Expressions:

Elizabeth Pople

4 Expressions:

Topics Covered^[1]

In case you missed something in class, or just want to review a specific topic covered in this Module, here is a list of topics covered:

Simplify Expressions using Order of Operations
Identify Terms, Coefficients, Like Terms
Evaluate Algebraic Expressions
Simplify Expressions by Combining Like Terms
Identify Expressions and Equations
Key Concepts

Simplify Expressions Using the Order of Operations^[2]

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

Example 1

Example 2

Steps

Algebraic

Steps

Algebraic

Some students say it simplifies to 49.

4 + 3 · 7

Some students say it simplifies to 25.

4 + 3 · 7

Since 4 + 3 gives 7.

7 · 7

Since 3 · 7 is 21.

21 + 4

And 7 · 7 is 49.

49

And 21 + 4 makes 25.

25

Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.

When simplifying mathematical expressions perform the operations in the following order:

1. Parentheses and other Grouping Symbols

Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

2. Exponents

Simplify all expressions with exponents.

3. Multiplication and Division

Perform all multiplication and division in order from left to right. These operations have equal priority.

4. Addition and Subtraction

Perform all addition and subtraction in order from left to right. These operations have equal priority.

This process is also known as P.E.M.D.A.S.

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.

Order of Operations
Please	Parentheses
Excuse	Exponents
My Dear	Multiplication and Division
Aunt Sally	Addition and Subtraction

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.

Doing the Manipulative Mathematics activity Game of 24 will give you practice using the order of operations.

Try it! – Order of Operations

Simplify:

a. 18 ÷ 9 · 2

b. 18 · 9 ÷ 2

Solution

Steps

Algebraic

a.

Are there any parentheses? No.

Are there any exponents? No.

Is there any multiplication or division? Yes.

Multiply and divide from left to right. Divide.

Multiply.

Steps

Algebraic

b.

Are there any parentheses? No.

Are there any exponents? No.

Is there any multiplication or division? Yes.

Multiply and divide from left to right.

Multiply.

Divide.

Try it! – Order of Operations

Simplify: 18 ÷ 6 + 4(5 − 2).

Solution

Steps

Algebraic

Parentheses? Yes, subtract first.

Exponents? No.

Multiplication or division? Yes.

Divide first because we multiply and divide left to right.

Any other multiplication or division? Yes.

Multiply.

Any other multiplication or division? No.

Any addition or subtraction? Yes.

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

Try it! – Order of Operations

Simplify: 5 + 23 + 3[6 − 3(4 − 2)].

Solution

Steps

Algebraic

5 + 23 + 3[6 − 3(4 − 2)].

Are there any parentheses (or other grouping symbol)? Yes.

Focus on the parentheses that are inside the brackets.

5 + 23 + 3[6 − 3(4 − 2)].

Subtract.

5 + 23 + 3[6 − 3(2)].

Continue inside the brackets and multiply.

5 + 23 + 3[6 − 6].

Continue inside the brackets and subtract.

5 + 23 + 3[0].

The expression inside the brackets requires no further simplification.

Are there any exponents? Yes.

Simplify exponents.

5 + 23 + 3[0].

Is there any multiplication or division? Yes.

Multiply.

5 + 8 + 3[0].

Is there any addition or subtraction? Yes.

Add.

5 + 8 + 0.

Add.

13 + 0.

13.

Try it! – Order of Operations

Simplify: 23 + 34 ÷ 3 − 52.

Solution

Steps

Algebraic

23 + 34 ÷ 3 − 52.

If an expression has several exponents, they may be simplified in the same step.

Simplify exponents.

23 + 34 ÷ 3 − 52.

Divide.

8 + 81 ÷ 3 − 25.

Add.

8 + 27 − 25.

Subtract.

35 − 25.

10.

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Identify Terms, Coefficients, and Like Terms^[3]

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are 7, y, 5x², 9a, and 13xy.

The constant that multiplies the variable(s) in a term is called the coefficient. We can 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. The table below gives the coefficients for each of the terms in the left column.

Term	Coefficient
7	7
9a	9
y	1
$5x^2$	5

Try it! – Identifying Terms, Coefficients, and Like Terms

Identify each term in the expression 9b + 15 $x^2$ + a + 6. Then identify the coefficient of each term.

