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 Table 1.2.

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.

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 in the table below. You’ll probably recognize some of them.

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, $\times ,$ is not used to show multiplication because that symbol may cause confusion. Does 3xy mean $3\times y$ (‘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.

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

$$\begin{array}{}\\ \\ a<b\text{is read “}a\text{is less than}b\text{”}\hfill \\ a\text{is to the left of}b\text{on the number line}\hfill \end{array}$$

$$\begin{array}{c}a>b\text{is read “}a\text{is greater than}b\text{”}\hfill \\ a\text{is to the right of}b\text{on the number line}\hfill \end{array}$$

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 (Table 1.3). 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.

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.

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. “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 an English phrase. Here are some examples of expressions:

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.

Here are some examples of equations.

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.

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.

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.

Table 1.4 shows how we read some expressions with exponents.

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:

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:

If you simplify this expression, what do you get?

Some students say 49,

Others say 25,

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.

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.

Let’s try an example.

Example 1.15

Simplify: ⓐ $4+3·7$ ⓑ $(4+3)·7.$

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.

Example 1.16

Simplify: $18÷6+4\left(5-2\right).$

Solution

Parentheses? Yes, subtract first.	$18÷6+4\left(5-2\right)$ $$
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.

Example 1.17

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.

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.

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

Example 1.18

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

Solution

ⓐ

Multiply.
Subtract.

ⓑ

Multiply.
Subtract.

Example 1.19

Evaluate the following for $x=4,$ 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.	$81$

Example 1.20

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

Solution

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

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.

Examples of terms are $7,y,5x^2,9a,\text{and}{b}^5.$

The constant that multiplies the variable is called the coefficient.

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

The 7 and the 4 are both constant terms.

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

The $n^2$ and the $9n^2$ 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 $9x^2$ are like terms.

Adding or subtracting terms forms an expression. In the expression $2x^2+3x+8,$ from Example 1.20, the three terms are $2x^2,3x,$ and 8.

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!

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

Example 1.24

How To Combine Like Terms

Simplify: $2x^2+3x+7+x^2+4x+5.$

Solution

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. Table 1.5 summarizes them.

Look closely at these phrases using the four operations:

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

Example 1.25

Translate each English phrase into an algebraic expression: ⓐ the difference of $17x$ and $5$ ⓑ the quotient of $10x^2$ 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.
   
   
   
2. ⓑ The key word is “quotient,” which tells us the operation is division.

This can also be written $10x^2\text{/}7\text{or}\frac{10x^2}{7}.$

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.

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.