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Mathematics LibreTexts

10.1: Use the Language of Algebra (Part 1)

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Learning Objectives
  • Use variables and algebraic symbols
  • Identify expressions and equations
  • Simplify expressions with exponents
  • Simplify expressions using the order of operations
Be Prepared!

Before you get started, take this readiness quiz.

  1. Add: 43+69. If you missed this problem, review Example 1.2.8.
  2. Multiply: (896)201. If you missed this problem, review Example 1.4.11.
  3. Divide: 7,263÷9. If you missed this problem, review Example 1.5.8.

Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is 20 years old and Alex is 23, so Alex is 3 years older than Greg. When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The 3 years between them always stays the same, so the age difference is the constant.

In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age g. Then we could use g+3 to represent Alex’s age. See Table 10.1.1.

Table 10.1.1
Greg’s age Alex’s age
12 15
20 23
35 38
g g + 3

Letters are used to represent variables. Letters often used for variables are x,y,a,b, and c.

Definition: Variables and Constants

A variable is a letter that represents a number or quantity whose value may change.

A constant is a number whose value always stays the same.

To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In Whole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.

Table 10.1.2
Operation Notation Say: The result is...
Addition a + b a plus b the sum of a and b
Subtraction a − b a minus b the difference of a and b
Multiplication a • b, (a)(b), (a)b, a(b) a times b the product of a and b
Division a ÷ b, a / b, ab, b¯)a a divided by b the quotient of a and b

In algebra, the cross symbol, ×, is not used to show multiplication because that symbol may cause confusion. Does 3xy mean 3×y (three times y) or 3xy (three times x times y)? To make it clear, use or parentheses for multiplication.

We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.

The sum of 5 and 3 means add 5 plus 3, which we write as 5+3.

The difference of 9 and 2 means subtract 9 minus 2, which we write as 92.

The product of 4 and 8 means multiply 4 times 8, which we can write as 48.

The quotient of 20 and 5 means divide 20 by 5, which we can write as 20÷5.

Example 10.1.1: translate to words

Translate from algebra to words:

  1. 12+14
  2. (30)(5)
  3. 64÷8
  4. xy

Solution

12 + 14
12 plus 14
the sum of twelve and fourteen
(30)(5)
30 times 5
the product of thirty and five
64 ÷ 8
64 divided by 8
the quotient of sixty-four and eight
x − y
x minus y
the difference of x and y
exercise 10.1.1

Translate from algebra to words.

  1. 18+11
  2. (27)(9)
  3. 84÷7
  4. pq
Answer a

18 plus 11; the sum of eighteen and eleven

Answer b

27 times 9; the product of twenty-seven and nine

Answer c

84 divided by 7; the quotient of eighty-four and seven

Answer d

p minus q; the difference of p and q

exercise 10.1.2

Translate from algebra to words.

  1. 4719
  2. 72÷9
  3. m+n
  4. (13)(7)
Answer a

47 minus 19; the difference of forty-seven and nineteen

Answer b

72 divided by 9; the quotient of seventy-two and nine

Answer c

m plus n; the sum of m and n

Answer d

13 times 7; the product of thirteen and seven

When two quantities have the same value, we say they are equal and connect them with an equal sign.

Definition: Equality Symbol

a=b is read a is equal to b

The symbol = is called the equal sign.

An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that b is greater than a, it means that b is to the right of a on the number line. We use the symbols "<" and ">" for inequalities.

Definition: Inequality

a<b is read a is less than b

a is to the left of b on the number line

The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.

a>b is read a is greater than b

a is to the right of b on the number line

The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.

The expressions a<b and a>b can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general, a<b is equivalent to b>a. For example, 7<11 is equivalent to 11>7. a>b is equivalent to b<a. For example, 17>4 is equivalent to 4<17.

When we write an inequality symbol with a line under it, such as ab, it means a<b or a=b. We read this a is less than or equal to b. Also, if we put a slash through an equal sign, , it means not equal.

We summarize the symbols of equality and inequality in Table 10.1.3.

Table 10.1.3
Algebraic Notation Say
a = b a is equal to b
a ≠ b a is not equal to b
a < b a is less than b
a > b a is greater than b
a ≤ b a is less than or equal to b
a ≥ b a is greater than or equal to b
Definition: Symbols < and >

The symbols < and > each have a smaller side and a larger side.

smaller side < larger side

larger side > smaller side

The smaller side of the symbol faces the smaller number and the larger faces the larger number.

