1.3 Evaluate, Simplify, and Translate Expressions
Learning Objectives
By the end of this section, you will be able to:
- Evaluate algebraic expressions
- Identify terms, coefficients, and like terms
- Simplify expressions by combining like terms
- Translate word phrases to algebraic 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.
EXAMPLE 1
Evaluate when
a. To evaluate, substitute for in the expression, and then simplify.
| Substitute. | |
| Add. |
When , the expression has a value of .
b. To evaluate, substitute for in the expression, and then simplify.
| Substitute. | |
| Add. |
When , the expression has a value of .
Notice that we got different results for parts a) and b) even though we started with the same expression. This is because the values used for were different. When we evaluate an expression, the value varies depending on the value used for the variable.
TRY IT 1.1
Evaluate:
Answer
- 10
- 19
TRY IT 1.2
Evaluate:
Answer
- 4
- 12
EXAMPLE 2
Evaluate
Remember means times , so means times .
a. To evaluate the expression when , we substitute for , and then simplify.
| Substitute 5 for x. | |
| Multiply. | |
| Subtract. |
b. To evaluate the expression when , we substitute for , and then simplify.
| Substitute 1 for x. | |
| Multiply. | |
| Subtract. |
Notice that in part a) that we wrote and in part b) we wrote . Both the dot and the parentheses tell us to multiply.
TRY IT 2.1
Evaluate:
Answer
- 13
- 5
TRY IT 2.2
Evaluate:
Answer
- 8
- 16
EXAMPLE 3
Evaluate when .
We substitute for , and then simplify the expression.
| Substitute 10 for x. | |
| Use the definition of exponent. | |
| Multiply. |
When , the expression has a value of .
TRY IT 3.1
Evaluate:
.
Answer
64
TRY IT 3.2
Evaluate:
.
Answer
216
EXAMPLE 4
.
In this expression, the variable is an exponent.
| Substitute 5 for x. | |
| Use the definition of exponent. | |
| Multiply. |
When , the expression has a value of .
TRY IT 4.1
Evaluate:
.
Answer
64
TRY IT 4.2
Evaluate:
.
Answer
81
EXAMPLE 5
.
This expression contains two variables, so we must make two substitutions.
| Substitute 5 for x and 2 for y. | |
| Multiply. | |
| Add and subtract left to right. |
When and , the expression has a value of .
TRY IT 5.1
Evaluate:
Answer
33
TRY IT 5.2
Evaluate:
Answer
10
EXAMPLE 6
.
We need to be careful when an expression has a variable with an exponent. In this expression, means and is different from the expression , which means .
| Substitute 4 for x. | |
| Simplify . | |
| Multiply. | |
| Add. |
TRY IT 6.1
Evaluate:
.
Answer
40
TRY IT 6.2
Evaluate:
.
Answer
9
I