Foundations

3 Use the Language of Algebra

Learning Objectives

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

  • 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

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, The Language of Algebra.

Use Variables and Algebraic Symbols

Suppose this year Greg is 20 years old and Alex is 23. You know that 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 variables and the 3 is a constant. The ages change (“vary”) but the 3 years between them always stays the same (“constant”). Since Greg’s age and Alex’s age will always differ by 3 years, 3 is the constant.

In algebra, we use letters of the alphabet to represent variables. So if we call Greg’s age g, then we could use g+3 to represent Alex’s age. See (Figure).

Greg’s age Alex’s age
12 15
20 23
35 38
{\text{g}} {\text{g}}+3

The letters used to represent these changing ages are called variables. The letters most commonly used for variables are x, y, a, b, and c.

Variable

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

Constant

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

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 ((Figure)). You’ll probably recognize some of them.

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,ab,\left(a\right)\left(b\right),\left(a\right)b,a\left(b\right) a times b the product of a and b
Division a÷b,a\text{/}b,\frac{a}{b},b\overline{)a} a divided by b the quotient of a and b, a is called the dividend, and b is called the divisor

We perform these operations on two numbers. When translating from symbolic form to English, or from English to symbolic form, pay attention to the words “of” and “and.”

  • The difference of 9 and 2 means subtract 9 and 2, in other words, 9 minus 2, which we write symbolically as 9-2.
  • The product of 4 and 8 means multiply 4 and 8, in other words 4 times 8, which we write symbolically as 4·8.

In algebra, the cross symbol, ×, is not used to show multiplication because that symbol may cause confusion. Does 3xy mean 3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}y (‘three times y’) or 3·x·y (three times x times y)? To make it clear, use · or parentheses for multiplication.


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

Equality Symbol

a=b is read “a is equal to b

The symbol ``='' is called the 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 “>.”

Inequality
\begin{array}{}\\ \\ a<b\phantom{\rule{0.2em}{0ex}}\text{is read ``}a\phantom{\rule{0.2em}{0ex}}\text{is less than}\phantom{\rule{0.2em}{0ex}}b\text{''}\hfill \\ a\phantom{\rule{0.2em}{0ex}}\text{is to the left of}\phantom{\rule{0.2em}{0ex}}b\phantom{\rule{0.2em}{0ex}}\text{on the number line}\hfill \end{array}

No Alt Text

\begin{array}{c}a>b\phantom{\rule{0.2em}{0ex}}\text{is read ``}a\phantom{\rule{0.2em}{0ex}}\text{is greater than}\phantom{\rule{0.2em}{0ex}}b\text{''}\hfill \\ a\phantom{\rule{0.2em}{0ex}}\text{is to the right of}\phantom{\rule{0.2em}{0ex}}b\phantom{\rule{0.2em}{0ex}}\text{on the number line}\hfill \end{array}

No Alt Text

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

Inequality Symbols Words
a\ne b a is not equal to b
a < b a is less than b
a\le b a is less than or equal to b
a > b a is greater than b
a\ge b a is greater than or equal to b

Translate from algebra into English:

17\le 268\ne 17-312>27÷3y+7<19

Solution

17\le 26
17 is less than or equal to 26

8\ne 17-3
8 is not equal to 17 minus 3

12>27÷3
12 is greater than 27 divided by 3

y+7<19
y plus 7 is less than 19

Translate from algebra into English:

14\le 2719-2\ne 812>4÷2x-7<1

14 is less than or equal to 27 19 minus 2 is not equal to 8 12 is greater than 4 divided by 2 x minus 7 is less than 1

Translate from algebra into English:

19\ge 157=12-515÷3<8y+3>6

19 is greater than or equal to 15 7 is equal to 12 minus 5 15 divided by 3 is less than 8 y plus 3 is greater than 6

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help to make clear which expressions are to be kept together and separate from other expressions. We will introduce three types now.

Grouping Symbols
\begin{array}{cccc}\text{Parentheses}\hfill & & & \phantom{\rule{0.15em}{0ex}}\left(\right)\hfill \\ \text{Brackets}\hfill & & & \phantom{\rule{0.15em}{0ex}}\left[\right]\hfill \\ \text{Braces}\hfill & & & \left\{\right\}\hfill \end{array}

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

\begin{array}{ccccccc}\hfill 8\left(14-8\right)\hfill & & & \hfill 21-3\left[2+4\left(9-8\right)\right]\hfill & & & \hfill 24÷\left\{13-2\left[1\left(6-5\right)+4\right]\right\}\hfill \end{array}


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.