Example 10.1.2: translate to words

Translate from algebra to words:

  1. 2035
  2. 11153
  3. 9>10÷2
  4. x+2<10

Solution

20 ≤ 35
20 is less than or equal to 35
11 ≠ 15 − 3
11 is not equal to 15 minus 3
9 > 10 ÷ 2
9 is greater than 10 divided by 2
x + 2 < 10
x plus 2 is less than 10
exercise 10.1.3

Translate from algebra to words.

  1. 1427
  2. 1928
  3. 12>4÷2
  4. x7<1
Answer a

fourteen is less than or equal to twenty-seven

Answer b

nineteen minus two is not equal to eight

Answer c

twelve is greater than four divided by two

Answer d

x minus seven is less than one

exercise 10.1.4

Translate from algebra to words.

  1. 1915
  2. 7=125
  3. 15÷3<8
  4. y3>6
Answer a

nineteen is greater than or equal to fifteen

Answer b

seven is equal to twelve minus five

Answer c

fifteen divided by three is less than eight

Answer d

y minus three is greater than six

Example 10.1.3: translate

The information in Figure 10.1.1 compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol =, in each expression to compare the fuel economy of the cars.

This table has two rows and six columns. The first column is a header column and it labels each row The first row is labeled “Car” and the second “Fuel economy (mpg)”. To the right of the ‘Car’ row are the labels: “Prius”, “Mini Cooper”, “Toyota Corolla”, “Versa”, “Honda Fit”. Each of these columns contains an image of the labeled car model. To the right of the “Fuel economy (mpg)” row are the algebraic equations: the letter p, the equals symbol, the number forty-eight; the letter m, the equals symbol, the number twenty-seven; the letter c, the equals symbol, the number twenty-eight; the letter v, the equals symbol, the number twenty-six; and the letter f, the equals symbol, the number twenty-seven.

Figure 10.1.1: (credit: modification of work by Bernard Goldbach, Wikimedia Commons)

  1. MPG of Prius _____ MPG of Mini Cooper
  2. MPG of Versa _____ MPG of Fit
  3. MPG of Mini Cooper _____ MPG of Fit
  4. MPG of Corolla _____ MPG of Versa
  5. MPG of Corolla_____ MPG of Prius

Solution

  MPG of Prius____MPG of Mini Cooper
Find the values in the chart. 48____27
Compare. 48 > 27
  MPG of Prius > MPG of Mini Cooper
  MPG of Versa____MPG of Fit
Find the values in the chart. 26____27
Compare. 26 < 27
  MPG of Versa < MPG of Fit
  MPG of Mini Cooper____MPG of Fit
Find the values in the chart. 27____27
Compare. 27 = 27
  MPG of Mini Cooper = MPG of Fit
  MPG of Corolla____MPG of Versa
Find the values in the chart. 28____26
Compare. 28 > 26
  MPG of Corolla > MPG of Versa
  MPG of Corolla____MPG of Prius
Find the values in the chart. 28____48
Compare. 28 < 48
  MPG of Corolla < MPG of Prius
exercise 10.1.5

Use Figure 10.1.1 to fill in the appropriate symbol, =, <, or >.

  1. MPG of Prius_____MPG of Versa
  2. MPG of Mini Cooper_____ MPG of Corolla
Answer a

>

Answer b

<

exercise 10.1.6

Use Figure 10.1.1 to fill in the appropriate symbol, =, <, or >.

  1. MPG of Fit_____ MPG of Prius
  2. MPG of Corolla _____ MPG of Fit
Answer a

<

Answer b

<

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. Table 10.1.4 lists three of the most commonly used grouping symbols in algebra.

Table 10.1.4
Common Grouping Symbols
parentheses ( )
brackets [ ]
braces { }

Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

8(148)213[2+4(98)]24÷132[1(65)+4]

Identify Expressions and Equations

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb.

