1.1 Use the Language of Algebra

By the end of this section, you will be able to:

  • Find factors, prime factorizations, and least common multiples
  • Use variables and algebraic symbols
  • Simplify expressions using the order of operations
  • Evaluate an expression
  • Identify and combine like terms
  • Translate an English phrase to an algebraic expression

Find Factors, Prime Factorizations, and Least Common Multiples

The numbers 2, 4, 6, 8, 10, 12 are called multiples of 2. A multiple of 2 can be written as the product of a counting number and 2.

Multiples of 2: 2 times 1 is 2, 2 times 2 is 4, 2 times 3 is 6, 2 times 4 is 8, 2 times 5 is 10, 2 times 6 is 12 and so on.

Similarly, a multiple of 3 would be the product of a counting number and 3.Multiples of 3: 3 times 1 is 3, 3 times 2 is 6, 3 times 3 is 9, 3 times 4 is 12, 3 times 5 is 15, 3 times 6 is 18 and so on.

We could find the multiples of any number by continuing this process.

Counting Number123456789101112
Multiples of 224681012141618202224
Multiples of 3369121518212427303336
Multiples of 44812162024283236404448
Multiples of 551015202530354045505560
Multiples of 661218243036424854606672
Multiples of 771421283542495663707784
Multiples of 881624324048566472808896
Multiples of 9918273645546372819099108

Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 3 into 15, we get a counting number. In fact, 15÷3 is 5, so 15 is 5⋅3.

If we were to look for patterns in the multiples of the numbers 2 through 9, we would discover the following divisibility tests:

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.

Since 8⋅9=72, we say that 8 and 9 are factors of 72. When we write 72=8⋅9, we say we have factored 72.

8 times 9 is 72. 8 and 9 are factors. 72 is the product.

Other ways to factor 72 are 1⋅72,2⋅36,3⋅24,4⋅18, and 6⋅12. The number 72 has many factors: 1,2,3,4,6,8,9,12,18,24,36, and 72.

The counting numbers from 2 to 20 are listed in the table with their factors. Make sure to agree with the “prime” or “composite” label for each!This table has three columns, 19 rows and a header row. The header row labels each column: number, factors and prime or composite. The values in each row are as follows: number 2, factors 1, 2, prime; number 3, factors 1, 3, prime; number 4, factors 1, 2, 4, composite; number 5, factors, 1, 5, prime; number 6, factors 1, 2, 3, 6, composite; number 7, factors 1, 7, prime; number 8, factors 1, 2, 4, 8, composite; number 9, factors 1, 3, 9, composite; number 10, factors 1, 2, 5, 10, composite; number 11, factors 1, 11, prime; number 12, factors 1, 2, 3, 4, 6, 12, composite; number 13, factors 1, 13, prime; number 14, factors 1, 2, 7, 14, composite; number 15, factors 1, 3, 5, 15, composite; number 16, factors 1, 2, 4, 8, 16, composite; number 17, factors 1, 17, prime; number 18, factors 1, 2, 3, 6, 9, 18, composite; number 19, factors 1, 19, prime; number 20, factors 1, 2, 4, 5, 10, 20, composite.

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.

A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful in many topics in this course.

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime. Otherwise it is easy to lose track of the prime numbers.

If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.

Solution

Step 1 is to find two factors whose product is 48 and use these numbers to create two branches. The two branches originating from 48 are formed by the factors 2 and 24.
Step 2 is to circle the prime factor. This completes that branch. In this case, 2 is circled as it is prime.
Step 3 is to treat the composite factor as a product, break it into two more factors and continue the process. 24 is not prime. It is broken into 4 and 6. 4 and 6 are not prime. 4 is broken into its factors 2 and 2, both of which are circled. 6 is not prime. It is broken into factors 2 and 3, both of which are circled.
Step 4 is to write the original composite number as the product of all the circled primes. 48 is 2 into 2 into 2 into 2 into 3.

One of the reasons we look at primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.

To find the least common multiple of two numbers we will use the Prime Factors Method. Let’s find the LCM of 12 and 18 using their prime factors.

Notice that the prime factors of 12 (2⋅2⋅3) and the prime factors of 18 (2⋅3⋅3) are included in the LCM (2⋅2⋅3⋅3). So 36 is the least common multiple of 12 and 18.

By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.

Use Variables and Algebraic Symbols

In algebra, we use a letter of the alphabet to represent a number whose value may change. We call this a variable and letters commonly used for variables are x,y,a,b,c.

A number whose value always remains the same is called a constant.

To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using. There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We’ll list the symbols used to indicate these operations below.

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

On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols “<” and “>”.

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help identify an expression, which can be made up of number, a variable, or a combination of numbers and variables using operation symbols. We will introduce three types of grouping symbols now.

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

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. A sentence has a subject and a verb. In algebra, we have expressions and equations.

Simplify Expressions Using the Order of Operations

To simplify an expression 

to do all the math possible. For example, to simplify 4⋅2+1 we would first multiply 4⋅2 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:

By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.

For example, consider the expression 4+3⋅7. Some students simplify this getting 49, by adding 4+3 and then multiplying that result by 7. Others get 25, by multiplying 3⋅7 first and then adding 4.

The same expression should give the same result. So mathematicians established some guidelines that are called the order of operations.

Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase “Please Excuse My Dear Aunt Sally”.

It’s good that “My Dear” goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.

Similarly, “Aunt Sally” goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.

Evaluate an Expression

In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

Identify and Combine Like Terms

Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.

Think of the coefficient as the number in front of the variable. The coefficient of the term 3x is 3. When we write x, the coefficient is 1, since x=1⋅x.

Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are like terms.

Look at the following 6 terms. Which ones seem to have traits in common?

We say,

7 and 4 are like terms.

5x and 3x are like terms.

n2 and 9n2 are like terms.

Translate an English Phrase to an Algebraic Expression

We listed many operation symbols that are used in algebra. Now, we will use them to translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. Table 1.2 summarizes them.

OperationPhraseExpression
Additiona plus b
the sum of a and b
a increased by b
b more than a
the total of a and b
b added to a
a+b
Subtractiona minus b
the difference of a and b
a decreased by b
b less than a
b subtracted from a
a−b
Multiplicationa times b
the product of a and b
twice a
a⋅b,ab,a(b),(a)(b)
2a
Divisiona divided by b
the quotient of a and b
the ratio of a and b
b divided into a
a÷b,a/b,ab,ba

Table1.2

Look closely at these phrases using the four operations:

The sum of a and b, the difference of a and b, the product of a and b, the quotient of a and b.

Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.

We look carefully at the words to help us distinguish between multiplying a sum and adding a product.

Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples.



The expressions in the next example will be used in the typical coin mixture problems we will see soon.

Section 1.1 Exercises

Practice Makes Perfect

Identify Multiples and Factors

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.

  1. 84
  2. 96
  3. 896
  4. 942
  5. 22,335
  6. 39,075
    Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

  1. 86
  2. 78
  3. 455
  4. 400
  5. 432
  6. 627
    In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.
  7. 8, 12
  8. 12, 16
  9. 28, 40
  10. 84, 90
  11. 55, 88
  12. 60, 72
    Simplify Expressions Using the Order of Operations

In the following exercises, simplify each expression.

  1. 23−12÷(9−5)
  2. 32−18÷(11−5)
  3. 2+8(6+1)
  4. 4+6(3+6)
  5. 20÷4+6(5−1)
  6. 33÷3+4(7−2)
  7. 3(1+9⋅6)−42
  8. 5(2+8⋅4)−72
  9. 2[1+3(10−2)]
  10. 5[2+4(3−2)]
  11. 8+2[7−2(5−3)]−32
  12. 10+3[6−2(4−2)]−24
    Evaluate an Expression

In the following exercises, evaluate the following expressions.

  1. When x=2,
    ⓐ x6
    ⓑ 4x
    ⓒ 2×2+3x−7
  2. When x=3,
    ⓐ x5
    ⓑ 5x
    ⓒ 3×2−4x−8
  3. When x=4,y=1
    x2+3xy−7y2
  4. When x=3,y=2
    6×2+3xy−9y2
  5. When x=10,y=7
    (x−y)2
  6. When a=3,b=8
    a2+b2
    Simplify Expressions by Combining Like Terms

In the following exercises, simplify the following expressions by combining like terms.

  1. 7x+2+3x+4
  2. 8y+5+2y−4
  3. 10a+7+5a−2+7a−4
  4. 7c+4+6c−3+9c−1
  5. 3×2+12x+11+14×2+8x+5
  6. 5b2+9b+10+2b2+3b−4
    Translate an English Phrase to an Algebraic Expression

In the following exercises, translate the phrases into algebraic expressions.

43.
ⓐ the difference of 5×2 and 6xy
ⓑ the quotient of 6y2 and 5x
ⓒ Twenty-one more than y2
ⓓ 6x less than 81×2
44.
ⓐ the difference of 17×2 and 5xy
ⓑ the quotient of 8y3 and 3x
ⓒ Eighteen more than a2;
ⓓ 11b less than 100b2
45.
ⓐ the sum of 4ab2 and 3a2b
ⓑ the product of 4y2 and 5x
ⓒ Fifteen more than m
ⓓ 9x less than 121×2
46.
ⓐ the sum of 3x2y and 7xy2
ⓑ the product of 6xy2 and 4z
ⓒ Twelve more than 3×2
ⓓ 7×2 less than 63×3
47.
ⓐ eight times the difference of y and nine
ⓑ the difference of eight times y and 9
48.
ⓐ seven times the difference of y and one
ⓑ the difference of seven times y and 1
49.
ⓐ five times the sum of 3x and y
ⓑ the sum of five times 3x and y
50.
ⓐ eleven times the sum of 4×2 and 5x
ⓑ the sum of eleven times 4×2 and 5x

  1. Eric has rock and country songs on his playlist. The number of rock songs is 14 more than twice the number of country songs. Let c represent the number of country songs. Write an expression for the number of rock songs.
  2. The number of women in a Statistics class is 8 more than twice the number of men. Let m represent the number of men. Write an expression for the number of women.
  3. Greg has nickels and pennies in his pocket. The number of pennies is seven less than three times the number of nickels. Let n represent the number of nickels. Write an expression for the number of pennies.
  4. Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.

Writing Exercises

55

Explain in your own words how to find the prime factorization of a composite number.56. 

Why is it important to use the order of operations to simplify an expression?57

Explain how you identify the like terms in the expression 8a2+4a+9−a2−1.8a2+4a+9−a2−1.58. 

Explain the difference between the phrases “4 times the sum of x and y” and “the sum of 4 times x and y”.

Self Check

ⓐ Use this checklist to evaluate your mastery of the objectives of this section.This table has 4 columns, 7 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: identify multiples and apply divisibility tests, find prime factorizations and least common multiples, use variables and algebraic symbols, simplify expressions using the order of operations, evaluate an expression, identify and combine like terms, translate English phrases to algebraic expressions. The remaining columns are blank.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no – I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.