Expression

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

An expression is like an English phrase. Here are some examples of expressions:

Expression Words English 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
\frac{x}{y} x divided by y the quotient of x and y

Notice that the English 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.

Equation

An equation is two expressions connected by an equal sign.

Here are some examples of equations.

Equation English 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.

Determine if each is an expression or an equation:

2\left(x+3\right)=104\left(y-1\right)+1x÷25y+8=40

Solution
2\left(x+3\right)=10 This is an equation—two expressions are connected with an equal sign.
4\left(y-1\right)+1 This is an expression—no equal sign.
x÷25 This is an expression—no equal sign.
y+8=40 This is an equation—two expressions are connected with an equal sign.

Determine if each is an expression or an equation: 3\left(x-7\right)=27 5\left(4y-2\right)-7.

equation expression

Determine if each is an expression or an equation: {y}^{3}÷14 4x-6=22.

expression equation

Suppose we need to multiply 2 nine times. We could write this as 2·2·2·2·2·2·2·2·2. This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write 2·2·2 as {2}^{3} and 2·2·2·2·2·2·2·2·2 as {2}^{9}. In expressions such as {2}^{3}, the 2 is called the base and the 3 is called the exponent. The exponent tells us how many times we need to multiply the base.

The number two is shown with a superscipted number three to the right of it. an arrow is drawn to the number two and labeled “base” while another arrow is drawn to the superscripted three and labeled “exponent”. This means multiply 2 by itself, three times, as in 2 times 2 times 2.

We read {2}^{3} as “two to the third power” or “two cubed.”

We say {2}^{3} is in exponential notation and 2·2·2 is in expanded notation.

Exponential Notation

{a}^{n} means multiply a by itself, n times.

a is shown with a superscripted n to the right of it. an arrow is drawn to a and labeled “base” while another arrow is drawn to the superscripted n and labeled “exponent”. Written below this is the equation a superscript n equals a times a times ellipsis times a, implying an indeterminate number of “a”s being multiplied. a bracket is drawn below the “a”s being multiplied and labeled “n factors”.

The expression {a}^{n} is read a to the {n}^{th} power.

While we read {a}^{n} as “a to the {n}^{th} power,” we usually read:

  • {a}^{2}a squared”
  • {a}^{3}a cubed”

We’ll see later why {a}^{2} and {a}^{3} have special names.

(Figure) shows how we read some expressions with exponents.

Expression In Words
{7}^{2} 7 to the second power or 7 squared
{5}^{3} 5 to the third power or 5 cubed
{9}^{4} 9 to the fourth power
{12}^{5} 12 to the fifth power

Simplify: {3}^{4}.

Solution
{3}^{4}
Expand the expression. 3·3·3·3
Multiply left to right. 9·3·3
Multiply. 27·3
Multiply. 81

Simplify: {5}^{3} {1}^{7}.

125 1

Simplify: {7}^{2} {0}^{5}.

49 0

Simplify Expressions Using the Order of Operations

To simplify an expression means to do all the math possible. For example, to simplify 4·2+1 we’d 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:

\begin{array}{c}\hfill 4·2+1\hfill \\ \hfill 8+1\hfill \\ \hfill 9\hfill \end{array}

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

Simplify an Expression

To simplify an expression, do all operations in the expression.

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

If you simplify this expression, what do you get?

Some students say 49,

\begin{array}{cccccc}& & & & & \hfill \phantom{\rule{0.8em}{0ex}}4+3·7\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}4+3\phantom{\rule{0.2em}{0ex}}\text{gives}\phantom{\rule{0.2em}{0ex}}7.\hfill & & & & & \hfill \phantom{\rule{0.8em}{0ex}}7·7\hfill \\ \text{And}\phantom{\rule{0.2em}{0ex}}7·7\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}49.\hfill & & & & & \hfill \phantom{\rule{0.8em}{0ex}}49\hfill \end{array}

Others say 25,

\begin{array}{cccccc}& & & & & \hfill 4+3·7\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}3·7\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}21.\hfill & & & & & \hfill 4+21\hfill \\ \text{And}\phantom{\rule{0.2em}{0ex}}21+4\phantom{\rule{0.2em}{0ex}}\text{makes}\phantom{\rule{0.2em}{0ex}}25.\hfill & & & & & \hfill 25\hfill \end{array}

Imagine the confusion in our banking system if every problem had several different correct answers!

