Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers Equivalent Expressions Two expressions that have the same value for all allowable replacements are called equivalent. d Use the distributive laws to factor expressions like 4x – 12 + 24y. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 1

1.7 Properties of Real Numbers The Identity Property of 0 For any real number a a + 0 = 0 + a = a (The number 0 is the additive identity.) a Find equivalent fraction expressions and simplify fraction expressions. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 2

1.7 Properties of Real Numbers The Identity Property of 1 For any real number a a  1 = 1  a = a (The number 1 is the multiplicative identity.) a Find equivalent fraction expressions and simplify fraction expressions. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 3

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers a Find equivalent fraction expressions and simplify fraction expressions. A Simplify: Solution Factor out the GCF of 40 and 24. Factoring the fraction expression. Removing a factor of 1 using the identity property of 1 8x/8x = 1 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 4

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 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 and y = 8. Solution We substitute 7 for x and 8 for y. x + y = 7 + 8 = 15 y + x = 8 + 7 = 15 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 5

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 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. Solution We substitute 7 for x and 8 for y. xy = 7(8) = 56 yx = 8(7) = 56 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 6

1.7 Properties of Real Numbers The Commutative Laws Addition: For any numbers a, and b, a + b = b + a. (We can change the order when adding without affecting the answer.) b Use the commutative and associative laws to find equivalent expressions. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 7

1.7 Properties of Real Numbers The Commutative Laws Multiplication. For any numbers a and b, ab = ba (We can change the order when multiplying without affecting the answer.) b Use the commutative and associative laws to find equivalent expressions. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 8

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers b Use the commutative and associative laws to find equivalent expressions. D Calculate and compare: 4 + (9 + 6) and (4 + 9) + 6. Solution OR 4 + (9 + 6) = 4 + 15 = 19 (4 + 9) + 6 = 13 + 6 = 19 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 9

1.7 Properties of Real Numbers The Associative Laws Addition: For any numbers a, b, and c, a + (b + c) = (a + b) + c. (Numbers can be grouped in any manner for addition.) b Use the commutative and associative laws to find equivalent expressions. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 10

1.7 Properties of Real Numbers The Associative Laws Multiplication. For any numbers a, b, and c, a (b c) = (a b) c (Numbers can be grouped in any manner for multiplication.) b Use the commutative and associative laws to find equivalent expressions. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 11

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers The Distributive Law of Multiplication over Addition For any numbers a, b, and c, a(b + c) = ab + ac. c Use the distributive laws to multiply expressions like 8 and x –y. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 12

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers The Distributive Law of Multiplication over Subtraction For any numbers a, b, and c, a(b  c) = ab  ac. c Use the distributive laws to multiply expressions like 8 and x –y. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 13

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers c Use the distributive laws to multiply expressions like 8 and x – y. E Multiply. 4(a + b). Solution 4(a + b) = 4 ∙ a + 4 ∙ b = 4a + 4b Using the distributive law of multiplication over addition. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 14

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 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: 1. 8(a – b) 2. (b – 7)c 3. –5(x – 3y + 2z) Solution 1. 8(a – b) = 8a – 8b 2. (b – 7)c = c(b – 7) = c ∙ b – c ∙ 7 = cb – 7c (continued) Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 15

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers c Use the distributive laws to multiply expressions like 8 and x – y. F The distributive property. 3. –5(x – 3y + 2z) = –5 ∙ x – (–5 ∙ 3)y + (–5 ∙ 2)z = –5x – (–15)y + (–10)z = –5x + 15y – 10z Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 16

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers Factor 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). d Use the distributive laws to factor expressions like 4x – 12 + 24y. To factor an expression is to find an equivalent expression that is a product. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 17

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers d Use the distributive laws to factor expressions like 4x – 12 + 24y. G Factor. a. 6x – 12 b. 8x + 32y – 8 Solution a. 6x – 12 = 6 ∙ x – 6 ∙ 2 = 6(x – 2) b. 8x + 32y – 8 = 8 ∙ x + 8 ∙ 4y – 8 ∙ 1 = 8(x + 4y – 1) Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 18

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers d Use the distributive laws to factor expressions like 4x – 12 + 24y. H Factor. Try to write just the answer, if you can. a. 7x – 7y b. 14z – 12x – 20 (continued) Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 19

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers d Use the distributive laws to factor expressions like 4x – 12 + 24y. H Factor. Try to write just the answer, if you can. a. 7x – 7y b. 14z – 12x – 20 Solution a. 7x – 7y = 7(x – y) b. 14z – 12x – 20 = 2(7z – 6x – 10) Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 20

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers e Collect like terms. 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. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 21

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers e Collect like terms. Terms in which the variable factors are exactly the same, such as 9x and –5x, are called like, or similar terms. Like Terms Unlike Terms 7x and 8x 8y and 9y2 3xy and 9xy 5ab and 4ab2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 22

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers e Collect like terms. I Combine like terms. Try to write just the answer. 1. 8x + 2x 2. 3x – 6x 3. 3a + 5b + 2 + a – 8 – 5b Solution 1. 8x + 2x = (8 + 2)x = 10x 2. 3x – 6x = (3 – 6)x = –3x (continued) Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 23

Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 1.7 Properties of Real Numbers e Collect like terms. I Combine like terms. Try to write just the answer. 1. 8x + 2x 2. 3x – 6x 3. 3a + 5b + 2 + a – 8 – 5b Solution 3. 3a + 5b + 2 + a – 8 – 5b = 3a + 5b + 2 + a + (–8) + (–5b) = 3a + a + 5b + (–5b) + 2 + (–8) = 4a + (–6) = 4a – 6 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 24