Adding and Subtracting Terms 1.6 OBJECTIVES 1.6 1. Identify terms and like terms 2. Combine like terms 3. Add algebraic expressions 4. Subtract algebraic expressions To find the perimeter of (or the distance around) a rectangle, we add 2 times the length and 2 times the width. In the language of algebra, this can be written as L W W Perimeter 2L 2W L We call 2L 2W an algebraic expression, or more simply an expression. Recall from Section 1.1 that an expression allows us to write a mathematical idea in symbols. It can be thought of as a meaningful collection of letters, numbers, and operation signs. Some expressions are 5x 2 3a 2b 4x 3 (2y) 1 In algebraic expressions, the addition and subtraction signs break the expressions into smaller parts called terms. Definitions: Term A term is a number, or the product of a number and one or more variables, raised to a power. In an expression, each sign ( or ) is a part of the term that follows the sign. Example 1 Identifying Terms NOTE This could also be written as 4x 3 2y 1 (a) 5x 2 has one term. (b) 3a 2b has two terms: 3a and 2b. Term Term (c) 4x 3 (2y) 1 has three terms: 4x 3, 2y, and 1. Term Term Term CHECK YOURSELF 1 List the terms of each expression. (a) 2b 4 (b) 5m 3n (c) 2s 2 3t 6 Note that a term in an expression may have any number of factors. For instance, 5xy is a term. It has factors of 5, x, and y. The number factor of a term is called the numerical coefficient. So for the term 5xy, the numerical coefficient is 5. 115
116 CHAPTER 1 THE LANGUAGE OF ALGEBRA Example 2 Identifying the Numerical Coefficient (a) 4a has the numerical coefficient 4. (b) 6a 3 b 4 c 2 has the numerical coefficient 6. (c) 7m 2 n 3 has the numerical coefficient 7. (d) Because 1 x x, the numerical coefficient of x is understood to be 1. CHECK YOURSELF 2 Give the numerical coefficient for each of the following terms. (a) 8a 2 b (b) 5m 3 n 4 (c) y If terms contain exactly the same letters (or variables) raised to the same powers, they are called like terms. Example 3 Identifying Like Terms (a) The following are like terms. 6a and 7a 5b 2 and b 2 10x 2 y 3 z and 6x 2 y 3 z 3m 2 and m 2 (b) The following are not like terms. Different letters Each pair of terms has the same letters, with each letter raised to the same power the numerical coefficients can be any number. 6a and 7b 5b 2 and b 3 Different exponents Different exponents 3x 2 y and 4xy 2 CHECK YOURSELF 3 Circle the like terms. 5a 2 b ab 2 a 2 b 3a 2 4ab 3b 2 7a 2 b Like terms of an expression can always be combined into a single term. Look at the following: 2x 5x 7x x x x x x x x x x x x x x x
ADDING AND SUBTRACTING TERMS SECTION 1.6 117 NOTE Here we use the distributive property from Section 1.2. NOTE You don t have to write all this out just do it mentally! Rather than having to write out all those x s, try 2x 5x (2 5)x 7x In the same way, 9b 6b (9 6)b 15b and 10a (4a) (10 (4))a 6a This leads us to the following rule. Step by Step: To Combine Like Terms To combine like terms, use the following steps. Step 1 Add or subtract the numerical coefficients. Step 2 Attach the common variables. Example 4 Combining Like Terms NOTE Remember that when any factor is multiplied by 0, the product is 0. Combine like terms.* (a) 8m 5m (8 5)m 13m (b) 5pq 3 4pq 3 5pq 3 (4pq 3 ) 1pq 3 pq 3 (c) 7a 3 b 2 7a 3 b 2 7a 3 b 2 (7a 3 b 2 ) 0a 3 b 2 0 CHECK YOURSELF 4 Combine like terms. (a) 6b 8b (b) 12x 2 3x 2 (c) 8xy 3 7xy 3 (d) 9a 2 b 4 9a 2 b 4 Let s look at some expressions involving more than two terms. The idea is just the same. Example 5 Combining Like Terms NOTE The distributive property can be used over any number of like terms. Combine like terms. (a) 5ab 2ab 3ab 5ab (2ab) 3ab (5 (2) 3)ab 6ab *When an example requires simplification of an expression, that expression will be screened. The simplification will then follow the equals sign.
