CHAPTER 5: ALGEBRA CHAPTER 5 CONTENTS

Transcription

1 CHAPTER 5: ALGEBRA Image from CHAPTER 5 CONTENTS 5. Introduction to Algebra 5. Algebraic Properties 5. Distributive Property 5.4 Solving Equations Using the Addition Property of Equality 5.5 Solving Equations Using the Multiplication Property of Equality 5.6 Solving Equations Using the Addition and Multiplication Properties of Equality 5.7 Translating English Sentences into Mathematical Equations and Solving 44

2 CCBC Math 08 Introduction to Algebra Section 5. pages 5. Introduction to Algebra When you think of algebra, you probably think about solving equations for x, where x is some unknown number. Solving equations is indeed a big part of algebra, but algebra is much more than that. It is the branch of mathematics that uses symbols (like x) to represent unknown values and also includes performing operations like adding, subtracting, multiply and dividing on these symbols. We will start with the basics. Below are definitions of terms that you will use as you continue to study algebra. Variable: A variable is a letter that represents an unknown quantity. Term: A term is a quantity that is added. A term consists of a number or a product of a number and variables (where the variables may be raised to a power). Algebraic Expression: An algebraic expression contains at least one variable and is formed by connecting numbers and variables using operations of addition, subtraction, multiplication, division, raising to powers, and/or taking roots (but has no equal sign). We will study one type of algebraic expression in this chapter called a polynomial. Polynomial: A polynomial contains at least one variable and is a sum of one or more different terms. The following are examples of algebraic expressions that are polynomials. 5x y + 7 can be written as a sum of terms 5x + (-y) x x can be written as a sum of terms.8 ( 7) x x x 5y + xy x 7 can be written as a sum of terms x + (-5y) + xy + (-x) + (-7) Note: In the algebraic expression x 5y + xy x 7, there are five terms: x, -5y, xy, -x, -7. Variable term: A variable term is a term that contains a variable. The variable terms in x 5y + xy x 7 are x, -5y, xy, and -x. Constant term: A constant term is a term that does not contain a variable. The constant term in x 5y + xy x 7 is -7. Coefficient: A coefficient is a number multiplied with a variable. When writing terms, the coefficient is placed to the left of the variable part. In the algebraic expression x 5y + xy x 7 = x + (-5y) + xy + (-x) + (-7) is the coefficient of the term x. -5 is the coefficient of the term -5y. is the coefficient of the term xy. - is the coefficient of the term -x. Note: When there is no number multiplied with a variable term, the coefficient is or -. 45

3 CCBC Math 08 Introduction to Algebra Section 5. pages Example : For the polynomial, x x xy 5, answer the following questions. First rewrite the polynomial as a sum of terms. What is/are the variable(s)? x x xy 5 x x ( xy) 5 x and y What are the terms? x, x, -xy, and 5 What is the constant term? 5 What is/are the variable term(s)? x, x and -xy What is the coefficient of the term xy? - Practice : For the polynomial, x 4y yz 5, answer the following questions. a. Write the polynomial as a sum of terms. b. What is/are the variable(s)? c. What are the terms? d. What is the constant term? e. What is/are the variable term(s)? f. What is the coefficient of the term yz? Answers: a. x ( 4 y) yz 5 d. 5 e. b. x, y, z c., 4, x y yz f. x, 4 y, yz, 5 Like Terms: Two terms are called like terms if the variable parts of the terms are the same. Like terms are terms that have the same variable(s) raised to the same power(s). Note: x is the same as x. Example : Determine if the following pairs of terms are like terms. a and a like terms (The variable parts are the same.) x and y unlike terms (The variables are not the same.) x and unlike terms (One is an x-term: the other a constant.) x and x unlike terms (The exponents are not the same.) x and 0.5x like terms (The variable parts are the same.) x and x like terms (The variable parts are the same.) 4 and like terms (Constant terms are also like terms.) ab and b unlike terms (The variable parts are not the same.) 46

4 CCBC Math 08 Introduction to Algebra Section 5. pages Practice : Determine if the following pairs of terms are like terms. a. 4x and 6x b. a and a c. b and d. e. y and.y xyand xy f. and 4 Answers: a. like terms b. unlike terms c. unlike terms d. like terms e. unlike terms f. like terms Combining like terms: When you combine like terms, you add the coefficients of the like terms and the answer is a like term the exact same kind of term. When you combine like terms, the variable part of the term will stay the same. For Examples 7, simplify, if possible, by combining like terms. Example : x + x These are like terms. = (+)x Simplify by adding the coefficients. = 5x Practice : 8y y Answer: 5y Example 4: y + 5 These two terms are unlike, so they cannot be combined into a single term. Practice 4: 9a + 6 Answer: 9a + 6 (not like terms) 47

5 CCBC Math 08 Introduction to Algebra Section 5. pages Example 5: xy 4xy y + 5y = ( 4)xy + (- +5)y Simplify by adding the coefficients of like terms. = -xy + 4y = -xy + 4y Note: -xy + 4y is equivalent to -xy + 4y. Either answer is acceptable, but the second is preferred. Practice 5: x + 7x Answer: 9x Example 6: a a 4 a 4 a 4 4 Add coefficients of like terms. Determine common denominators to add fractions. 5 a Simplify. 4 Note: We could write 5 as the mixed number, but when doing algebra problems, simplified improper fractions are preferred. Practice 6: 4 x x Answer: x 5 Example 7: 0.8x 5.x y 0.y (0.8 5.) x ( 0.) y Add coefficients of like terms. 4.x.y Simplify. Practice 7:.a 0.4a 8b 0.5b Answer:.6a + 8.5b 48

