a multiplication problem. We multiply the hourly wage by the
number of hours worked.
5 X $4.50
0r 5($4.50) = $22.50
Solution This is a division problem. We must divide the cost of the boat by
the number of people.
$4,200 -5- 3 = $1,400
Often problems involve more than one arithmetic operation.
We could also write this as
Solution This problem involves both multiplication and subtraction. We first
need to find 20% of $529. To do this we multiply $529 by the decimal equivalent
of 20%.
.20($529) = $105.80
Remember, to change a percent to a decimal move the decimal point
two places to the left
This is the amount of the discount. To find the sale price we now , must subtract $105.80 from the regular price.
$529.00 - $105.80 = $423.20
GROUPING SYMBOLS
OBJECTIVES Upon completing this section you should be able to:
1. Perform operations in the correct order indicated by grouping symbols.
2. Perform operations in a specific order when grouping symbols are not used.
Parentheses ( ), brackets [ ], and braces j } are all used as grouping
symbols in algebra. A number expression enclosed in a grouping symbol is treated
as if it were a single number.
Example 1 12 - (3 + 4) indicates that the
sum (3 + 4) is subtracted from 12. 12 - (3 + 4) = 12 - 7 = 5. Note that without
parentheses, 12 - 3 + 4 = 13.
Example 2 7(5 + x) indicates that the sum (5
+ x) is multiplied by 7.
In general, operations within the grouping symbols are performed first. Brackets
and braces can be used instead of parentheses.
12 - [3 + 4] and 12 - {3 + 4} mean the same as 12 - (3 + 4).
Some applications may require the use of grouping symbols.
Example 3 A bank account has a balance of
$567.19. Checks are written for $18.50, $24.95, and $129.40. What is the new
balance?
Parentheses are most commonly used when no other grouping symbols are involved.
Solution We need to add the amounts of the three checks and subtract the sum
from the original balance. We therefore write
$567.19 - ($18.50 + $24.95 + $129.40) = $567.19 - $172.85 = $394.34.
The first step is to simplify what is inside the parentheses.
As mentioned earlier, an expression may have a different value if parentheses
are not used. We saw that
12 - (3 + 4) = 5,
but it was stated that without parentheses
12 - 3 + 4 = 13.
This is due to the following rule:
If an expression without grouping symbols contains only additions and
subtractions, these operations are performed in order from left to right.
If an expression contains operations other than just addition and
subtraction, we use the following rule:
If no grouping symbols occur in an expression, multiplication and division
are performed from left to right and then addition and subtraction from left to
right.
Notice that when adding three numbers, the order in which they are added does
not affect the sum.
3 + (8 + 4) = 3 + 12 = 15 (3 + 8) + 4 = 11 + 4 = 15 This characteristic is
called the associative property. Is multiplication associative?
Sometimes more than one set of grouping symbols is needed in an expression.
When this occurs we use brackets or braces along with parentheses for
clarification. For instance, 5 + [7 - (2 + 1)] could be written using only
parentheses, but 5 + (7 - (2 + 1)) is not as clear at first glance. Therefore,
we alternate the symbols to avoid confusion. To evaluate such an expression, we
use the following rule:
When simplifying an expression containing grouping symbols within grouping
symbols, remove the innermost set of symbols first.
In using this rule do not try to do more than one operation at a time.
Example 10 To evaluate 5 + [7 - (2 + 1)] we
simplify the innermost set of symbols, namely (2 + 1). Writing (2 + 1) as 3, we
now obtain
Remember that the different symbols are all used for the same purpose-to group
numbers.
Again, make sure you do only one operation at a time. This is very important in
order to avoid errors.
Notice that we start with (5 - 2) since it is the innermost set of symbols.
ALGEBRAIC EXPRESSIONS
When letters are used to represent numbers, they are called literal numbers.
If a number expression contains one or more literal numbers, it is called a
literal expression or algebraic expression.
OBJECTIVES Upon completing this section you should be able to:
1. Write literal expressions involving operations of arithmetic.
2. Identify terms and factors of an expression.
Example 1 x + 5 represents "the sum of x and
5." a - b represents "the difference of a and b." 4y represents "the product of
4 and y."