The same expression should give the same result. So mathematicians early on established some guidelines that are called the Order of Operations.

Perform the Order of Operations.
  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.
Doing the Manipulative Mathematics activity “Game of 24” give you practice using 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.”

\begin{array}{cccc}\mathbf{\text{P}}\text{arentheses}\hfill & & & \mathbf{\text{P}}\text{lease}\hfill \\ \mathbf{\text{E}}\text{xponents}\hfill & & & \mathbf{\text{E}}\text{xcuse}\hfill \\ \mathbf{\text{M}}\text{ultiplication}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{D}}\text{ivision}\hfill & & & \mathbf{\text{M}}\text{y}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{D}}\text{ear}\hfill \\ \mathbf{\text{A}}\text{ddition}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{S}}\text{ubtraction}\hfill & & & \mathbf{\text{A}}\text{unt}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{S}}\text{ally}\hfill \end{array}

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.

Let’s try an example.

Simplify: 4+3·7 \left(4+3\right)·7.

Solution


.
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply first. .
Add. .
.


.
Are there any parentheses? Yes. .
Simplify inside the parentheses. .
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply. .

Simplify: 12-5·2 \left(12-5\right)·2.

2 14

Simplify: 8+3·9 \left(8+3\right)·9.

35 99

Simplify: 18÷6+4\left(5-2\right).

Solution
Parentheses? Yes, subtract first. 18÷6+4\left(5-2\right)
\phantom{\rule{0.2em}{0ex}}.
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. .

Simplify: 30÷5+10\left(3-2\right).

16

Simplify: 70÷10+4\left(6-2\right).

23

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

Simplify: 5+{2}^{3}+3\left[6-3\left(4-2\right)\right].

Solution
.
Are there any parentheses (or other grouping symbol)? Yes.
Focus on the parentheses that are inside the brackets. .
Subtract. .
Continue inside the brackets and multiply. .
Continue inside the brackets and subtract. .
The expression inside the brackets requires no further simplification.
Are there any exponents? Yes. .
Simplify exponents. .
Is there any multiplication or division? Yes.
Multiply. .
Is there any addition or subtraction? Yes.
Add. .
Add. .

Simplify: 9+{5}^{3}-\left[4\left(9+3\right)\right].

86

Simplify: {7}^{2}-2\left[4\left(5+1\right)\right].

1

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.

Evaluate an Expression

To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.

Evaluate 7x-4, when x=5 and x=1.

Solution


. .
.
Multiply. .
Subtract. .


. .
.
Multiply. .
Subtract. .

Evaluate 8x-3, when x=2 and x=1.

13 5

Evaluate 4y-4, when y=3 and y=5.

8 16

Evaluate x=4, when {x}^{2} {3}^{x}.

Solution


{x}^{2}
. .
Use definition of exponent.  4·4
Simplify. 16


{3}^{\text{x}}
. .
Use definition of exponent. 3·3·3·3
Simplify. 81

Evaluate x=3, when {x}^{2} {4}^{x}.

964

Evaluate x=6, when {x}^{3} {2}^{x}.

21664

Evaluate 2{x}^{2}+3x+8 when x=4.

Solution
2{x}^{2}+3x+8
. .
Follow the order of operations. 2\left(16\right)+3\left(4\right)+8
32+12+8
52

Evaluate 3{x}^{2}+4x+1 when x=3.

40

Evaluate 6{x}^{2}-4x-7 when x=2.

9

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.

Term

A term is a constant, or the product of a constant and one or more variables.

Examples of terms are 7,y,5{x}^{2},9a,\text{and}\phantom{\rule{0.2em}{0ex}}{b}^{5}.

The constant that multiplies the variable is called the coefficient.



Coefficient

The coefficient of a term is the constant that multiplies the variable in a term.

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.

Identify the coefficient of each term: 14y 15{x}^{2} a.

Solution

The coefficient of 14y is 14.

The coefficient of 15{x}^{2} is 15.

The coefficient of a is 1 since a=1\phantom{\rule{0.2em}{0ex}}a.

Identify the coefficient of each term: 17x 41{b}^{2} z.