6 CCBC Math 08 Introduction to Algebra Section 5. pages Evaluating Expressions Consider the following expression: x + 5. What does the expression mean? The variable x holds the place for an unknown number. So, the expression above says to multiply times an unknown number named x and then add 5. Note: Although there is no operation symbol between the and the x, the operation between the and the x is multiplication. Now, suppose that we are told that x is the number. What is the value of x + 5 if x is? By changing the x to a we get: Remember the order of operations!! x +5 = () + 5 Substitute for x. = Multiply first. = Add. So, if x =, x + 5 =. This process is evaluating the expression x + 5 if x =. Example 8: Evaluate 6x + 7 if x =. By replacing the x with, we get: 6x + 7 = 6() + 7 Substitute for x. = + 7 Multiply first. = 9 Add. Practice 8: Evaluate 4x 7 if x =. Answer:

7 CCBC Math 08 Introduction to Algebra Section 5. pages Example 9: Evaluate ( x 4) if x = 8. By replacing the x with 8, we get: ( x 4) ( 8 4) () 6 6 Substitute 8 for x. Simplify inside the parentheses. Write as a fraction in order to multiply. Divide out common factors in the numerator and denominator. Multiply. Practice 9: Evaluate ( x 4) if x =. Answer: Example 0: Evaluate (x + ) + 0 if x = -5. By replacing the x with -5, we get: (x + ) + 0 = (-5 + ) + 0 Substitute -5 for x. = (-) + 0 First, simplify inside the parentheses. = 8 Add. Practice 0: Evaluate (a ) + when a = 6. Answer: 50

8 CCBC Math 08 Introduction to Algebra Section 5. pages Example : Evaluate 0.4( + x) if x = -.. By replacing the x with -., we get: 0.4( + x) = 0.4( + (-.)) Substitute -. for x. =0.4(0.8) First, simplify inside the parentheses. = 0. Practice : Evaluate 0.5( + x) if x = -.4. Answer: Example : Evaluate x( + x) if x = 4. By replacing each x with 4, we get: x( + x) = 4( + 4) Substitute 4 for x. = 4(6) Simplify inside the parentheses. = 4 Multiply. Practice : Evaluate y(y 7) if y =. Answer: - Let s look at another expression: x + 4y. This expression is different from the other expressions we ve looked at because it has an x and y variable. This expression says to multiply with an unknown number called x and multiply 4 with another unknown number called y, then add those two products. Example : Evaluate x + 4y if x = and y = 5. By replacing the x with and the y with 5, we get: x + 4y = () + 4(5) Substitute for x and 5 for y. = Multiply. = 6 5

9 CCBC Math 08 Introduction to Algebra Section 5. pages Practice : Evaluate 4x + y if x = 6 and y = 5. Answer: 4 Example 4: Evaluate 5x + y if x = - and y = -. By replacing the x with - and the y with -, we get: 5x + y = 5(-) + (-) Substitute. = -0 + (-6) Multiply. = -6 Add. Practice 4: Evaluate x + 6y if x = 4 and y = 7. Answer: Example 5: Evaluate x + 8y if x = and y = -. By replacing the x with and the y with -, we get: x 8y 8( -) ( 8) ( 8) ( 8) 6 Substitute. Write as the fraction in order to multiply. Divide out common factors in the numerator and denominator. Multiply. Add. Practice 5: Evaluate 4x + y if x = and y = -. Answer: - 5

10 CCBC Math 08 Introduction to Algebra Section 5. pages Example 6: Evaluate (x 5 y) if x = 4 and y = 6. By replacing the x with 4 and the y with 6, we get: ( x 5 y ) (( 4 ) 5( 6 )) Substitute. ( 0) Simplify inside the parentheses. (4) Multiply. Practice 6: Evaluate (x 6 y) if x = and y =. Answer: 8 Example 7: Evaluate x if x = -. By replacing the x with -, we get: x = (x) Rewrite with parentheses around the variable. = (-) Substitute - for x inside the parentheses. = (-)(-) Write in expanded form. = 9 Multiply. Practice 7: Evaluate x if x = -5. Answer: 5 5

11 CCBC Math 08 Introduction to Algebra Section 5. pages Example 8: Evaluate ( x y) if x = -6 and y = 4. By replacing the x with -6 and the y with 4, we get: (x + y) = (-6 + 4) Substitute -6 for x and 4 for y. = (-) Simplify inside the parentheses. = (-)(-) Write in expanded form. = 4 Multiply. Practice 8: Evaluate ( x ) y if x = - and y = 5. Answer: 8 Watch All: 54

12 CCBC Math 08 Introduction to Algebra Section 5. pages 5. Introduction Exercises For problems 9, simplify by combining like terms.. 8x 5x + 7. x 7x 5 9. x x + x x 0x + y + y 5. -x + 4x + x x + 0.7x ab +.ab a + 0.a x x 5 5 x x x x 4 0. Evaluate + 5x if x =.. Evaluate 4 + ( + x) if x = -5.. Evaluate x(x + 7) if x = -.. Evaluate 8(x + ) if x = Evaluate 4x + 5y if x = and y =. 5. Evaluate x + 7y if x = - and y = Evaluate 4 x if x = Evaluate (4 x ) y if x = - and y = Evaluate x y if x and y. 9. Evaluate 0x + y if x = 0. and y = Evaluate ( x 4) if x =. 55

13 CCBC Math 08 Introduction to Algebra Section 5. pages 5. Introduction Exercises Answers. x x 4. -8x x + 5y 5. 4x 6..9x ab.9a x x x 4 56

