## Algebra: Expressions and Equations

Evaluating Expressions
An arithmetic expression consists of numbers and operations using parentheses, exponents, multiplication, division, addition, and subtraction. An algebraic expression is like an arithmetic expression, but it contains at least one variable. A variable is a letter that represents a number.

Students have evaluated arithmetic expressions in the past. They have used the Commutative, Associative, Distributive, Identity, and Zero properties of addition and multiplication to evaluate arithmetic expressions. This chapter reviews these skills and then extends them to evaluating algebraic expressions in which given quantities are substituted for the variables in the expressions. Students should understand that the properties they have applied to arithmetic expressions also apply to algebraic expressions.

The order of operations provides rules for evaluating or simplifying expressions that ensure that each arithmetic expression has exactly one value. The order of operations is as follows.

• Always work in parentheses first.
• Rewrite any exponents or powers.
• Multiply and divide from left to right.
• Add and subtract from left to right.

Students might use the first letters of each word in the phrase “Pretty Please, My Dear Aunt Sally” to remind them of the correct order of operations: Parentheses, Powers, Multiply, Divide, Add, Subtract. It is also helpful for students to remember that multiplication and division are inverse operations and addition and subtraction are inverse operations. In using the order of operations, inverse operations occur at the same level.

Multiplication and division are at the same level. Addition and subtraction are at the same level. When there are several operations at the same level, work from left to right to perform the operations.

Expressions can always be evaluated by following the correct order of operations. However, using the Commutative, Associative, and Distributive properties may simplify the process or make it possible to complete the evaluation mentally.

Examples:

5 + 87 + 3 = 92 + 3 = 95
Work from left to right
5 + 87 + 3 = 5 + (87 + 3)
= 5 + 90 = 95
Use the Associative Property.
76 × 97 + 3 × 76 = 7,372 + 228
= 7,600
76 × 97 + 3 × 76 = 76 × 97 + 76 × 3
Use the Commutative Property.
= 76 × (97 + 3)
Use the Distributive Property.
= 76 × 100 = 7,600
(This may look longer, but can be done mentally.)

Equations
The equality of two expressions gives an equation. To solve an equation means to find the value of the variable that will make the equation true. Some students may want to solve an equation by inspection or by using a guess-and-check strategy because they can “see” what the answer is. Explain that not all equations are easy to solve, so it is necessary to learn each step in finding the solution to simple equations before moving on to more difficult ones.

To solve an equation in one variable, the first step is to get the variable by itself on one side of the equals sign. This process is called #8220;isolating the variable.” Both sides of an equation are equal, or in balance. When solving an equation, it is necessary to keep that equality, or balance. Therefore, operations performed on one side of the equals sign must also be performed on the other side of the equals sign.

When solving equations in this chapter, the first step should be to simplify each side of the equation as far as possible. Then inverse operations can be used to solve the equation. Students should be encouraged to check that their answer makes the equation true by substituting their answer back into the equation.

Examples:

5 + x − 2.3 = 9.3 + 3 × 1.1

x + 2.7 = 12.6
Simplify
x + 2.7 − 2.7 = 12.6 − 2.7
Use inverse operations.
x + 0 = 9.9
Simplify
x = 9.9

Check: 5 + x − 2.3 = 9.3 + 3 × 1.1

5 + 9.9 − 2.3 9.3 + 3 × 1.1
Check by substituting into the equation.
14.9 − 2.3 9.3 + 3.3

12.6 = 12.6
The solution checks.

n × 1.2 = (3.8 + 4.2) × 3

n × 1.2 = 24
Simplify
n × 1.2 ÷ 1.2 = 24 ÷ 1.2
Use inverse operations.
n × 1 = 20
Simplify
n = 20

Check: n × 1.2 = (3.8 + 4.2) × 3

20 × 1.2 8 × 3
Check by substituting into the equation.
24.0 = 24
The solution checks.

Teaching Model 12.3: Solve Addition and Subtraction Equations