Solution

The expression has four terms. They are 9b, $15x^2$ , a, and 6.

The coefficient of 9b is 9.

The coefficient of 15 $x^2$ is 15.

Remember that if no number is written before a variable, the coefficient is 1. So, the coefficient of a is 1.

The coefficient of a constant is the constant, so the coefficient of 6 is 6.

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

5x, 7, $n^2$ , 4, 3x, 9 $n^2$

Which of these terms are like terms?

The terms 7 and 4 are both constant terms.
The terms 5x and 3x are both terms with x.
The terms $n^2$ and 9 $n^2$ both have $n^2$ .

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms 5x, 7, $n^2$ , 4, 3x, 9 $n^2$ ,

7 and 4 are like terms.

5x and 3x are like terms.

$n^2$ and 9 $n^2$ are like terms.

Like Terms

Terms that are either constants or have the same variables with the same exponents are like terms.

Try it! – Identify the Like Terms

Identify the like terms:

a. $y^3$ , $7x^2$ , 14, 23, $4y^3$ , $9x$ , $5x^2$ b. $4x^2 + 2x + 5x^2+6x+40x+8xy$

Solution

a. $y^3$ , $7x^2$ , 14, 23, $4y^3$ , $9x$ , $5x^2$

Look at the variables and exponents. The expression contains $y^3$ , $x^2$ , $x$ , and constants.
The terms $y^3$ and 4 $y^3$ are like terms because they both have $y^3$ .
The terms 7 $x^2$ and 5 $x^2$ are like terms because they both have $x^2$ .
The terms 14 and 23 are like terms because they are both constants.
The term 9x does not have any like terms in this list since no other terms have the variable x raised to the power of 1.

b. $4x^2 + 2x + 5x^2+6x+40x+8xy$

Look at the variables and exponents. The expression contains the terms $4x^2$ , 2x, $5x^2$ , 6x, 40x, and 8xy
The terms $4x^2$ and $5x^2$ are like terms because they both have $x^2$ .
The terms $2x$ , $6x$ , and $40x$ are like terms because they all have $x$ .
The term 8xy has no like terms in the given expression because no other terms contain the two variables xy.

Evaluate Algebraic Expressions^[4]

In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

Try it! – Evaluating Algebraic Expressions

Evaluate x + 7 when

a. x = 3

b. x = 12

Solution

a. To evaluate, substitute 3 for x in the expression, and then simplify.

Steps

Algebraic

x + 7

Substitute.

3 + 7

Add.

10

When x = 3, the expression x + 7 has a value of 10.

b. To evaluate, substitute 12 for x in the expression, and then simplify.

Steps

Algebraic

x + 7

Substitute.

12 + 7

Add.

19

When x = 12, the expression x + 7 has a value of 19.

Notice that we got different results for parts a. and b. even though we started with the same expression. This is because the values used for x were different. When we evaluate an expression, the value varies depending on the value used for the variable.

Try it! – Evaluating Algebraic Expressions

Evaluate 9x − 2, when

a. x = 5
b. x = 1

Solution

Remember ab means a times b, so 9x means 9 times x.

a. To evaluate the expression when x = 5, we substitute 5 for x, and then simplify.

Steps

Algebraic

9x – 2

Substitute 5 for x

9 * 5 – 2

Multiply.

45 – 2

Subtract.

43

b. To evaluate the expression when x = 1, we substitute 1 for x, and then simplify.

Steps

Algebraic

9x – 2

Substitute 1 for x

9(1) – 2

Multiply.

9 – 2

Subtract.

7

Notice that in part a. that we wrote 9 ⋅ 5 and in part b. we wrote 9(1). Both the dot and the parentheses tell us to multiply.

Try it! – Evaluating Algebraic Expressions

Evaluate $x^2$ when x = 1

Solution

We substitute 10 for x, and then simplify the expression.

Steps

Algebraic

$x^2$

Substitute 10 for x

$10^2$

Use the definition of exponent.

10 * 10

Multiply.

100

When x=10, the expression $x^2$ has a value of 100.

Try it! – Evaluating Algebraic Expressions

Evaluate $2^x$ when x = 5.

Solution

In this expression, the variable is an exponent.

Steps

Algebraic

Use the definition of exponent.

Multiply.

When x = 5, the expression $2^x$ has a value of 32.