14 CCBC Math 08 Algebraic Properties Section 5. pages 5. Algebraic Properties Your friend s birthday is next weekend, and you are sending her a birthday card. As usual, you will put a return address label in the upper left corner of the envelope and a stamp in the upper right corner. Does the order in which you place these label then stamp or stamp then label on the envelope matter? No, the end result will be the same regardless of which you put on first and which you put on second. Now you are ready to go out to mail the card, but you need to put your socks and shoes on. Does the order in which you place these on your feet matter? I think so! It may not work out so well if you place your shoes on first, then your socks! The point is that sometimes the order in which we perform tasks matters and sometimes it does not. How about in math? Does the order in which we perform mathematical operations matter? We begin this section of Chapter 5 by investigating that question and stating our results in the form of Algebra Properties. In fact there are five Algebra Properties that will be presented in this section. You will learn to recognize each property and also how to apply each property to make problem solving easier. Commutative Property of Addition You have invited some of your friends to get together Friday night for pizza at your house. Your friend Mike volunteered to bring the pizza and your friend Sara volunteered to bring soda. You and your friends agree to split equally the cost of the food and drinks. To do that you need the total amount that Mike and Sara paid. Mike said that the pizza cost $6.89, and Sara said that the soda cost $6.9. When you calculate the total cost, does it matter in which order you write down the two costs? No. Pizza : $6.89 Soda : $6.9 TOTAL : $4.8 Soda : $6.9 Pizza : $6.89 TOTAL : $4.8 Adding the prices in this order: Pizza Cost + Soda Cost = $4.8. Adding the prices in this order: Soda Cost + Pizza Cost = $4.8. So, the order in which the amounts were added did not matter. This is always true in any addition problem regardless of the numbers being added. In algebra, we call this property the Commutative Property of Addition. And since variables represent numbers in algebra, the property applies to all algebraic terms, not just numbers. So, the property simply says that the order in which we add does not matter. We will get the same answer (called the sum) regardless of the order in which we add the terms. 57

15 CCBC Math 08 Algebraic Properties Section 5. pages COMMUTATIVE PROPERTY OF ADDITION WORDS SYMBOLS EXAMPLE Changing the order in which two terms are added does not change the sum (answer). a b b a Commutative Property of Multiplication One way to interpret a multiplication problem such as ab is to calculate a groups of b. You went to the store and purchased bags of apples with 5 apples in each bag. You have groups of 5 apples. To determine how many apples you bought you would multiply as follows: 5 5 apples Your friend went to a different store and purchased 5 bags of apples with apples in each bag. Your friend has 5 groups of apples. To determine how many apples your friend bought you would multiply as follows: 5 5 apples You and your friend purchased the same number of apples. The way they were packaged was the only difference. Notice that the order in which we multiply factors does not matter. We will get the same answer (called the product) regardless of the order in which we multiply factors. COMMUTATIVE PROPERTY OF MULTIPLICATION WORDS SYMBOLS EXAMPLE Changing the order in which two factors are multiplied does not change the product (answer). ab b a

16 CCBC Math 08 Algebraic Properties Section 5. pages Example : Addition. Practice : Addition. Rewrite each expression using the Commutative Property of Multiplication or x + 5 Answer: 5 + x (x)(8) Answer: (8)(x) which equals 8x 4 + 9x Answer: 9x + 4 yx Answer: xy x Answer: -x + -5x + 6 Answer: 6 + (-5x) (-)() Answer: ()(-) Rewrite each expression using the Commutative Property of Multiplication or a. y + 9 b. (c)( 7) c. + 6w d. (a)(c) e. 4x 5 = 4x + (-5) Answers: a. 9 + y b. (-7)(c) = -7c c. 6w + (-) = 6w d. (c)(a) e x The real beauty and power of the Commutative Property of Multiplication is seen when we multiply three or more factors. For instance, to solve the problem 5 7, we could follow our typical order of operations and work left to right. First we must multiply 5 with 7. Take a moment to perform that multiplication. 5 7 Next, complete the problem by multiplying 85 with. 85 The final answer is Did you get that? Wasn t all that multiplication fun? Now we are going to rework the exact same problem, but this time, let s use the Commutative Property of Multiplication. The Commutative Property of Multiplication says that the order of the factors does not matter. Switch the order of the second and third numbers. 5 7 Work from left to right just as we did last time. First multiply 5 with. That s 0, a simple product to figure out. Finally, multiplying a whole number by 0 is as simple as placing a 0 at the end of that whole number

17 CCBC Math 08 Algebraic Properties Section 5. pages So, while our first method involved a mildly time consuming multiplication problem, you must admit that the second method was much quicker and easier. Now do you see the beauty and power of the Commutative Property of Multiplication? Calculate the problem using the appropriate Commutative Property. Example : = Commute the nd and rd terms. = Add the st two terms. = 79 Calculate the problem using the appropriate Commutative Property. Practice : Answer: = = 9 Calculate the problem using the appropriate Commutative Property. Example : Commute the nd and rd factors. 0 9 Multiply the st two factors. = 80 Calculate the problem using the appropriate Commutative Property. Practice : 54 Answer: Simplify the problem using the appropriate Commutative Property, then combine like terms. Example 4: 6x + + 4x = 6x + 4x + Using the Commutative Property of Addition, rewrite the expression with the like terms next to each other. = 0x + Combine like terms. Simplify the problem using the appropriate Commutative Property, then combine like terms. Practice 4: 5a 8 a Answer: -8a

18 CCBC Math 08 Algebraic Properties Section 5. pages Simplify the problem using the appropriate Commutative Property, then combine like terms. Example 5: x + 4y 0x + 7y = x + 4y + (-0 x) + 7y Write as a sum of terms. = x + (-0x) + 4y + 7y Using the Commutative Property of Addition, rewrite the expression with the like terms next to each other. = -7x + y Combine like terms. Note: The sign (+/-) to the left of a term is part of the term and thus will move with the term when using the Commutative Property. Simplify the problem using the appropriate Commutative Property, then combine like terms. Practice 5: 8x y + x 5y Answer: 0x 8y Simplify the problem using the appropriate Commutative Property, then combine like terms. Example 6: -x x + x 6 =(-x) (-7) + 8x + x + (-6) Write as a sum of terms. = -x + 8x + x (-7) + (-6) Using the Commutative Property of Addition, rewrite the expression with the like terms next to each other. = 7x 9 Combine like terms. Simplify the problem using the appropriate Commutative Property, then combine like terms. Practice 6: -7a a + 4a 5 Answer: -a 6

