# The Language of Algebra

## Evaluate, Simplify, and Translate Expressions


### Evaluate Algebraic Expressions

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.

### Identify Terms, Coefficients, and 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. Some examples of terms are 
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  is  When we write  the coefficient is  since   gives the coefficients for each of the terms in the left column.
An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together.  gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.
Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?
Which of these terms are like terms?
1. The terms  and  are both constant terms.
2. The terms  and  are both terms with 
3. The terms  and  both have 
Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms 

### Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think  would simplify to? If you thought  you would be right!
We can see why this works by writing both terms as addition problems.
Add the coefficients and keep the same variable. It doesn’t matter what  is. If you have  of something and add  more of the same thing, the result is  of them. For example,  oranges plus  oranges is  oranges. We will discuss the mathematical properties behind this later.
The expression  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.
Now it is easier to see the like terms to be combined.
When any of the terms have negative coefficients, the procedure is the same, except that you have to subtract instead of adding to combine like terms. 

### Translate Words to Algebraic Expressions

In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized in .
Look closely at these phrases using the four operations:
1. the sum of  and 
2. the difference of  and 
3. the product of  and 
4. the quotient of  and 
Each phrase tells you to operate on two numbers. Look for the words  and  to find the numbers.
How old will you be in eight years? What age is eight more years than your age now? Did you add  to your present age? Eight more than means eight added to your present age.
How old were you seven years ago? This is seven years less than your age now. You subtract  from your present age. Seven less than means seven subtracted from your present age.
Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase to an algebraic expression. We’ll need to be clear about what the expression will represent. We’ll see how to do this in the next two examples.

### Key Concepts

1. Combine like terms.


### Practice Makes Perfect

Evaluate Algebraic Expressions 
In the following exercises, evaluate the expression for the given value.
Identify Terms, Coefficients, and Like Terms 
In the following exercises, list the terms in the given expression.
In the following exercises, identify the coefficient of the given term.
In the following exercises, identify all sets of like terms.
Simplify Expressions by Combining Like Terms 
In the following exercises, simplify the given expression by combining like terms.
Translate English Phrases into Algebraic Expressions 
In the following exercises, translate the given word phrase into an algebraic expression.
In the following exercises, write an algebraic expression.

### Everyday Math

In the following exercises, use algebraic expressions to solve the problem.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.


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