When we refer to the difference of two numbers, the second number is always
subtracted from the first.
Example 2 Write a literal expression for
"the sum of x and 5, divided by 7."
Solution
(x + 5) represents "the sum of x and 5," so we have (x + 5) / 7. Notice that
parentheses are necessary if we write the above as (x + 5) / 7.
Notice that the comma is very important in example 2. Without the comma the
statement would read as "the sum of x and 5 divided by 7." The literal
expression would be
Example 3 Write an algebraic expression
for "the difference of a and b, multiplied by the sum of a and b."
x + (5 -4- 7) or .V + -y .
Solution The difference of a and b is (o - b)\ the sum of a and b is (a + b), so
we have (a - b)(a + b).
Notice again the importance of the use of the comma in example 3.
Example 4 Express c - (a + b) in words.
Solution Since (a + b) is in parentheses, it is "the sum of a and b." So we
write c - (a + b) as "the difference of c, and the sum of a and b"
When an algebraic expression is composed of parts connected by addition or
subtraction signs, these parts are called the terms of the expression.
Example 5 a + b has two terms.
In a + b
the terms are a and b.
Example 6 2x + 5y + 3 has three terms:
2x, 5y and 3.
When an algebraic expression is composed of parts to be multiplied, these
parts are called the factors of the expression.
Example 7 ab has two factors, a and b.
It is very important to be able to distinguish between terms and factors. Rules that apply to terms will not, in general, apply to
factors. When naming terms or factors, it is
necessary to regard the entire expression.
From now on through all of algebra you
will be using the words term and factor. Make sure you understand the
definitions.
Example 8 The expression 3x + 2y + 7 has
three terms. Note that a term may contain factors (for example, the first term
of this expression, 3x, contains two factors)-but the entire expression is made
up of terms.
It should be noted that there are more than just four factors in the expression
5xyz. The obvious factors are 5, x, y, and z. But 5x is also a factor. Other
factors are 5xy, xy, 5yz, xz, 5z, and so on.
An exponent is sometimes referred to as a "power." For example, 5^{3}
could be referred to as "five to the third power."
Unless parentheses are used, the exponent only affects the factor directly
preceding it.
Example 9 5xyz is one term made up of
factors.
Example 10 3(a + b) is one term having two
factors. Note here that the factor (a + b) has two terms, but the entire
expression is made up of factors.
An exponent is a numeral used to indicate how many times a factor is to be used
in a product.
An exponent is usually written as a smaller (in size) numeral slightly above and
to the right of the factor affected by the exponent.
Stopped here
Note the difference between 2x^{3} and (2x)^{3}. From the use
of parentheses as grouping symbols we see that
2x^{3} means 2(x)(x)(x), whereas (2x)^{3} means (2x)(2x)(2x) or
8x^{3}.
In an expression such as 5x^{4}
5 is the numerical coefficient or coefficient,
x is the base,
4 is the exponent.
Many students make the error of multiplying the base by the exponent.
For example, they will say
3^{4} = 12 instead of the correct answer
3^{4}= (3)(3)(3)(3) = 81.
Note that only the base is affected by the exponent.
Example 13 In the expression ax^{3}
a is the coefficient,
x is the base,
3 is the exponent.
ax^{3} means (a)(x)(x)(x).
Example 14 (ax)^{3}
Solution 1 is the coefficient (understood), ax is the base (because of the
parentheses), 3 is the exponent.
(ax)^{3} means (ax)(ax)(ax).
When we write a literal number such as x, it will be understood that the
coefficient is one and the exponent is one. This can be very important in many
operations.
x means 1x^{1}.
If an expression has grouping symbols, the operations therein are performed
first.
The coefficient, 1, is understood and usually we do not bother to write it in
the expression.
Recall the order of operations.