14 41 1

Identify the coefficient of each term: 9p 13{a}^{3} {y}^{3}.

9 13 1

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

\begin{array}{cccccccccccccccc}5x\hfill & & & 7\hfill & & & {n}^{2}\hfill & & & 4\hfill & & & 3x\hfill & & & 9{n}^{2}\hfill \end{array}

The 7 and the 4 are both constant terms.

The 5x and the 3x are both terms with x.

The {n}^{2} and the 9{n}^{2} are both terms with {n}^{2}.

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

  • 7 and 4 are like terms.
  • 5x and 3x are like terms.
  • {x}^{2} and 9{x}^{2} are like terms.
Like Terms

Terms that are either constants or have the same variables raised to the same powers are called like terms.

Identify the like terms: {y}^{3}, 7{x}^{2}, 14, 23, 4{y}^{3}, 9x, 5{x}^{2}.

Solution

{y}^{3} and 4{y}^{3} are like terms because both have {y}^{3}; the variable and the exponent match.

7{x}^{2} and 5{x}^{2} are like terms because both have {x}^{2}; the variable and the exponent match.

14 and 23 are like terms because both are constants.

There is no other term like 9x.

Identify the like terms: 9, 2{x}^{3}, {y}^{2}, 8{x}^{3}, 15, 9y, 11{y}^{2}.

9 and 15, {y}^{2} and 11{y}^{2}, 2{x}^{3} and 8{x}^{3}

Identify the like terms: 4{x}^{3}, 8{x}^{2}, 19, 3{x}^{2}, 24, 6{x}^{3}.

19 and 24, 8{x}^{2} and 3{x}^{2}, 4{x}^{3} and 6{x}^{3}

Adding or subtracting terms forms an expression. In the expression 2{x}^{2}+3x+8, from (Figure), the three terms are 2{x}^{2},3x, and 8.

Identify the terms in each expression.

  1. 9{x}^{2}+7x+12
  2. 8x+3y
Solution

The terms of 9{x}^{2}+7x+12 are 9{x}^{2}, 7x, and 12.

The terms of 8x+3y are 8x and 3y.

Identify the terms in the expression 4{x}^{2}+5x+17.

4{x}^{2},5x,17

Identify the terms in the expression 5x+2y.

5x, 2y

If there are like terms in an expression, you can simplify the expression by combining the like terms. What do you think 4x+7x+x would simplify to? If you thought 12x, you would be right!

\begin{array}{c}\hfill 4x+7x+x\hfill \\ \hfill x+x+x+x\phantom{\rule{1em}{0ex}}+x+x+x+x+x+x+x\phantom{\rule{1em}{0ex}}+x\hfill \\ \hfill 12x\hfill \end{array}

Add the coefficients and keep the same variable. It doesn’t matter what x is—if you have 4 of something and add 7 more of the same thing and then add 1 more, the result is 12 of them. For example, 4 oranges plus 7 oranges plus 1 orange is 12 oranges. We will discuss the mathematical properties behind this later.

Simplify: 4x+7x+x.

Add the coefficients. 12x

How To Combine Like Terms

Simplify: 2{x}^{2}+3x+7+{x}^{2}+4x+5.

Solution

Three lines of instructions are listed in a column on the left side of the image while four algebraic expressions are listed on the right. The first line of instruction on the left says: “Step 1. Identify like terms.” Across from step 1 in the right column is the algebraic expression: 2x squared plus 3x plus 7 plus x squared plus 4x plus 5. One line down on the right, the same algebraic expression is repeated, except each of the terms appears in one of three colors to illustrate that these are like terms: 2x squared and x squared appear as red, illustrating that these are like terms; 3x and 4x appear as blue, illustrating that these are also like terms; 7 and 5 appear as green, illustrating that these are like terms as well.The second line of instruction on the left says: “Step 2. Rearrange the expression so the like terms are together. Across from step 2 in the right column is the original algebraic expression with terms reordered so that like terms appear side by side: 2x squared plus x2, both written in red, plus 3x plus 4x, both written n blue, plus 7 plus 5, both written in green.The third line of instruction on the left says: “Step 3. Combine like terms.” Across from step 3 in the right column is the algebraic expression with like terms combined: 3x squared in red, plus 7x in blue, plus 12 in green.

Simplify: 3{x}^{2}+7x+9+7{x}^{2}+9x+8.