In algebra, we have expressions and equations. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:

Table 10.1.5
Expression Words Phrase
3 + 5 3 plus 5 the sum of three and five
n - 1 n minus one the difference of n and one
6 • 7 6 times 7 the product of six and seven
xy x divided by y the quotient of x and y

Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:

Table 10.1.6
Equation Sentence
3 + 5 = 8 The sum of three and five is equal to eight.
n − 1 = 14 n minus one equals fourteen.
6 • 7 = 42 The product of six and seven is equal to forty-two.
x = 53 x is equal to fifty-three.
y + 9 = 2y − 3 y plus nine is equal to two y minus three.
Definition: Expressions and Equations

An expression is a number, a variable, or a combination of numbers and variables and operation symbols.

An equation is made up of two expressions connected by an equal sign.

Example 10.1.4: expression or equation

Determine if each is an expression or an equation:

  1. 166=10
  2. 42+1
  3. x÷25
  4. y+8=40

Solution

(a) 16 − 6 = 10 This is an equation—two expressions are connected with an equal sign.
(b) 4 • 2 + 1 This is an expression—no equal sign.
(c) x ÷ 25 This is an expression—no equal sign.
(d) y + 8 = 40 This is an equation—two expressions are connected with an equal sign.
exercise 10.1.7

Determine if each is an expression or an equation:

  1. 23+6=29
  2. 737
Answer a

equation

Answer b

expression

exercise 10.1.8

Determine if each is an expression or an equation:

  1. y÷14
  2. x6=21
Answer a

expression

Answer b

equation

Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify 42+1 we’d first multiply 42 to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

42+18+19

Suppose we have the expression 222222222. We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write 222 as 23 and 222222222 as 29. In expressions such as 23, the 2 is called the base and the 3 is called the exponent. The exponent tells us how many factors of the base we have to multiply.

The image shows the number two with the number three, in superscript, to the right of the two. The number two is labeled as “base” and the number three is labeled as “exponent”.

means multiply three factors of 2

We say 23 is in exponential notation and 222 is in expanded notation.

Definition: Exponential Notation

For any expression an, a is a factor multiplied by itself n times if n is a positive integer.

At the top of the image is the letter a with the letter n, in superscript, to the right of the a. The letter a is labeled as “base” and the letter n is labeled as “exponent”. Below this is the letter a with the letter n, in superscript, to the right of the a set equal to n factors of a.

The expression an is read a to the nth power.

For powers of n=2 and n=3, we have special names. a2 is read as "a squared" a3 is read as "a cubed" Table 10.1.7 lists some examples of expressions written in exponential notation.

Table 10.1.7
Exponential Notation In Words
72 7 to the second power, or 7 squared
53 5 to the third power, or 5 cubed
94 9 to the fourth power
125 12 to the fifth power
Example 10.1.5: exponential form

Write each expression in exponential form:

  1. 16161616161616
  2. 99999
  3. xxxx
  4. aaaaaaaa

Solution

(a) The base 16 is a factor 7 times. 167
(b) The base 9 is a factor 5 times. 95
(c) The base x is a factor 4 times. x4
(d) The base a is a factor 8 times. a8
exercise 10.1.9

Write each expression in exponential form: 4141414141

Answer

415

exercise 10.1.10

Write each expression in exponential form: 777777777

Answer

79

Example 10.1.6: expanded form

Write each exponential expression in expanded form:

  1. 86
  2. x5

Solution

  1. The base is 8 and the exponent is 6, so 86 means 888888
  2. The base is x and the exponent is 5, so x5 means xxxxx
exercise 10.1.11

Write each exponential expression in expanded form:

  1. 48
  2. a7
Answer a

44444444

Answer b

aaaaaaa

exercise 10.1.12

Write each exponential expression in expanded form:

  1. 88
  2. b6
Answer a

88888888

Answer b

bbbbbb

To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

Example 10.1.7: simplify

Simplify: 34.

Solution

Expand the expression. 34 = 3 • 3 • 3 • 3
Multiply left to right. 9 • 3 • 3 = 27 • 3
Multiply. 81
exercise 10.1.13

Simplify:

  1. 53
  2. 17
Answer a

125

Answer b

1

exercise 10.1.14

Simplify:

  1. 72
  2. 05
Answer a

49

Answer b

0

Contributors and Attributions

  • Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.

This page titled 10.1: Use the Language of Algebra (Part 1) is shared under a not declared license and was authored, remixed, and/or curated by OpenStax.

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