1-2 ALGEBRAIC EXPRESSIONS AND

MODELSToday’s Objective:

I will evaluate and simplify algebraic expressions by combining like terms.

Evaluating Algebraic Expressions

Numerical expressions: consist of numbers, operations, and grouping symbols.

Exponents: used to represent repeated factors in multiplication (23 = 2•2•2)

The number 2 is the base.The number 3 is the exponent.The expression 23 is a power.

Evaluating Powers

1) (- 4)2 = (- 4)•(- 4) = 16

2) - 42 = -(4•4) = -16

Notice how parentheses are used in part 1) to indicate that the base is - 4. In the expression - 42 , however the base is 4, not - 4.

An order of operations helps avoid confusion when evaluating expressions.

ORDER OF OPERATIONS

1st, do operations that occur within grouping symbols ( ), [ ], and { }.

Next, evaluate powers (exponents2).Then, do multiplications (x and •) and

divisions (/,―, and ÷) from left to right.Finally, do additions (+) and

subtractions (-) from left to right.

Using Order of Operations

3) -6 + 3(-3 + 7)2

First step? Parenthesis (-3 + 7) = 4 Next step? Powers: (4) 2 = 16 Next step? Multiplication: 3(16) =

48 Next step? Addition: -6 + 48 = 42

4) 24 – 8 • 12 ÷ 4 First step? Multiplication: 8 • 12 = 96 Next step? Division: 96 ÷ 4 = 24 Final step? Subtraction: 24 – 24 = 0

Evaluating an Algebraic Expression

Variable: a letter that represents one or more numbers.

Value of a variable: any number used to replace a variable.

Algebraic expression: an expression involving variables.

Evaluating Algebraic

Expressions

Write Algebraic

Expression

Substitute Values of Variables

Simplify!

Evaluating Algebraic Expressions

5) 6x + 9 when x = 4 First step? Substitute 4 for x. 6(4) + 9 Next step? Multiplication: 6(4) = 24 Final step? Addition: 24 + 9 = 33

6) x2 + 5 – x when x = 5 First step? Substitute 5 for x. (5)2 + 5 – 5 Next step? Powers: (5)2 = 25 Next step? Addition: 25 + 5 = 30 Final step? Subtraction: 30 – 5 = 25

Simplifying Algebraic Expressions

6x + 4 – x Terms : parts that are added together (6x, 4, & -x). Coefficient: the # in front of a term that is a

product of a # and a power of a variable (6 & -1). Like terms: terms that have the same variable

part (6x & -x). Like terms that have variables can be combined

by adding the coefficients (6x + (-x) = 5x) Constant: terms with numbers but no variables

(4).

Simplifying by Combining Like Terms

7) 7x2 + 12x – x2 – 40x First step? Combine like terms. What’s the highest

power? 2: Add like terms. 7x2 – x2 = 6x2

What’s the next power? 1: Add like terms. 12x – 40x = -28x Simplify: 6x2 – 28x

8) 12(n – 3) + 4(n – 13) First step? Distributive Property. 12(n – 3) = 12n – 36

and 4(n – 13) = 4n – 52 Combine like terms. 12n + 4n = 16n and

-36 – 52 = -88Simplify: 16n - 88

APPLICATION: Hawaii’s Population

For 1980 – 1998, the population (in thousands) ofHawaii can be modeled by13.2t + 965 where t is the number of years since 1980.What was the population of

Hawaii in 1998? Given: 13.2t + 965 What is the value of t? t = 1998 – 1980 = 18 Substitute.

13.2(18) + 965 237.6 + 965 = 1202.6 What does 1202.6

mean? Is this the answer? Population is in

thousands so multiply by 1000.

The population of Hawaii in 1998 is 1,202,600.

APPLICATION: Used cars

You buy a used car with 37,148 miles on the odometer. You plan to drive the car 15,000 miles each year. Write an expression for the number of miles at the end of each year. Evaluate the expression to find the number of miles after 3 years.

How many current miles? (37,148)

How many per year? (15,000)

So the expression is:37,148 + 15,000tAfter 3 years, there would be 37,148 + 15,000(3) = 37,148 + 45,000 = 82,148miles in 3 years.

Practice ON YOUR OWN!

Evaluate the Powers.

1) (- 6)2

2) - 62

Use Order of Operations.

3) -6 + 3(-3 + 7)2

4) 24 – 8 • 12 ÷ 4

Evaluate the Expressions.

5) 6x + 9 when x = 4

6) x2 + 5 – x when x = 5

Simplify the Expressions.

7) 7x2 + 12x – x2 – 40x

8) 12(n – 3) + 4(n – 13)