19 CCBC Math 08 Algebraic Properties Section 5. pages Associative Property of Addition George and Angela work at the CCBC bookstore and are preparing to stock the shelves before the semester starts. In the stock room, George counts the number of Calculus books and puts them in a box. Then he counts the number of Calculus books and puts them in the same box. Then he counts the number of Calculus books, but puts them in a second box since the first box is full. Before going on break, George calculates the total number of Calculus books as shown below and records his total. First Box Calculus + Calculus Second Box Calculus First Box (40 + 0) Second Box 5 First Box 70 Second Box 5 Image from Microsoft Office Clip Art Total of Both Boxes 85 6

20 CCBC Math 08 Algebraic Properties Section 5. pages While George is on break, Angela begins to pick up the first box of Calculus books to carry it out and place the books on the shelf. But she finds it much too heavy to pick up. She notices that there aren t many books in the second box. So Angela takes the Calculus books out of the first box and places them in the second box to distribute the weight more evenly. She then carries each box out. Angela takes the Calculus books out of the first box, counts them, and places them on the shelf. She then takes the Calculus books and the Calculus books out of the second box, counts them, and places them on the shelf. Angela counts the books as shown below and records her total. First Box Calculus Second Box Calculus + Calculus First Box 40 Second Box (0 + 5) First Box 40 Second Box 45 Image from Microsoft Office Clip Art Total of Both Boxes 85 George obtained the total number of books by adding: (40 + 0) + 5. Angela obtained the total number of books by adding: 40 + (0 + 5). Notice that George and Angela both had the same total of 85 books, and that is exactly what we would expect. Pay particular attention to how the two methods differ. It is NOT the order of the numbers that differs. The order was the same in both methods. Both George and Angela started with the number 40, then followed with the number 0, and ended with the number 5. What IS different is the grouping of the numbers. George grouped 40 and 0 together in parentheses because he had the Calculus I and II books in the same box. Angela, on the other hand, grouped the Calculus II and III books in the same box, so she grouped the numbers 0 and 5 together in parentheses. So, the grouping of the numbers did not matter in the addition problem. This is always true in any addition problem regardless of the numbers being added. In algebra, we call this property 6

21 CCBC Math 08 Algebraic Properties Section 5. pages the Associative Property of Addition. As with the Commutative Property of Addition, it applies to all algebraic terms, not just numbers. So, the property simply says that the grouping in an addition problem does not matter. We will get the same answer (called the sum) regardless of how the terms are grouped. When you use the Associative Property of Addition, think of the word association which means group. This will help you to remember that the property involves a change in grouping. ASSOCIATIVE PROPERTY OF ADDITION WORDS SYMBOLS EXAMPLE Changing the grouping of terms being added does not change the sum (answer). ( a b) c a ( b c) Notice that the order of the terms is the same on both sides of the equation. (4 6) 4 (6 ) There is an Associative Property of Multiplication as well. The grouping of the factors in a multiplication problem does not matter. We will get the same answer (called the product) regardless of how the factors are grouped. ASSOCIATIVE PROPERTY OF MULTIPLICATION WORDS SYMBOLS EXAMPLE Changing the grouping of factors being multiplied does not change the product (answer). (ab)c = a(bc) Notice that the order of the factors is the same on both sides of the equation. (46) 4 (6)

22 CCBC Math 08 Algebraic Properties Section 5. pages Rewrite each expression using the appropriate Associative Property. Example 7: (7 + x) + 8 Answer: 7 + (x + 8) Note: The order of the terms is the same in the original expression as it is in the answer. What has changed is the grouping. The 7 and x are grouped together in the original expression, whereas the x and 8 are grouped together in the answer. Practice 7: (y + 8) + 4 Answer: y + (8 + 4) Example 8: (9 y) Answer: (9) y Note: The order of the factors is the same in the original expression as it is in the answer. What has changed is the grouping. The 9 and y are grouped together in the original expression, whereas the and 9 are grouped together in the answer. Practice 8: 5 (7 a) Answer: (57) a Show that each statement is true using the appropriate Associative Property. Example 9: (6 + ) + 5 = 6 + ( + 5) (6 ) 5 6 ( 5) Practice 9: (7 + 4) + 8 = 7 + (4 + 8) Answer: 9 = 9 Example 0: 4 () (4) 4 () (4) Practice 0: (79) (7) 9 Answer: 6 =

23 CCBC Math 08 Algebraic Properties Section 5. pages We demonstrated the power and beauty of the Commutative Property just a few pages back. Let s illustrate the power and beauty of the Associative Property with the problem To make this computation, we follow the order of operations discussed in Chapter. We start by adding the fractions in the parentheses To add fractions you need a Least Common Denominator. The Least Common Denominator is Perform the pair of multiplications inside the parentheses The two fractions inside the parentheses have a common denominator. So, we add those fractions To add these two fractions, we need a common denominator. The Least Common Denominator, once again, is Perform the multiplication Perform the addition. 4 Simplify Convert to a mixed fraction. 66

24 CCBC Math 08 Algebraic Properties Section 5. pages Hopefully, you haven t lost sight of our purpose here. We are trying to show the usefulness of the Associative Property. Next, we will rework the same problem after applying the Associative Property of Addition. Compute by using the Associative Property of Addition First apply the Associative Property of Addition in order to regroup the terms Following the order of operations, we add the fractions inside the parentheses. This is easy to do because they have the same denominator. 8 8 A fraction with the same numerator and denominator is. Write answer as a mixed number What a difference in these two methods! The first method took 9 steps and quite a bit of time and work. The second method took only 4 steps and hardly any time or work. So, are you now impressed by the power of the Associative Property? As shown in some of the examples below, the property is also very useful in simplifying algebraic expressions in order to get them in proper form and make them easier to work with. 67

