Use the Language of Algebra

In algebra, we use a letter of the alphabet to represent a number whose value may change or is unknown. Commonly used symbols are $a$, $b$, $c$, $m$, $n$, $x$, and $y$. Further discussion of constants and variables appears later in this section.

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 $2$ and a counting number.

Similarly, a multiple of $3$ would be the product of a counting number and $3$.

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

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 $15$ by $3$, we get a counting number. In fact, $15 \div 3$ is $5$, so $15$ is $5 \cdot 3$.

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

Example. Is $5{,}625$ divisible by (a) $2$? (b) $3$? (c) $5$ or $10$? (d) $6$?

(a) Is $5{,}625$ divisible by $2$? Does it end in $0$, $2$, $4$, $6$, or $8$? No, so $5{,}625$ is not divisible by $2$.

(b) Is $5{,}625$ divisible by $3$? The sum of the digits is $5 + 6 + 2 + 5 = 18$. Is $18$ divisible by $3$? Yes, so $5{,}625$ is divisible by $3$.

(c) Is $5{,}625$ divisible by $5$ or $10$? The last digit is $5$, so $5{,}625$ is divisible by $5$ but not by $10$.

(d) Is $5{,}625$ divisible by $6$? Is it divisible by both $2$ and $3$? No — $5{,}625$ is not divisible by $2$, so $5{,}625$ is not divisible by $6$.

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 $8 \cdot 9 = 72$, we say that $8$ and $9$ are factors of $72$. When we write $72 = 8 \cdot 9$, we say we have factored $72$.

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

Some numbers, such as $72$, have many factors. Other numbers have only two factors. A prime number is a counting number greater than $1$ whose only factors are $1$ and itself.

The counting numbers from $2$ to $20$ are listed below with their factors. 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, 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.

Example. Find the prime factorization of $48$.

We find two factors whose product is $48$, say $2$ and $24$, and use them to create two branches. Since $2$ is prime, we circle it and that branch is complete. Since $24$ is not prime, we break it into two more factors, $4$ and $6$. Neither $4$ nor $6$ is prime, so we break each into two factors: $4$ into $2$ and $2$, and $6$ into $2$ and $3$. Now $2$, $2$, $2$, and $3$ are all prime, so we circle them.

We write the composite number as the product of all the circled primes:

We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer. If we first factored $48$ in a different way, for example as $6 \cdot 8$, the result would still be the same.

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.

Example. Find the least common multiple (LCM) of $12$ and $18$ using the prime factors method.

First, we write each number as a product of primes:

Then we list the primes of each number and match primes vertically when possible. Where a number is missing a prime, we leave a gap in that column. Bring down the primes in each column, and the LCM is the product of these factors.

Notice that the prime factors of $12$ ($2 \cdot 2 \cdot 3$) and the prime factors of $18$ ($2 \cdot 3 \cdot 3$) are included in the LCM ($2 \cdot 2 \cdot 3 \cdot 3$). 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 “$>$”.

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. 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 $4 < 17$.

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.

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

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 by 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 $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$. This is tedious and it can be hard to keep track of all those $2$s, so we use exponents. We write $2 \cdot 2 \cdot 2$ as $2^3$ and $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ 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.”

While we read $a^n$ as “$a$ to the $n^{\text{th}}$ power,” we usually read $a^2$ as “$a$ squared” and $a^3$ as “$a$ cubed.”

We’ll see later why $a^2$ and $a^3$ have special names. The table below 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 $4 \cdot 2 + 1$ we would first multiply $4 \cdot 2$ 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 $4 + 3 \cdot 7$. Some students simplify this getting $49$, by adding $4 + 3$ and then multiplying that result by $7$. Others get $25$, by multiplying $3 \cdot 7$ 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.

Example. Simplify: $18 \div 6 + 4(5 - 2)$.

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

Example. Simplify: $5 + 2^3 + 3[6 - 3(4 - 2)]$.

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. Evaluate when $x = 4$: (a) $x^2$ (b) $3^x$ (c) $2x^2 + 3x + 8$.