10{x}^{2}+16x+17

Simplify: 4{y}^{2}+5y+2+8{y}^{2}+4y+5.

12{y}^{2}+9y+7

Combine Like Terms.
  1. Identify like terms.
  2. Rearrange the expression so like terms are together.
  3. Add or subtract the coefficients and keep the same variable for each group of like terms.

Translate an English Phrase to an Algebraic Expression

In the last section, we listed many operation symbols that are used in algebra, then we translated expressions and equations into English phrases and sentences. Now we’ll reverse the process. We’ll translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. (Figure) summarizes them.

Operation Phrase Expression
Addition a 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
Subtraction a minus b
the difference of a and b
a decreased by b
b less than a
b subtracted from a
a-b
Multiplication a times b
the product of a and b
twice a
a·b,ab,a\left(b\right),\left(a\right)\left(b\right)

2a
Division a divided by b
the quotient of a and b
the ratio of a and b
b divided into a
a÷b,a\text{/}b,\frac{a}{b},b\overline{)a}

Look closely at these phrases using the four operations:

Four phrases are shown. The first reads “the sum of a and b”, where the words “of” and “and” are written in red. The second reads “the difference of a and b”, where the words “of” and “and” are written in red. The third reads “the product of a and b”, where the words “of” and “and” are written in red. The fourth reads “the quotient of a and b”, where the words “of” and “and” are written in red.

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

Translate each English phrase into an algebraic expression: the difference of 17x and 5 the quotient of 10{x}^{2} and 7.

Solution
  1. The key word is difference, which tells us the operation is subtraction. Look for the words of and and to find the numbers to subtract.
    The phrase “the difference of 17x and 5”, where the words “of” and “and” are written in red, is written above the phrase “17 x minus 5”. a final phrase written below reads “17 x, minus sign, 5”.

  2. The key word is “quotient,” which tells us the operation is division.

The phrase “the quotient of 10x squared and 7”, where the words “of” and “and” are written in red, is written above the expression “divide 10x squared by 7”. The expression written below reads “10x squared, division sign,v7”.

This can also be written 10{x}^{2}\text{/}7\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\frac{10{x}^{2}}{7}.

Translate the English phrase into an algebraic expression: the difference of 14{x}^{2} and 13 the quotient of 12x and 2.

14{x}^{2}-1312x÷2

Translate the English phrase into an algebraic expression: the sum of 17{y}^{2} and 19 the product of 7 and y.

17{y}^{2}+19 7y

How old will you be in eight years? What age is eight more years than your age now? Did you add 8 to your present age? Eight “more than” means 8 added to your present age. How old were you seven years ago? This is 7 years less than your age now. You subtract 7 from your present age. Seven “less than” means 7 subtracted from your present age.

Translate the English phrase into an algebraic expression: Seventeen more than y Nine less than 9{x}^{2}.

Solution
  1. The key words are more than. They tell us the operation is addition. More than means “added to.”
    \begin{array}{c}\hfill \text{Seventeen more than}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill \text{Seventeen added to}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill y+17\hfill \end{array}
  2. The key words are less than. They tell us to subtract. Less than means “subtracted from.”
    \begin{array}{}\\ \\ \\ \hfill \phantom{\rule{5em}{0ex}}\text{Nine less than}\phantom{\rule{0.2em}{0ex}}9{x}^{2}\hfill \\ \hfill \phantom{\rule{5em}{0ex}}\text{Nine subtracted from}\phantom{\rule{0.2em}{0ex}}9{x}^{2}\hfill \\ \hfill \phantom{\rule{5em}{0ex}}9{x}^{2}-9\hfill \end{array}

Translate the English phrase into an algebraic expression: Eleven more than x Fourteen less than 11a.

x+1111a-14

Translate the English phrase into an algebraic expression: 13 more than z 18 less than 8x.

z+138x-18

Translate the English phrase into an algebraic expression: five times the sum of m and n the sum of five times m and n.

Solution

There are two operation words—times tells us to multiply and sum tells us to add.