25 CCBC Math 08 Algebraic Properties Section 5. pages Simplify each expression using the appropriate Associative Property. Example : (x + ) + 6 x ( 6) Regroup like terms using Associative Property of Addition. x 9 Simplify inside the parentheses. Practice : (y + 8) + 4 Answer: y + Example : 8(4x) 8 (4 x) (8 4) x Regroup using Associative Property of Multiplication. x Simplify inside the parentheses. Practice : 5 (7 a) Answer: 5a Example : 5x + (x + 0) (5x x) 0 7x 0 Regroup like terms using Associative Property of Addition. Simplify inside the parentheses by combining like terms. Practice : 9a + (4a + 6) Answer: a Example 4: x 4 x 4 x 4 x 4 x 7 Regroup using the Associative Property of Addition. Write mixed numbers using addition. Rewrite using Commutative Property of Addition. Add whole numbers together and fractions together. x (7 ) Simplify. x 8 68

26 CCBC Math 08 Algebraic Properties Section 5. pages Practice 4: 5x Answer: 5x + Note: The operations of subtraction and division do not have a Commutative Property or an Associate Property. Subtraction is NOT commutative: Subtraction is NOT associative: 00 (0 0) (00 0) Division is NOT commutative: Division is NOT associative: 64 (8 4) (64 8)

27 CCBC Math 08 Algebraic Properties Section 5. pages Identity Property Nick is a dialysis patient. Each time he goes to the dialysis center for treatment, the nurse begins by weighing Nick. It is very important to compare Nick s previous weight with his current weight in order to determine how much fluid to take off during the dialysis treatment. This time, the nurse weighs Nick and records the following in her chart. Date Previous Weight Weight Gain/Loss 08/ Current Weight Since Nick had no weight gain this time, his current weight is , which, of course, equals 40. Thus, Nick s current weight is the same as his previous weight. In any addition problem, adding 0 to a number results in the same number. In algebra, this fact is called the Identity Property of Addition. When you use the Identity Property of Addition, think of the word identical. Adding 0 to a term produces the identical term. IDENTITY PROPERTY OF ADDITION WORDS SYMBOLS EXAMPLE The sum of any number and 0 is the same number. The sum of any term and 0 is the same term. a 0 0 a a x 0 0 x x Note: The Additive Identity Element is 0. There is also an Identity Property of Multiplication. Does it work exactly the same? For instance, does 6(0) = 6? No, we know that 6(0) = 0. So, 0 is not the multiplicative identity. The question is: 6 = 6? Hopefully you see that the number must fill in the box to make a true statement. Multiplying a number or expression by results in that same number or expression. That is exactly what the Identity Property of Multiplication states. IDENTITY PROPERTY OF MULTIPLICATION WORDS SYMBOLS EXAMPLE The product of any factor and is that same factor. aa a 66 6 xx x Note: The Multiplicative Identity Element is. 70

28 CCBC Math 08 Algebraic Properties Section 5. pages Example 5: Complete each equation using the appropriate Identity Property. 7x + 0 =? 5x() =? Answer: 7x + 0 = 7x Answer: 5x() = 5x 0 +? = 9x Answer: 0 + 9x = 9x x(? ) = x Answer: x( ) = x Practice 5: Complete each equation using the appropriate Identity Property. a. 8a + 0 =? b. 9y() =? c. 0 +? = w d. 6a(? ) = 6a Answers: a. 8a b. 9y c. w d. Inverse Property of Addition The Student Government Association is sponsoring a dance, spending $64 on a DJ, decorations, and refreshments. How much money does the SGA need to make from ticket sales in order to break even that is, to pay all of their expenses (without making any profit)? The money the SGA has already spent is represented as a negative number, -64. The goal to break even is represented as the number 0. A positive 64 is needed to offset the -64. Thus, the SGA will need to make $64 from ticket sales in order to break even. This can be written as: In fact, the sum of any number and its opposite is 0. In algebra, we state this as the Inverse Property of Addition. Moreover, -64 is said to be the opposite of 64, and 64 is said to be the opposite of -64. Another term for opposite is additive inverse. 7

29 CCBC Math 08 Algebraic Properties Section 5. pages INVERSE PROPERTY OF ADDITION WORDS SYMBOLS EXAMPLE The sum of a number and its opposite is 0. The sum of a term and its opposite is 0. a ( a) 0 4 ( 4) 0 x + (-x) = 0 Inverse Property of Multiplication There is also an Inverse Property of Multiplication. It would be used if we multiplied two numbers together and obtained the answer, which is the multiplicative identity. The two numbers that multiply together equaling are reciprocals of each other. Recall we first learned about reciprocals in chapter when we divided fractions. For instance, The number 8 is the reciprocal of 8, and 8 is the reciprocal of. Another term for reciprocal 8 is multiplicative inverse. INVERSE PROPERTY OF MULTIPLICATION WORDS SYMBOLS EXAMPLE The product of a number and its reciprocal is. The product of a term and its reciprocal is. a a 5 5 x x 7

30 CCBC Math 08 Algebraic Properties Section 5. pages Complete each equation using the appropriate Inverse Property. Example 6: Complete each equation using the appropriate Inverse Property. 9 +? = 0 Answer: = 0? = Answer: x + (-x) =? Answer: x + (-x) = 0 - +? = 0 Answer: - + = 0? 4 Answer: 4 4-7y + 7y =? Answer: -7y + 7y = 0 4 x? Answer: 4x 4x Practice 6: Complete each equation using the appropriate Inverse Property. a. 7 +? = 0 b.? = c. w + (-w) =? d. -5 +? = 0 e.? 5 f. -x + x =? g. 4 x? Answers: a. -7 b. c. 0 d. 5 e. 5 f. 0 g. 4x Watch All: 7