Try it! – Evaluating Algebraic Expressions

Evaluate 3x + 4y − 6 when x = 10 and y = 2.

Solution

This expression contains two variables, so we must make two substitutions.

Steps

Algebraic

Multiply.

Add and subtract left to right.

When x = 10 and y = 2, the expression 3x + 4y − 6 has a value of 32.

Try it! – Evaluating Algebraic Expressions

Evaluate $2x^2$ + 3x + 8 when x = 4.

Solution

We need to be careful when an expression has a variable with an exponent. In this expression, 2 $x^2$ means 2 ⋅ x ⋅ x and is different from the expression $</em>(2x)^2<em>$ , which means 2x ⋅ 2x.

Steps

Algebraic

2 $x^2$ + 3x + 8

Substitute 4 for each x

2[latex](4)^2[\latex] + 3(4) + 8

Simplify 42.

2(16) + 3(4) + 8

Multiply.

32 + 12 + 8

Add.

52

Simplify Expressions by Combining Like Terms^[5]

We can simplify an expression by combining the like terms. What do you think 3x+6x would simplify to? If you thought 9x, you would be right!

We can see why this works by writing both terms as addition problems.

The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x. Add the coefficients and keep the same variable. It doesn’t matter what x is. If you have 3 of something and add 6 more of the same thing, the result is 9 of them. For example, 3 oranges plus 6 oranges is 9 oranges. We will discuss the mathematical properties behind this later.

The expression 3x+6x has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So, we could rearrange the following expression before combining like terms.

The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together. Now it is easier to see the like terms to be combined.

How To:

Identify like terms.
Rearrange the expression so like terms are together.
Add the coefficients of the like terms.

Try it! – Combining Like Terms

Simplify the expression: 3x + 7 + 4x + 5.

Solution

Steps

Algebraic

3x + 7 + 4x + 5

Identify the like terms.

Rearrange the expression, so the like terms are together.

Add the coefficients of the like terms.

The original expression is simplified to…

7x + 12

Identify Expressions and Equations^[6]

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:

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

x ÷ y

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:

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.

Expression

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

Equation

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

Try it! – Identifying Expressions and Equations

Determine if each is an expression or an equation:

a. 16 − 6 = 10

b. 4 · 2 + 1

c. x ÷ 25

d. y + 8 = 40

Solution

Problem Number

Explanation

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.

Key Concepts

Algebraic Notation

Understanding Algebraic Notation
Operation	Notation	Say:	The result is…
Addition	a + b	a plus b	the sum of a and b
Multiplication	a · b, (a)(b), (a)b, a(b)	a times b	The product of a and b
Subtraction	a − b	a minus b	the difference between a and b
Division	a ÷ b, a/b, ab, ba	a divided by b	The quotient of a and b

Algebraic inequality symbols

Understanding Algebraic Notation
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

Equality Symbol
- a = b is read as a is equal to b
- The symbol = is called the equal sign.
Order of Operations When simplifying mathematical expressions perform the operations in the following order:
- Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
- Exponents: Simplify all expressions with exponents.
- Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
- Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.
Combine like terms.
1. Identify like terms.
2. Rearrange the expression so like terms are together.
3. Add the coefficients of the like terms

Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction ↵
Section material derived from Openstax Prealgebra: The Language of Algebra-Use the Language of Algebra ↵
Section material derived from Openstax Prealgebra: The Language of Algebra-Evaluate Simplify and Translate Expressions ↵
Section material derived from Openstax Prealgebra: The Language of Algebra-Evaluate Simplify and Translate Expressions ↵
Section material derived from Openstax Prealgebra: The Language of Algebra-Evaluate Simplify and Translate Expressions ↵
Section material derived from Openstax Prealgebra: The Language of Algebra-Use the Language of Algebra ↵

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Simplify Expressions Using the Order of Operations[2]

Access Additional Online Resources

Identify Terms, Coefficients, and Like Terms[3]

Evaluate Algebraic Expressions[4]

Simplify Expressions by Combining Like Terms[5]

How To:

Identify Expressions and Equations[6]

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Simplify Expressions Using the Order of Operations^[2]

Identify Terms, Coefficients, and Like Terms^[3]

Evaluate Algebraic Expressions^[4]

Simplify Expressions by Combining Like Terms^[5]

Identify Expressions and Equations^[6]