  1. Because we are multiplying 5 times the sum we need parentheses around the sum of m and n, \left(m+n\right). This forces us to determine the sum first. (Remember the order of operations.)
    \begin{array}{}\\ \\ \hfill \text{five times the sum of}\phantom{\rule{0.2em}{0ex}}m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\hfill \\ \hfill 5\phantom{\rule{0.2em}{0ex}}\left(m+n\right)\hfill \end{array}
  2. To take a sum, we look for the words “of” and “and” to see what is being added. Here we are taking the sum of five times m and n.
    \begin{array}{c}\hfill \text{the sum}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}\text{five times}\phantom{\rule{0.2em}{0ex}}m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\hfill \\ \hfill 5m+n\hfill \end{array}

Translate the English phrase into an algebraic expression: four times the sum of p and q the sum of four times p and q.

4\left(p+q\right)4p+q

Translate the English phrase into an algebraic expression: the difference of two times x and 8, two times the difference of x and 8.

2x-82\left(x-8\right)

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 length of a rectangle is 6 less than the width. Let w represent the width of the rectangle. Write an expression for the length of the rectangle.

Solution
Write a phrase about the length of the rectangle. 6 less than the width
Substitute w for “the width.” 6 less than w
Rewrite “less than” as “subtracted from.” 6 subtracted from w
Translate the phrase into algebra. w-6

The length of a rectangle is 7 less than the width. Let w represent the width of the rectangle. Write an expression for the length of the rectangle.

w-7

The width of a rectangle is 6 less than the length. Let l represent the length of the rectangle. Write an expression for the width of the rectangle.

l-6

June has dimes and quarters in her purse. The number of dimes is three less than four times the number of quarters. Let q represent the number of quarters. Write an expression for the number of dimes.

Solution
Write a phrase about the number of dimes. three less than four times the number of quarters
Substitute q for the number of quarters. 3 less than 4 times q
Translate “4 times q.” 3 less than 4q
Translate the phase into algebra. 4q-3

Geoffrey has dimes and quarters in his pocket. The number of dimes is eight less than four times the number of quarters. Let q represent the number of quarters. Write an expression for the number of dimes.

4q-8

Lauren has dimes and nickels in her purse. The number of dimes is three more than seven times the number of nickels. Let n represent the number of nickels. Write an expression for the number of dimes.

7n+3

Key Concepts

  • Notation                      The result is…

    \begin{array}{cccc}\phantom{\rule{1.8em}{0ex}}\circ \phantom{\rule{0.6em}{0ex}}a+b\hfill & & & \phantom{\rule{3em}{0ex}}\text{the sum of}\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b\hfill \\ \phantom{\rule{1.8em}{0ex}}\circ \phantom{\rule{0.6em}{0ex}}a-b\hfill & & & \phantom{\rule{3em}{0ex}}\text{the difference of}\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b\hfill \\ \phantom{\rule{1.8em}{0ex}}\circ \phantom{\rule{0.6em}{0ex}}a·b,ab,\left(a\right)\left(b\right)\phantom{\rule{0.2em}{0ex}}\left(a\right)b,a\left(b\right)\hfill & & & \phantom{\rule{3em}{0ex}}\text{the product of}\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b\hfill \\ \phantom{\rule{1.8em}{0ex}}\circ \phantom{\rule{0.6em}{0ex}}a÷b,a\text{/}b,\frac{a}{b},b\overline{)a}\hfill & & & \phantom{\rule{3em}{0ex}}\text{the quotient of}\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b\hfill \end{array}
  • Inequality

    \begin{array}{cccc}\phantom{\rule{1.8em}{0ex}}\circ \phantom{\rule{0.6em}{0ex}}a<b\phantom{\rule{0.2em}{0ex}}\text{is read}\phantom{\rule{0.2em}{0ex}}\text{``}a\phantom{\rule{0.2em}{0ex}}\text{is less than}\phantom{\rule{0.2em}{0ex}}b\text{''}\hfill & & & a\phantom{\rule{0.2em}{0ex}}\text{is to the left of}\phantom{\rule{0.2em}{0ex}}b\phantom{\rule{0.2em}{0ex}}\text{on the number line}\hfill \\ \phantom{\rule{1.8em}{0ex}}\circ \phantom{\rule{0.6em}{0ex}}a>b\phantom{\rule{0.2em}{0ex}}\text{is read}\phantom{\rule{0.2em}{0ex}}\text{``}a\phantom{\rule{0.2em}{0ex}}\text{is greater than}\phantom{\rule{0.2em}{0ex}}b\text{''}\hfill & & & a\phantom{\rule{0.2em}{0ex}}\text{is to the right of}\phantom{\rule{0.2em}{0ex}}b\phantom{\rule{0.2em}{0ex}}\text{on the number line}\hfill \end{array}
  • Inequality Symbols                 Words