31 CCBC Math 08 Algebraic Properties Section 5. pages Zero Property of Multiplication There is one final property to learn. It is called the Zero Property of Multiplication and works only with multiplication. What does 7 0 equal? How about 8 0? How about -64 0? Hopefully, your answer was 0 for all three of these problems. We know that any number multiplied by 0 will give us an answer of 0. The Zero Property of Multiplication states this fact. ZERO PROPERTY OF MULTIPLICATION WORDS SYMBOLS EXAMPLE The product of a number and 0 is 0. The product of a term and 0 is 0. a0 0a x0 0x 0 You have learned five properties in this section. Before you attempt the practice problems, take some time to review the summary of the properties. Make sure that you understand the explanations of the properties and how they apply to addition and multiplication. PROPERTY DESCRIPTION ADDITION MULTIPLICATION Commutative Associative Changing the order does not change the answer. Regrouping does not change the answer. a b b a ab b a ( a b) c a ( b c) ( ab) c a( b c) Identity The sum of any term and 0 is that same term. The product of any factor and is that same factor. a0 a and 0a a a a and a a Inverse The sum of a number and its opposite is 0. The product of a number and its reciprocal is. a ( a) 0 a a Zero The product of a number and 0 is 0. a 0 = 0 and 0 a = 0 74

32 CCBC Math 08 Algebraic Properties Section 5. pages 5. Algebraic Properties Exercises Match the expression or equation to the correct property it illustrates.. 0 A. Commutative Property of Addition. B. Commutative Property of Multiplication. x 4 4 x C. Associative Property of Addition 4. x() x D. Associative Property of Multiplication 5. E. Identity Property of Addition 6. (4 + 0) + 6 = 4 + (0 + 6) F. Identity Property of Multiplication G. Additive Identity Element (-7) = 0 H. Multiplicative Identity Element I. Inverse Property of Addition 0. (9 x) 5x (9x 5 x) J. Inverse Property of Multiplication = 9 K. Zero Property of Multiplication For problems 5, rewrite each expression using the given property. Expression Property x Commutative. x Commutative 4. (x + y) + z Associative 5. x (9 y) Associative 75

33 CCBC Math 08 Algebraic Properties Section 5. pages For problems 6 5, complete each equation using the given property. Equation Property 6. 8a =? ? = 6x 8. Identity Identity 5 6 x? Identity 9. 7.x + 0 =? Identity 0.? Inverse. 5 +? Inverse = 0 a. b? Inverse. x +(-x) =? Inverse =? 5. x? = 0 Zero Zero For problems 6, and 7, the statements below illustrate the Associative Property. Show that the statements are true. 6. (5. +.) = 5. + ( ) 7. 0 (85) (08) 5 76

34 CCBC Math 08 Algebraic Properties Section 5. pages For problems 8 4, simplify each expression using the given property. Problem Property Commutative Commutative x + 5 Commutative. a a Commutative. 6x + + 8x Commutative. (4x + 4) + 8 Associative 4..6x + (7.x + 8) Associative 5. 5 x 5 Associative 6. ( 4) 5 Associative Associative Associative 77

35 CCBC Math 08 Algebraic Properties Section 5. pages 5. Algebraic Properties Exercises Answers. G 7. K. J 8. I. A 9. B 4. F 0. D 5. H. E 6. C. 5x x + (y + z). x 5. ( x 9) y 6. 8a 8a = 0 a b x = 6x. b a x x. x + (-x) = x + 0 = 7.x x (5 ) ( 7 4) 7. 0 (8 5) (0 8)

36 CCBC Math 08 Algebraic Properties Section 5. pages (.6x 7. x) 8 0.9x x x x 0. x8 5 x or 8 5 x x 6. (45) a a 5 5a x 8x 4x x (4 8) 4x 4 79

37 CCBC Math 08 Distributive Property Section 5. 5 pages 5. Distributive Property A property used to simplify algebraic expressions with parentheses is the Distributive Property. When this property is used, multiplication is distributed over addition or subtraction. Distributive Property: For any real numbers a, b, and c, a ( b c ) ab ac The Distributive Property allows us to multiply each term inside the parentheses by the term on the outside of the parentheses. So, to simplify the expression a( b c), first distribute the a to the b; this means to multiply a with b. Then distribute the a to the c; again, this means to multiply a with c. Finally, add the products together. The result is shown below. a ( b c ) ab ac An alternative to this would be to simplify using the order of operations. If the two terms inside the parentheses are like terms, then we add them first and then multiply that sum by a. Example : Simplify: (4 + 7) Method : Order of operations method Method : Distributive property (4 + 7) = () (4 + 7) = (4) + (7) = = + Note: The result is the same using either methods. = Practice : Simplify: 5(8 + 6) Method : Order of operations method Method : Distributive property Answer: 5(4) Example : Distribute. ( x ) Note: It is not possible to use the order of operations method here because it is not possible to combine x and. They are not like terms so they cannot be combined into a single term. Therefore, the Distributive Property is the only approach to simplify this problem. ( x) ( x) () Distribute the through the parentheses. x 6 Simplify. 80

38 CCBC Math 08 Distributive Property Section 5. 5 pages Practice : Distribute. 4(y + ) Answer: 4y Example : Distribute. ( y 5) ( y) ( )(5) Distribute - through the parentheses. 6 y ( 5) Simplify. 6y 5 Rewrite addition of a negative as subtraction. Practice : Distribute. (9a + 4) Answer: -8a 8 Example 4: Distribute. 7(m 9) 7( m) ( 7)(9) Distribute -7 through the parentheses. 4 m( 6) Simplify. 4m 6 Rewrite subtraction of a negative as addition. Practice 4: Distribute. (c 7) Answer: -c Example 5: Distribute. (z 4) ( z) (4) 4 z 4 z 8 z Distribute through the parentheses. Write integers as fractions. Divide out common factors in the numerator and denominator. Simplify. Practice 5: Distribute. (8x 5) Answer: x 8

