Two expressions that have the same value for all allowable replacements are called equivalent. 1.7...

Two expressions that have the same value for all allowable replacements are called equivalent.

1.7 Properties of Real Numbers

Equivalent Expressions

Slide 1Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

For any real number a

a + 0 = 0 + a = a

(The number 0 is the additive identity.)

1.7 Properties of Real Numbers

The Identity Property of 0

Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

For any real number a

a 1 = 1 a = a

(The number 1 is the multiplicative identity.)

1.7 Properties of Real Numbers

The Identity Property of 1

Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 40.

24

x

x

5

3

Factor out the GCF of 40 and 24.

Factoring the fraction expression.

8x/8x = 1Removing a factor of 1 using the identity property of 1

Solution

1.7 Properties of Real Numbers

a Find equivalent fraction expressions and simplify fraction expressions.

A Simplify:

Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

x

x

24

40

x

x

83

85

x

x

83

85

13

5

EXAMPLE

SolutionWe substitute 7 for x and 8 for y.

x + y = 7 + 8 = 15

1.7 Properties of Real Numbers

b Use the commutative and associative laws to find equivalent expressions.

B Evaluate x + y and y + x when x = 7 andy = 8.

Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

y + x = 8 + 7 = 15

EXAMPLESolutionWe substitute 7 for x and 8 for y.

xy = 7(8) = 56

yx = 8(7) = 56

1.7 Properties of Real Numbers

b Use the commutative and associative laws to find equivalent expressions.

C Evaluate xy and yx when x = 7 and y = 8.

Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Addition: For any numbers a, and b,a + b = b + a.

(We can change the order when adding without affecting the answer.)

1.7 Properties of Real Numbers

The Commutative Laws

Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Multiplication. For any numbers a and b,

ab = ba(We can change the order when

multiplying without affecting the answer.)

1.7 Properties of Real Numbers

The Commutative Laws

Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE4 + (9 + 6) and (4 + 9) + 6.

Solution OR

1.7 Properties of Real Numbers

b Use the commutative and associative laws to find equivalent expressions.

D Calculate and compare:

Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

(4 + 9) + 6 = 13 + 6 = 19

4 + (9 + 6) = 4 + 15 = 19

Addition: For any numbers a, b, and c,

a + (b + c) = (a + b) + c.

(Numbers can be grouped in any manner for addition.)

1.7 Properties of Real Numbers

The Associative Laws

Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Multiplication. For any numbers a, b, and c,

a (b c) = (a b) c

(Numbers can be grouped in any manner for multiplication.)

1.7 Properties of Real Numbers

The Associative Laws

Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

For any numbers a, b, and c,

a(b + c) = ab + ac.

1.7 Properties of Real Numbers

The Distributive Law of Multiplication over Addition

Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

For any numbers a, b, and c,

a(b c) = ab ac.

1.7 Properties of Real Numbers

The Distributive Law of Multiplication over Subtraction

Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLESolution

4(a + b) = 4 ∙ a + 4 ∙ b

= 4a + 4b

Using the distributive law of multiplication over addition.

1.7 Properties of Real Numbers

c Use the distributive laws to multiply expressions like 8 and x – y.

E Multiply. 4(a + b).

Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE

1. 8(a – b) 2. (b – 7)c 3. –5(x – 3y + 2z)

Solution1. 8(a – b) = 8a – 8b2. (b – 7)c = c(b – 7)

= c ∙ b – c 7∙ = cb – 7c

1.7 Properties of Real Numbers

c Use the distributive laws to multiply expressions like 8 and x – y.

F Use the distributive law to write an expression equivalent to each of the following:

(continued)

Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE

3. –5(x – 3y + 2z) = –5 ∙ x – (–5 3)∙ y + (–5 2)∙ z = –5x – (–15)y + (–10)z = –5x + 15y – 10z

1.7 Properties of Real Numbers

c Use the distributive laws to multiply expressions like 8 and x – y.

F The distributive property.

Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

To factor an expression is to find an equivalent expression that is a product.

Factoring is the reverse of multiplying. To factor, we can use the distributive laws in reverse:ab + ac = a(b + c) and ab – ac = a(b – c).

1.7 Properties of Real Numbers

Factor

Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLEa. 6x – 12 b. 8x + 32y – 8

Solutiona. 6x – 12 = 6 ∙ x – 6 2∙

= 6(x – 2)

b. 8x + 32y – 8 = 8 ∙ x + 8 4∙ y – 8 1∙ = 8(x + 4y – 1)

1.7 Properties of Real Numbers

d Use the distributive laws to factor expressions like4x – 12 + 24y.

G Factor.

Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE

a. 7x – 7y b. 14z – 12x – 20

1.7 Properties of Real Numbers

d Use the distributive laws to factor expressions like4x – 12 + 24y.

H Factor. Try to write just the answer, if you can.

(continued)

Slide 19Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE

a. 7x – 7y b. 14z – 12x – 20

Solutiona. 7x – 7y = 7(x – y)

b. 14z – 12x – 20 = 2(7z – 6x – 10)

1.7 Properties of Real Numbers

d Use the distributive laws to factor expressions like4x – 12 + 24y.

H Factor. Try to write just the answer, if you can.

Slide 20Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

A term is a number, a variable, a product of numbers and/or variables, or a quotient of two numbers and/or variables.

Terms are separated by addition signs. If there are subtraction signs, we can find an equivalent expression that uses addition signs.

The process of collecting like terms is based on the distributive laws.

1.7 Properties of Real Numbers

e Collect like terms.

Slide 21Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Terms in which the variable factors are exactly the same, such as 9x and –5x, are called like, or similar terms.

Like Terms Unlike Terms7x and 8x 8y and 9y2

3xy and 9xy 5ab and 4ab2

1.7 Properties of Real Numbers

e Collect like terms.

Slide 22Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE

1. 8x + 2x 2. 3x – 6x 3. 3a + 5b + 2 + a – 8 – 5b

Solution1. 8x + 2x = (8 + 2)x

= 10x

2. 3x – 6x = (3 – 6)x = –3x

1.7 Properties of Real Numbers

e Collect like terms.

I Combine like terms. Try to write just the answer.

(continued)

Slide 23Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE

1. 8x + 2x 2. 3x – 6x 3. 3a + 5b + 2 + a – 8 – 5b

Solution3. 3a + 5b + 2 + a – 8 – 5b = 3a + 5b + 2 + a + (–8) + (–5b) = 3a + a + 5b + (–5b) + 2 + (–8) = 4a + (–6) = 4a – 6

1.7 Properties of Real Numbers

e Collect like terms.

I Combine like terms. Try to write just the answer.

Slide 24Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.