    \begin{array}{cccc}\phantom{\rule{1.8em}{0ex}}\circ \phantom{\rule{0.6em}{0ex}}a\ne b\hfill & & & \phantom{\rule{11em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{not equal to}}\phantom{\rule{0.2em}{0ex}}b\hfill \\ \phantom{\rule{1.8em}{0ex}}\circ \phantom{\rule{0.6em}{0ex}}a<b\hfill & & & \phantom{\rule{11em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{less than}}\phantom{\rule{0.2em}{0ex}}b\hfill \\ \phantom{\rule{1.8em}{0ex}}\circ \phantom{\rule{0.6em}{0ex}}a\le b\hfill & & & \phantom{\rule{11em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{less than or equal to}}\phantom{\rule{0.2em}{0ex}}b\hfill \\ \phantom{\rule{1.8em}{0ex}}\circ \phantom{\rule{0.6em}{0ex}}a>b\hfill & & & \phantom{\rule{11em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{greater than}}\phantom{\rule{0.2em}{0ex}}b\hfill \\ \phantom{\rule{1.8em}{0ex}}\circ \phantom{\rule{0.6em}{0ex}}a\ge b\hfill & & & \phantom{\rule{11em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{greater than or equal to}}\phantom{\rule{0.2em}{0ex}}b\hfill \end{array}
  • Grouping Symbols
    • Parentheses \left(\text{}\phantom{\rule{0.2em}{0ex}}\right)
    • Brackets \left[\text{}\phantom{\rule{0.2em}{0ex}}\right]
    • Braces \left\{\text{}\phantom{\rule{0.2em}{0ex}}\right\}
  • Exponential Notation
    • {a}^{n} means multiply a by itself, n times. The expression {a}^{n} is read a to the {n}^{th} power.
  • Order of Operations: 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.
  • Combine Like Terms
    1. Identify like terms.
    2. Rearrange the expression so like terms are together.
    3. Add or subtract the coefficients and keep the same variable for each group of like terms.

Practice Makes Perfect

Use Variables and Algebraic Symbols

In the following exercises, translate from algebra to English.

16-9

16 minus 9, the difference of sixteen and nine

3·9

28÷4

28 divided by 4, the quotient of twenty-eight and four

x+11

\left(2\right)\left(7\right)

2 times 7, the product of two and seven

\left(4\right)\left(8\right)

14<21

fourteen is less than twenty-one

17<35

36\ge 19

thirty-six is greater than or equal to nineteen

6n=36

y-1>6

y minus 1 is greater than 6, the difference of y and one is greater than six

y-4>8

2\le 18÷6

2 is less than or equal to 18 divided by 6; 2 is less than or equal to the quotient of eighteen and six

a\ne 1·12

In the following exercises, determine if each is an expression or an equation.

9·6=54

equation

7·9=63

5·4+3

expression

x+7

x+9

expression

y-5=25

Simplify Expressions Using the Order of Operations

In the following exercises, simplify each expression.

{5}^{3}

125

{8}^{3}

{2}^{8}

256

{10}^{5}

In the following exercises, simplify using the order of operations.

3+8·5\left(3+8\right)·5

43 55

2+6·3\left(2+6\right)·3

{2}^{3}-12÷\left(9-5\right)

5

{3}^{2}-18÷\left(11-5\right)

3·8+5·2

34

4·7+3·5

2+8\left(6+1\right)

58

4+6\left(3+6\right)

4·12\text{/}8

6

2·36\text{/}6

\left(6+10\right)÷\left(2+2\right)

4

\left(9+12\right)÷\left(3+4\right)

20÷4+6·5

35

33÷3+8·2

{3}^{2}+{7}^{2}

58

{\left(3+7\right)}^{2}

3\left(1+9·6\right)-{4}^{2}

149

5\left(2+8·4\right)-{7}^{2}

2\left[1+3\left(10-2\right)\right]

50

5\left[2+4\left(3-2\right)\right]

Evaluate an Expression

In the following exercises, evaluate the following expressions.