39 CCBC Math 08 Distributive Property Section 5. 5 pages Example 6: 5 Distribute. x z 5 5 ( x) z x z x 5 5 x z 5 x z 5 z Distribute through the parentheses. 5 Write integers as fractions. Divide out common factors in the numerator and denominator. Rewrite subtraction of a negative as addition. Practice 6: 8 Distribute. (6 h k) Answer: h k Example 7: Distribute. 0.7(4x 0.0) 0.7(4 x) ( 0.7)(0.0).8 x ( 0.0).8x 0.0 Distribute -0.7 through the parentheses. Multiply. Rewrite subtraction of a negative as addition. Practice 7: Distribute..(7a 0.4) Answer: -9.a Watch All: 8

40 CCBC Math 08 Distributive Property Section 5. 5 pages 5. Distributive Property Exercises Distribute.. ( x 4). (4 x ). ( m ). ( 9 p ). 7( x 4) 4. 4 ( 5) 5 k 4. ( k 5) 5. ( 6) h 5. 7(4z ) (y 7) 7. (4 x ) a 7. 74x y p 9. 0 m x p z k (. x.)..5(4..7 y) 8

41 CCBC Math 08 Distributive Property Section 5. 5 pages 5. Distributive Property Exercises Answers. x. x. m 6. 6p. 7x k k 9 54 h 5. 8z x 6. y a x p x y 4 m p z x k y 84

42 CCBC Math 08 Chapter 5 Mid-Chapter Review Sections 5. to 5. CHAPTER 5 Mid-Chapter Review Simplify by combining like terms.. a 6a 8. x 4x 8. 9x x y 5y 4. xy 5xy x 6x x5. x x x x x Evaluate each expression. 8. x if x x if x 0. 6( x ) if x. xx ( ) if x 5. 5x 6y if x 07. and y 8.. x 6 if 0 4 x 4. 0a b if a and b 5 Simplify using the Distributive Property. 5. 9(7x 6) 6. 5(x 8) 7. ( x ) 8. 5(x 0 75) x 6 k a b Use the property named to rewrite each expression.. 6x Commutative Property. x 5 Commutative Property 4. ( a b) c Associative Property 5. 9 (4 x) Associative Property Use the property to complete each expression. 6. Identity Property 7. 8x 0 Identity Property 8. 4 Identity Property Identity Property 9. a a 0. 0 Inverse Property. 7x 7x Inverse Property. 6 Inverse Property 6. n Inverse Property Inverse Property Zero Property Use the property named to simplify each expression a 6 Commutative Property 7. 6x7 4x Commutative Property Commutative Property 9. a Commutative Property Associative Property 4. x(4x 8) Associative Property 4. 5 ( ) Associative Property x 4 a Associative Property Associative Property 85

43 CCBC Math 08 Chapter 5 Mid-Chapter Review Sections 5. to 5. M i d - C h a p t e r 5 R e v i e w A n s w e r s. 9a 4. x 5. 7x 8y 4. xy 5x x x x x x x 6 8. x x 0. 9k. a b 4. 6x. 5x 4. a ( b c) 5. (94) x x0 8x a a x 7x n n a 8 6 5a x 4x 7 0x a 6 a 6a (x 4 x) 8 6x 8 4. ( ) x 4 4 x x a a a 86

44 CCBC Math 08 Solving Equations Using the Addition Property of Equality Section pages 5.4 Solving Equations Using the Addition Property of Equality What is an equation? An equation is a mathematical statement that two expressions are equal. For example, + 5 = 7 is an equation. Note: An equation contains an equal sign, = ; an expression does not contain an equal sign. We solve equations, and we simplify expressions. EQUATIONS VERSUS EXPRESSIONS EQUATION An equation contains an equal sign. EXPRESSION An expression does not contain an equal sign. Equations are solved. Expressions are simplified. Example: x + 5 = 7 Example: x We will see later in this section that the solution to the equation x + 5 = 7 is x = We have learned to simplify x by combining like terms: x + In this section, we will learn how to solve linear equations in one variable; such an equation can have no solution, exactly one solution, or infinitely many solutions. In this course, we will only consider linear equations in one variable that have exactly one solution. In general, solving an equation involves finding all numbers that can replace the variable to make the equation a true statement. For example, a solution of the equation x + = 7 is the number 4, because replacing x with 4 in the equation gives a true statement. x This is a true statement. Thus, x = 4 is a solution of x + = 7. 87

45 CCBC Math 08 Solving Equations Using the Addition Property of Equality Section pages Example : Is x = a solution of the equation x 8 =? Let s think about this problem first. If we say in the place of x, we would read eleven minus 8 on the left side of the equation. Does minus 8 equal? Yes. Now let s do this algebraically. Remember, a solution of an equation is a number that when used in place of the variable makes the equation a true statement. To see if is a solution of x 8 =, we replace x with in the equation and see if the result is a true statement. x 8 8 This is a true statement. Thus, x = is the solution of x 8 =. Practice : Is y = 7 the solution to the equation y = 4? Answer: true Example : Is a = -7 a solution of the equation a 4 = -? Let s think about this problem first. If we replace a with -7, we would read the left side of the equation as Negative seven minus 4. What does -7 4 equal? Remembering the rules for signed numbers we get -. This does not equal the right side of the equation. Therefore, -7 is not a solution for this equation. Now, let s use algebra to determine this. To see if -7 is a solution of a 4 = -, we replace a with -7 in the equation and see if the result is a true statement. a This is a false statement.. Thus, a = -7 is not the solution of a 4 = -. Practice : Is c = 9 the solution to the equation c + = 7? Answer: false 88