118 CHAPTER 1 THE LANGUAGE OF ALGEBRA Only like terms can be combined. (b) 8x 2x 5y (8 (2)) x 5y 6x 5y Like terms Like terms NOTE With practice you won t be writing out these steps, but doing it mentally. (c) 5m 8n 4m 3n (5m 4m) (8n (3n)) 9m 5n Here we have used the associative and commutative properties. (d) 4x 2 2x 3x 2 x (4x 2 (3x 2 )) (2x x) x 2 3x As these examples illustrate, combining like terms often means changing the grouping and the order in which the terms are written. Again all this is possible because of the properties of addition that we introduced in Section 1.2. CHECK YOURSELF 5 Combine like terms. (a) 4m 2 3m 2 8m 2 (b) 9ab 3a 5ab (c) 4p 7q 5p 3q As you have seen in arithmetic, subtraction can be performed directly. As this is the form used for most of mathematics, we will use that form throughout this text. Just remember, by using negative numbers, you can always rewrite a subtraction problem as an addition problem. Example 6 Combining Like Terms Combine the like terms. (a) 2xy 3xy 5xy (b) 5a 2b 7b 8a (2 3 5)xy (5a 8a) (2b 7b) 4xy 3a 5b CHECK YOURSELF 6 Combine like terms. (a) 4ab 5ab 3ab 7ab CHECK YOURSELF ANSWERS (b) 2x 7y 8x y 1. (a) 2b 4 ; (b) 5m, 3n; (c) 2s 2, 3t, 6 2. (a) 8; (b) 5; (c) 1 3. The like terms are 5a 2 b, a 2 b, and 7a 2 b 4. (a) 14b; (b) 9x 2 ; (c) xy 3 ; (d) 0 5. (a) 9m 2 ; (b) 4ab 3a; (c) 9p 4q 6. (a) ab; (b) 6x 8y
Name 1.6 Exercises Section Date List the terms of the following expressions. 1. 5a 2 2. 7a 4b ANSWERS 1. 3. 4x 3 4. 3x 2 5. 3x 2 3x 7 6. 2a 3 a 2 a Circle the like terms in the following groups of terms. 7. 5ab, 3b, 3a, 4ab 2. 3. 4. 5. 6. 7. 8. 9m 2, 8mn, 5m 2, 7m 9. 4xy 2, 2x 2 y, 5x 2, 3x 2 y, 5y, 6x 2 y 10. 8a 2 b, 4a 2, 3ab 2, 5a 2 b, 3ab, 5a 2 b Combine the like terms. 11. 3m 7m 12. 6a 2 8a 2 13. 7b 3 10b 3 14. 7rs 13rs 15. 21xyz 7xyz 16. 4mn 2 15mn 2 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 17. 9z 2 3z 2 18. 7m 6m 19. 19. 5a 3 5a 3 20. 13xy 9xy 21. 19n 2 18n 2 22. 7cd 7cd 23. 21p 2 q 6p 2 q 24. 17r 3 s 2 8r 3 s 2 25. 10x 2 7x 2 3x 2 26. 13uv 5uv 12uv 20. 21. 22. 23. 24. 25. 26. 119
ANSWERS 27. 28. 29. 30. 31. 27. 9a 7a 4b 28. 5m 2 3m 6m 2 29. 7x 5y 4x 4y 30. 6a 2 11a 7a 2 9a 31. 4a 7b 3 2a 3b 2 32. 5p 2 2p 8 4p 2 5p 6 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 2 33. 34. 3 m 3 4 3 m 13 35. 36. 5 x 2 3 5 x 5 37. 2.3a 7 4.7a 3 38. 5.8m 4 2.8m 11 Perform the indicated operations. 39. Find the sum of 5a 4 and 8a 4. 40. Find the sum of 9p 2 and 12p 2. 41. Subtract 12a 3 from 15a 3. 42. Subtract 5m 3 from 18m 3. 43. Subtract 4x from the sum of 8x and 3x. 44. Subtract 8ab from the sum of 7ab and 5ab. 45. Subtract 3mn 2 from the sum of 9mn 2 and 5mn 2. 46. Subtract 4x 2 y from the sum of 6x 2 y and 12x 2 y. 1 5 a 2 4 5 a 17 12 y 7 7 12 y 3 Use the distributive property to remove the parentheses in each expression. Then simplify by combining like terms. 51. 52. 47. 2(3x 2) 4 48. 3(4z 5) 9 49. 5(6a 2) 12a 50. 7(4w 3) 25w 51. 4s 2(s 4) 4 52. 5p 4( p 3) 8 120
ANSWERS 53. Write a paragraph explaining the difference between n 2 and 2n. 53. 54. Complete the explanation: x 3 and 3x are not the same because... 54. 55. Complete the statement: x 2 and 2x are different because... 55. 56. Write an English phrase for each algebraic expression below: 56. (a) 2x 3 5x (b) (2x 5) 3 (c) 6(n 4) 2 57. Work with another student to complete this exercise. Place,, or in the blank in these statements. 57. 58. 1 2 2 1 2 3 3 2 3 4 4 3 4 5 5 4 What happens as the table of numbers is extended? Try more examples. What sign seems to occur the most in your table?,, or? Write an algebraic statement for the pattern of numbers in this table. Do you think this is a pattern that continues? Add more lines to the table and extend the pattern to the general case by writing the pattern in algebraic notation. Write a short paragraph stating your conjecture. 58. Work with other students on this exercise. n 2 1 Part 1: Evaluate the three expressions, n, n2 1 using odd values of n: 2 2 1, 3, 5, 7, etc. Make a chart like the one below and complete it. n 1 3 5 7 9 11 13 a n 2 1 2 b n c n 2 1 2 a 2 b 2 c 2 Part 2: The numbers a, b, and c that you get in each row have a surprising relationship to each other. Complete the last three columns and work together to discover this relationship. You may want to find out more about the history of this famous number pattern. 121
ANSWERS a. b. c. d. Getting Ready for Section 1.7 [Section 0.3] Write the following using exponential notation. (a) 4 4 4 (b) 6 6 6 6 6 6 e. (c) 3 3 3 3 3 (d) (2) (2) (2) f. (e) (8) (8) (8) (8) (f) 9 9 9 9 9 9 9 9 Answers 1. 5a, 2 3. 4x 3 5. 3x 2, 3x, 7 7. 5ab, 4ab 9. 2x 2 y, 3x 2 y, 6x 2 y 11. 10m 13. 17b 3 15. 28xyz 17. 6z 2 19. 0 21. n 2 23. 15p 2 q 25. 6x 2 27. 2a 4b 29. 3x y 31. 2a 10b 1 33. 2m 3 35. 2x 7 37. 7a 10 39. 13a 4 41. 3a 3 43. 7x 45. 11mn 2 47. 6x 8 49. 42a 10 51. 6s 12 53. 55. 57. a. 4 3 b. 6 6 c. 3 5 d. (2) 3 e. (8) 4 f. 9 8 122