7x+8 when x=2

22

8x-6 when x=7

{x}^{2} when x=12

144

{x}^{3} when x=5

{x}^{5} when x=2

32

{4}^{x} when x=2

{x}^{2}+3x-7 when x=4

21

6x+3y-9 when
x=6,y=9

{\left(x-y\right)}^{2} when
x=10,y=7

9

{\left(x+y\right)}^{2} when x=6,y=9

{a}^{2}+{b}^{2} when a=3,b=8

73

{r}^{2}-{s}^{2} when r=12,s=5

2l+2w when
l=15,w=12

54

2l+2w when
l=18,w=14

Simplify Expressions by Combining Like Terms

In the following exercises, identify the coefficient of each term.

8a

8

13m

5{r}^{2}

5

6{x}^{3}

In the following exercises, identify the like terms.

{x}^{3},8x,14,8y,5,8{x}^{3}

{x}^{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}8{x}^{3},14\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}5

6z,3{w}^{2},1,6{z}^{2},4z,{w}^{2}

9a,{a}^{2},16,16{b}^{2},4,9{b}^{2}

16\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}4,16{b}^{2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}9{b}^{2}

3,25{r}^{2},10s,10r,4{r}^{2},3s

In the following exercises, identify the terms in each expression.

15{x}^{2}+6x+2

15{x}^{2},6x,2

11{x}^{2}+8x+5

10{y}^{3}+y+2

10{y}^{3},y,2

9{y}^{3}+y+5

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

10x+3x

13x

15x+4x

4c+2c+c

7c

6y+4y+y

7u+2+3u+1

10u+3

8d+6+2d+5

10a+7+5a-2+7a-4

22a+1

7c+4+6c-3+9c-1

3{x}^{2}+12x+11+14{x}^{2}+8x+5

17{x}^{2}+20x+16

5{b}^{2}+9b+10+2{b}^{2}+3b-4

Translate an English Phrase to an Algebraic Expression

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

the difference of 14 and 9

14-9

the difference of 19 and 8

the product of 9 and 7

9·7

the product of 8 and 7

the quotient of 36 and 9

36÷9

the quotient of 42 and 7

the sum of 8x and 3x

8x+3x

the sum of 13x and 3x

the quotient of y and 3

\frac{y}{3}

the quotient of y and 8

eight times the difference of y and nine

8\left(y-9\right)

seven times the difference of y and one

Eric has rock and classical CDs in his car. The number of rock CDs is 3 more than the number of classical CDs. Let c represent the number of classical CDs. Write an expression for the number of rock CDs.

c+3

The number of girls in a second-grade class is 4 less than the number of boys. Let b represent the number of boys. Write an expression for the number of girls.

Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let n represent the number of nickels. Write an expression for the number of pennies.

2n-7

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.

Everyday Math

Car insurance Justin’s car insurance has a ?750 deductible per incident. This means that he pays ?750 and his insurance company will pay all costs beyond ?750. If Justin files a claim for ?2,100.

  1. how much will he pay?
  2. how much will his insurance company pay?

?750 ?1,350

Home insurance Armando’s home insurance has a ?2,500 deductible per incident. This means that he pays ?2,500 and the insurance company will pay all costs beyond ?2,500. If Armando files a claim for ?19,400.

  1. how much will he pay?
  2. how much will the insurance company pay?

Writing Exercises

Explain the difference between an expression and an equation.

Answers may vary

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

Explain how you identify the like terms in the expression 8{a}^{2}+4a+9-{a}^{2}-1.

Answers may vary

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.

A table is shown that is composed of four columns and six rows. The header row reads, from left to right, “I can …”, “Confidently”, “With some help” and “No – I don’t get it!”. The phrases in the first column read “use variables and algebraic symbols.”, “simplify expressions using the order of operations.”, “evaluate an expression.”, “identify and combine like terms.”, and “translate English phrases to algebraic expressions.”

After reviewing this checklist, what will you do to become confident for all objectives?

Glossary

coefficient
The coefficient of a term is the constant that multiplies the variable in a term.
constant
A constant is a number whose value always stays the same.
equality symbol
The symbol “\text{=}” is called the equal sign. We read a=b as “a is equal to b.”
equation
An equation is two expressions connected by an equal sign.
evaluate an expression
To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
expression
An expression is a number, a variable, or a combination of numbers and variables using operation symbols.
like terms
Terms that are either constants or have the same variables raised to the same powers are called like terms.
simplify an expression
To simplify an expression, do all operations in the expression.
term
A term is a constant or the product of a constant and one or more variables.
variable
A variable is a letter that represents a number whose value may change.

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Elementary Algebra by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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