46 CCBC Math 08 Solving Equations Using the Addition Property of Equality Section pages Addition Property of Equality We want to develop a process for solving linear equations in one variable. One property needed for solving the equations in this section is called the Addition Property of Equality. Let a, b, and c be expressions representing real numbers. Then, if a = b, then a + c = b + c. This property states that when you add an expression to one side of an equation, you must also add the same expression to the other side of the equation. Adding the same expression to both sides of an equation will produce an equivalent equation. Therefore, adding the same expression to both sides of an equation will not change the equation s solution. ADDITION PROPERTY OF EQUALITY WORDS SYMBOLS Adding the same expression to both sides of an equation will produce an equivalent equation. If a = b, then a + c = b + c Additive Inverse Recall the Inverse Property of Addition that we studied in a previous section. INVERSE PROPERTY OF ADDITION WORDS SYMBOLS EXAMPLE The sum of a number and its opposite is 0. The sum of a term and its opposite is 0. a ( a) 0 4 ( 4) 0 x + (-x) = 0 Additive Inverse: The additive inverse of a is -a. Similarly, the additive inverse of -a is a. We can say that -a is the opposite of a. We also say that the inverse operation of addition is subtraction and the inversion operation of subtraction is addition. Note: When two opposites are added together the answer is 0. 89

47 CCBC Math 08 Solving Equations Using the Addition Property of Equality Section pages Example : Solve p = 5. Then, check the solution. Let s try to do this by thinking about what the equation says. What number minus is equal to 5? The answer is 6. Now, we will learn how to solve this equation algebraically. To solve the equation means to determine the value of the variable that makes the equation a true statement. To do this, we want to get the variable on one side of the equation by itself; we call this isolating the variable. On the left side of the equal sign, is subtracted from p. To isolate p, we perform the inverse operation; the inverse of subtraction is addition. Thus, we add to each side of the equation. p 5 p + 5+ p 06 Use the Addition Property of Equality. Add to both sides of the equation. Simplify. p 6 Typically, we do not write the third step: p + 0 = 6. However, even if unwritten, we are using the idea of the additive identity: adding a number and 0 (p + 0 in this problem) produces p, thereby isolating the variable. Check: Substitute 6 for p in the original equation. p This is a true statement. Thus, p = 6 is a solution for the equation p = 5. Practice : Solve h + 8 =. Then, check the solution. Answer: h = -6 90

48 CCBC Math 08 Solving Equations Using the Addition Property of Equality Section pages Example 4: Solve 9 = m 4. Then, check the solution. First, let s try to do this by thinking about what the equation says. Negative nine equals what number minus 4? This one is a little tricky, but if we remember our rules for signed numbers we would see that the answer is -5. Now, let s use algebra to isolate the variable to solve this equation. On the right side of the equal sign, 4 is subtracted from m. To isolate m, we perform the inverse operation; the inverse of subtraction is addition. Thus, we add 4 to each side of the equation. 9 m m m 0 Use the Addition Property of Equality. Add 4 to both sides of the equation. Simplify. 5 m Typically, we do not write the third step: -5 = m + 0. However, even when it is unwritten, we are using the idea of the additive identity: adding a number and 0 (m + 0 in this problem) produces m, thereby isolating the variable. Check: Substitute -5 for m in the original equation. 9 m This is a true statement. Therefore, m = -5 is the solution of 9 = m 4. Practice 4: Solve 8 = w. Then, check the solution. Answer: w = -7 Example 5: Solve x +.6 = 4. Then, check the solution. On the left side of the equal sign,.6 is added to x. To isolate x, we perform the inverse operation; the inverse of addition is subtraction. Thus, we subtract.6 from each side of the equation. x.6 4 x Subtract.6 from both sides of the equation. x 0.4 Simplify. x.4 9

49 CCBC Math 08 Solving Equations Using the Addition Property of Equality Section pages Check: Substitute.4 for x in the original equation. x This is a true statement. Therefore, x =.4 is the solution of x.6 4. Practice 5: Solve c = 9. Then, check the solution. Answer: c =.4 Example 6: Solve h. Then, check the solution. 0 5 On the left side of the equal sign, is subtracted from h. To isolate h, we perform the inverse 0 operation; the inverse of subtraction is addition. Thus, we add to each side of the equation. 0 h 0 5 h h Add to both sides of the equation. 0 Write fractions with a common denominator. 7 h 0 Add numerators to simplify. Check: Substitute 7 for h in the original equation. 0 h This is a true statement. Therefore, h is the solution of 0 9 h. 0 5

50 CCBC Math 08 Solving Equations Using the Addition Property of Equality Section pages Practice 6: Solve 8 x. Then, check the solution. 9 Answer: x 9 Watch All: 9

51 CCBC Math 08 Solving Equations Using the Addition Property of Equality Section pages 5.4 Solving Equations using the Addition Property Exercises Determine whether the given value is a solution to the equation.. 5 = w 4 w = 9. = - + z z = 7. x 8 = 0 x = = -6 + h h = 5. = x + 7 x = 9. Solve each equation and check the solution. 6. n 6 = 7. x = 9 8. a = - 9. x 6 = = y. a.9 = = p b 8 d 9 5 h r 4. = = -7 + x Solve each of the following equations. 8. y = 7 9. x 6 = = a 94

52 CCBC Math 08 Solving Equations Using the Addition Property of Equality Section pages 5.4 Solving Equations using the Addition Property Exercises Answers. True Statement. False Statement. True Statement 4. True Statement 5. False Statement 6. n = 9 7. x = 8. a = 9. x = - 0. y = -9. a = p = -.7. b = d 9 7 h 8 6. r = x = 7 8. y = 8 9. x = 0. a = - 95