EVALUATING LITERAL EXPRESSIONS
OBJECTIVES Upon completing this section you should be able to:
1. Substitute numbers for letters in literal expressions.
2. Evaluate the expression once the substitutions have been made.
The principle of substitution states that any quantity may be
substituted for its equal in any process. This principle is used extensively in
algebra and we will use it here to evaluate literal expressions. In this process
we substitute numbers for letters and find a numerical value.
Example 1 Evaluate x + 3 if x = 5.
We substitute 5 for x in the expression obtaining
5 + 3 = 8.
Example2 Evaluate 4a - 1 if a = 3.
We substitute 3 for a in the expression obtaining
4(3) -1 = 12-1
= 11.
Remember that in a literal expression the letters are merely holding a place for
various numbers that may be assigned to them. For that reason, these letters are
sometimes called place holders or variables.
To find the perimeter of a rectangle we need to add up all four sides, but
since the opposite sides are equal, we only need to double the length and double
the width, and then find their sum.
The height of a triangle is called the altitude of the triangle. It is the
perpendicular distance from the base to the opposite corner, which is called the
vertex.
One of the most common uses of evaluating literal expressions is in working with
formulas.
Example 7 The perimeter (distance around) of
a rectangle is found by using the formula P = 2l + 2w, where l represents the
length and w represents the width.
If the length of a rectangle is 10 and the width is 6, we may find the
perimeter by substituting 10 for l and 6 for w.
P = 2(10) + 2(6)
P = 2(10) + 2(6) = 20+12 = 32.
Example 8 The area of a triangle is found
by using the formula
where b represents the base of the triangle and h represents the height.
If we want to find the area of a triangle where the base is 10 and the height is
8, we would let b = 10 and h = 8. Substituting these values in the formula, we
obtain
COMBINING LIKE TERMS
OBJECTIVES Upon completing this section, you should be able to:
1. Identity like terms.
2. Combine like terms.
In the preceding sections we have presented symbols used in algebraic
expressions. We now proceed to the operations of addition and subtraction on
certain of these expressions.
Like terms are terms that have exactly the same literal factors.
Example 1 5x and 3x are like terms since
they have the same literal factors, (x is the literal factor in each term.)
Example 2 5x and 3y are not like terms since
the literal factors are not the same, (x and y are the literal factors.)
Example 3 3x^{2}y and x^{2}y
are like terms since they have the same literal
factors (x^{2} and y).
Example 4 3x^{2}y^{2} and
2xy are not like terms. They have different literal factors. (Note that x^{2}
and x are not the same.)
Only like terms can be combined. When two terms are like terms, combine
the numerical coefficients to obtain the coefficient of the like common factors.
Example 5 If we add the like terms 5x +
3x, we combine the coefficients 5 and 3 obtaining
5x + 3x = (5 + 3)x = 8x.
Many students make the mistake of interchanging capital and lowercase letters
in an expression. In algebra a capital "A" and a lowercase "a" are considered
just as different as x and y. Therefore, terms such as 5A and 3a would not be
considered like terms. Be careful. Be consistent with the letters you use.
Notice that the numerical coefficient of x is 1 and in this example must be
added to the 7 of lx.
Example 11 12 ab + 4 ba - 6 ab = 10 ab
By the commutative law of multiplication 4ba = 4ab.
Key Words
Parentheses ( ), brackets [ ], and braces{ } are all
used as grouping symbols in algebra.
Literal numbers are letters used to represent numbers.
An algebraic expression or literal expression is a number
expression containing one or more literal numbers.
Terms are those parts of an algebraic expression that are being added or
subtracted. Factors are those parts of an algebraic expression that are being
multiplied.
An exponent indicates the number of times a factor is to be used in a
product.
The principle of substitution states that any quantity may be substituted
for its equal in any process.
Like terms are terms that have exactly the same literal factors.
Procedures
If no grouping symbols occur in an expression, multiplication and
division are performed from left to right, and then addition and subtraction
from left to right.
When simplifying an expression containing grouping symbols within
grouping symbols, remove the innermost set of symbols first.
To evaluate a literal expression substitute the given values for the
literal numbers and perform the indicated operations.
To combine like terms combine the numerical coefficients and use this
result as the coefficient of the common literal factors.