(a) Replace $x$ with $4$, then use the definition of exponent:

(b) Replace $x$ with $4$, then use the definition of exponent:

(c) Replace each $x$ with $4$, then follow the order of operations:

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 $7$, $y$, $5x^2$, $9a$, 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 $x = 1 \cdot x$.

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.
• $n^2$ and $9n^2$ are like terms.

If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.

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

First, identify the like terms: $2x^2$ and $x^2$; $3x$ and $4x$; $7$ and $5$. Next, rearrange the expression so the like terms are together. Then combine the like terms by adding the coefficients:

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 table below 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.

• the sum of $a$ and $b$
• the difference of $a$ and $b$
• the product of $a$ and $b$
• the quotient of $a$ and $b$

Example. Translate each English phrase into an algebraic expression: (a) the difference of $14x$ and $9$ (b) the quotient of $8y^2$ and $3$ (c) twelve more than $y$ (d) seven less than $49x^2$.

(a) 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 difference of $14x$ and $9$ becomes $14x$ minus $9$, which is $14x - 9$.

(b) The key word is quotient, which tells us the operation is division. The quotient of $8y^2$ and $3$ means divide $8y^2$ by $3$, which is $8y^2 \div 3$. This can also be written $8y^2/3$ or $\tfrac{8y^2}{3}$.

(c) The key words are more than. They tell us the operation is addition. More than means “added to.” Twelve more than $y$ is twelve added to $y$, which is $y + 12$.

(d) The key words are less than. They tell us to subtract. Less than means “subtracted from.” Seven less than $49x^2$ is seven subtracted from $49x^2$, which is $49x^2 - 7$.

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

Example. Translate the English phrase into an algebraic expression: (a) eight times the sum of $x$ and $y$ (b) the sum of eight times $x$ and $y$.

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

(a) Because we are multiplying $8$ times the sum, we need parentheses around the sum of $x$ and $y$, $(x + y)$. This forces us to determine the sum first. (Remember the order of operations.) Eight times the sum of $x$ and $y$ is $8(x + y)$.

(b) 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 eight times $x$ and $y$. The sum of eight times $x$ and $y$ is $8x + y$.

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.

Example. The length of a rectangle is $14$ less than the width. Let $w$ represent the width of the rectangle. Write an expression for the length of the rectangle.

Write a phrase about the length: “$14$ less than the width.” Substitute $w$ for “the width.” Rewrite less than as subtracted from: “$14$ subtracted from $w$.” Translate the phrase into algebra:

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

Write a phrase about the number of dimes: “seven less than four times the number of quarters.” Substitute $q$ for the number of quarters: “$7$ less than $4$ times $q$.” Translate $4$ times $q$: “$7$ less than $4q$.” Translate the phrase into algebra:

Key terms

multiple of a number — a number that is the product of a counting number and $n$. divisible by a number — describes $m$ when $m$ is a multiple of $n$. factors — in $a \cdot b = m$, the integers $a$ and $b$ that multiply to give $m$. prime number — a counting number greater than $1$ whose only factors are $1$ and itself. composite number — a counting number greater than $1$ that is not prime. prime factorization — the product of prime numbers that equals a given number. least common multiple (LCM) — the smallest number that is a multiple of two given numbers. variable — a letter that represents a number whose value may change. constant — a number whose value always stays the same. expression — a number, a variable, or a combination of numbers and variables using operation symbols. equation — two expressions connected by an equal sign. base / exponent — in $a^n$, the base $a$ is multiplied by itself $n$ times, where $n$ is the exponent. term — a constant or the product of a constant and one or more variables. coefficient — the constant that multiplies the variable in a term. like terms — terms that are constants or have the same variables raised to the same powers.

This section is adapted from Intermediate Algebra 2e, Section 1.1: Use the Language of Algebra by Lynn Marecek and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the multiples and factor tables as accessible markdown tables, the factor-tree diagram as an accessible inline graphic, and the operation/inequality/exponent references as tables and typeset math; omitted the Be Prepared quiz, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.