Algebra unit 1.7

UNIT 1.7 COMMUTATIVE, UNIT 1.7 COMMUTATIVE, ASSOCIATIVE AND DISTRIBUTIVE ASSOCIATIVE AND DISTRIBUTIVE

PROPERTIESPROPERTIES

Warm UpAdd.

1. 427 + 35 2. 1.06 + 0.74

3.

Multiply.

4. 25(8)

6.

5. 1.3(22) 28.6200

10

462 1.80

Use the Commutative, Associative, and Distributive Properties to simplify expressions.

Combine like terms.

Objectives

termlike termscoefficient

Vocabulary

The Commutative and Associative Properties of Addition and Multiplication allow you to rearrange an expression to simplify it.

Example 1A: Using the Commutative and Associative Properties

Simplify.

11(5)

55

Use the Commutative Property.

Use the Associative Property to make groups of compatible numbers.

Simplify.

Example 1B: Using the Commutative and Associative Properties

45 + 16 + 55 + 4

45 + 55 + 16 + 4

(45 + 55) + (16 + 4)

(100) + (20)

120



Helpful Hint

Compatible numbers help you do math

mentally. Try to make multiples of 5

or 10. They are simpler to use when

multiplying.

Check It Out! Example 1a

Simplify.

21



Check It Out! Example 1b

Simplify.

410 + 58 + 90 + 2

410 + 90 + 58 + 2

(410 + 90) + (58 + 2)

(500) + (60)

560



Check It Out! Example 1c

Simplify.

28



12 • 7 • 8

12 • 8 • 7

( )12 • 8 7

4 • 7

The Distributive Property is used with Addition to Simplify Expressions.

The Distributive Property also works with subtraction because subtraction is the same as adding the opposite.

Example 2A: Using the Distributive Property with Mental Math

Write the product using the Distributive Property. Then simplify.

5(59)

5(50 + 9)

5(50) + 5(9)

250 + 45

295

Rewrite 59 as 50 + 9.

Use the Distributive Property.

Multiply.

Add.

8(33)

8(30 + 3)

8(30) + 8(3)

240 + 24

264



Multiply.

Add.

Example 2B: Using the Distributive Property with Mental Math



9(52)

9(50) + 9(2)

9(50 + 2)

450 + 18

468



Multiply.

Add.



12(98)

1176

Rewrite 98 as 100 – 2.


Multiply.

Subtract.

12(100 – 2)

1200 – 24

12(100) – 12(2)


Check It Out! Example 2c

7(34)

7(30 + 4)

7(30) + 7(4)

210 + 28

238



Multiply.

Add.


The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms.

4x – 3x + 2

Like terms Constant

A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1.

1x2 + 3x

Coefficients

Using the Distributive Property can help you combine like terms. You can factor out the common factors to simplify the expression.

7x2 – 4x2 = (7 – 4)x2

= (3)x2

= 3x2

Factor out x2 from both terms.

Perform operations in parenthesis.

Notice that you can combine like terms by adding or subtracting the coefficients and keeping the variables and exponents the same.

Example 3A: Combining Like Terms

Simplify the expression by combining like terms.

72p – 25p

72p – 25p

47p

72p and 25p are like terms.

Subtract the coefficients.

Example 3B: Combining Like Terms


A variable without a coefficient has a coefficient of 1.

Write 1 as .

Add the coefficients.

and are like terms.

Example 3C: Combining Like Terms


0.5m + 2.5n

0.5m + 2.5n

0.5m + 2.5n

0.5m and 2.5n are not like terms.

Do not combine the terms.

Check It Out! Example 3

Simplify by combining like terms.

3a. 16p + 84p

16p + 84p

100p16p + 84p are like terms.

Add the coefficients.

3b. –20t – 8.5t2

–20t – 8.5t2 20t and 8.5t2 are not like terms.

–20t – 8.5t2 Do not combine the terms.

3m2 + m3 3m2 and m3 are not like terms.

3c. 3m2 + m3

Do not combine the terms.3m2 + m3

Example 4: Simplifying Algebraic ExpressionsSimplify 14x + 4(2 + x). Justify each step.

14x + 4(2) + 4(x) Distributive Property

Multiply.Commutative Property

Associative Property

Combine like terms.

14x + 8 + 4x

(14x + 4x) + 8

14x + 4x + 8

18x + 8

14x + 4(2 + x)1. 2. 3. 4. 5. 6.

Procedure Justification

6(x) – 6(4) + 9 Distributive Property

Multiply.Combine like terms.

6x – 24 + 9

6x – 15

6(x – 4) + 91. 2. 3. 4.



Simplify 6(x – 4) + 9. Justify each step.

–12x – 5x + x + 3a Commutative Property

Combine like terms.–16x + 3a

–12x – 5x + 3a + x1. 2. 3.



Simplify −12x – 5x + 3a + x. Justify each step.

Lesson Quiz: Part I

Simplify each expression.

1. 165 +27 + 3 + 5

2.

Write each product using the Distributive Property. Then simplify.

3. 5($1.99)

4. 6(13)

200

8

5($2) – 5($0.01) = $9.95

6(10) + 6(3) = 78

Lesson Quiz: Part II

Simplify each expression by combining like terms. Justify each step with an operation or property.

7. 301x – x

8. 24a + b2 + 3a + 2b2

5.

300x

27a + 3b2

6. 14c2 – 9c 14c2 – 9c

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Algebra unit 1.7

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Transcript of Algebra unit 1.7