# Algebra

## Algebraic Expressions


### Learning Objectives

After completing this section, you should be able to:
1. Convert between written and symbolic algebraic expressions and equations.
2. Simplify and evaluate algebraic expressions.
3. Add and subtract algebraic expressions.
4. Multiply and divide algebraic expressions.
Algebraic expressions are the building blocks of algebra. While a numerical expression (also known as an arithmetic expression) like  can represent only a single number, an algebraic expression such as  can represent many different numbers. This section will introduce you to algebraic expressions, how to create them, simplify them, and perform arithmetic operations on them.

### Algebraic Expressions and Equations

Xavier and Yasenia have the same birthday, but they were born in different years. This year Xavier is 20 years old and Yasenia is 23, so Yasenia is three years older than Xavier. When Xavier was 15, Yasenia was 18. When Xavier will be 33, Yasenia will be 36. No matter what Xavier’s age is, Yasenia’s age will always be 3 years more.
In the language of algebra, we say that Xavier's age and Yasenia's age are variable and the 3 is a constant. The ages change, or vary, so age is a variable. The 3 years between them always stays the same or has the same value, so the age difference is the constant. In algebra, letters of the alphabet are used to represent variables. The letters most often used for variables are , , , , , and . Suppose we call Xavier's age . Then we could use  to represent Yasenia's age, as shown in the table below.
To write algebraically, we need some symbols as well as numbers and variables. The symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division are summarized in , along with words we use for the operations and the result.
We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.
1. The sum  5  3 means add 5 plus 3, which we write as .
2. The difference  9  2 means subtract 9 minus 2, which we write as .
3. The product  4  8 means multiply 4 times 8, which we can write as .
4. The quotient  20  5 means divide 20 by 5, which we can write as .
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.  and  used expressions. An expression is like an English phrase. Notice that the English phrases do not form a complete sentence because the phrase does not have a verb. The following table has examples of expressions, which are numbers, variables, or combinations of numbers and variables using operation symbols.
An equation is two expressions linked with an equal sign (the symbol =). When two quantities have the same value, we say they are equal and connect them 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. So,  is read “ is equal to .” The following table has some examples of equations.

### Simplifying and Evaluating Algebraic Expressions

To simplify an expression means to do all the math possible. For example, to simplify  we would first multiply  to get 8 and then add 1 to get 9. We have 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. Consider . Do you add first or multiply first? Do you get different answers?
Early on, mathematicians realized the need to establish some guidelines when performing arithmetic operations to ensure that everyone would get the same answer. Those guidelines are called the order of operations and are listed in the table below.
In the last example, 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.

### Operations of Algebraic Expressions

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, , 5, 9, and . The constant that multiplies the variable is called the coefficient. Think of the coefficient as the number in front of the variable. Consider the algebraic expressions 5, which has a coefficient of 5, and 9, which has a coefficient of 9. If there is no number listed in front of the variable, then the coefficient is 1 since .
Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they 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.
Before looking at multiplying algebraic expressions we look at the Distributive Property, which says that to multiply a sum, first you multiply each term in the sum and then you add the products. For example,  can also be solved as . If we use a variable, then . 
We can extended this example to , which can also be solved as . If we use variables, then .
In 2012, Andrew Hacker wrote an opinion piece in the New York Times Magazine suggesting that teaching algebra in high school was a waste of time. Keith Devlin, a British mathematician, was asked to comment on Hacker's article by his students in his Stanford University Continuing Studies course "Mathematics: Making the Invisible Visible" on iTunes University. Devlin concludes that Hacker was displaying his ignorance of what algebra is.

### Check Your Understanding


### Key Terms

1. variable
2. constant
3. expression
4. equation
5. equal sign
6. term
7. coefficient
8. like terms
9. Distributive Property

### Key Concepts

1. Algebra is useful because it allows us to understand many situations in real life by modeling them with expressions.
2. Algebraic expressions are the building blocks of algebra. From algebraic expressions we can create algebraic equations.
3. Algebraic expressions are the building blocks of algebra. From algebraic expressions we can create algebraic equations.
4. Algebraic expressions are often simplified and evaluated using the four arithmetic operations.

### Videos

1. Q&A: Why We Teach Algebra

### Formulas

1.  Distributive Property: 