Algebra Chapter 15 PDF

FOR ONLINE USE ONLY DO NOT DUPLICATE Chapter Fifteen Algebra Introduction Algebra is a branch of mathematics which uses both alphabetical letters and...

FOR ONLINE USE ONLY DO NOT DUPLICATE Chapter Fifteen Algebra Introduction Algebra is a branch of mathematics which uses both alphabetical letters and LY numerals. The alphabetical letters represent either numbers or objects. In this chapter, you will learn about algebraic expressions. You will also learn how to use arithmetic operations to represent algebraic expressions and N their simplifications. O Algebraic expressions Algebra is used in everyday life to simplify mathematical operations. For SE example; suppose that, each of four families has one chicken; If we represent a chicken by the alphabet k, then, the total number of chickens owned by the four families is U k + k + k + k = 4k. Similarly, suppose that, the four families keep goats as follows: the first family E has 2 goats, the second family has 3 goats, the third family has 2 goats, and N the fourth family has 4 goats. If we represent a goat by the alphabet m, then the total number of goats owned by the four families is LI 2m + 3m + 2m + 4m = 11m. N Again, if the four families have children as follows: the first family has 2 O children, the second family has 3 children, the third family has 3 children, and the fourth family has 4 children. If we represent a child by the alphabet R w, then the total number of children in the four families is 2w + 3w + 3w + 4w = 12w. FO 298 MATH std 5.indd 298 30/07/2021 14:50  FOR ONLINE USE ONLY DO NOT DUPLICATE The algebraic expressions obtained for the four families are 4k, 11m and 12w. These are examples of algebraic expressions in which k represents chicken, m represents goats, and w represents children. In the term of the expression such as 4k, the number 4 is called the coefficient of k and the alphabet k is called the variable. Similarly, in the term 11m, 11 is the coefficient of m and m is the variable. Also, in the term 12w, 12 is the coefficient of w and w is the variable. A term may also have two or more LY variables, such as in the expression 8km, where 8 is the coefficient of k and m, and k and m are variables. N Example 1 O Identify the coefficient and the variable in each of the following terms: (a) 6a (b) y SE Answer U (a) In the term 6a; 6 is the coefficient of a and a is the variable. (b) For the term y; 1 is the coefficient of y and y is the variable. E Example 2 N Write the variables and coefficient in each of the following terms: LI (a) 3pq (b) 5wz N Answer O (a) In the term 3pq; 3 is the coefficient, while p and q are variables. (b) In the term 5wz; 5 is the coefficient, while w and z are variables. R Algebraic expressions are constructed using terms such as 4k, 11m and FO 12w. Algebraic expressions may be combined using arithmetic operations. The following are examples of algebraic expressions: 4k + 6k; 11m – m; 6m × 3; 12w ÷ 2n. 299 MATH std 5.indd 299 30/07/2021 14:50  FOR ONLINE USE ONLY DO NOT DUPLICATE Exercise1 Answer the following questions: 1. Write the variables in each of the following terms: (a) 7mn (b) 5p (c) n (d) 10tp (e) 3yw LY 2. Write the coefficient in each of the following terms: (a) 3pq (b) 4ab (c) pk N (d) m (e) a O Addition of algebraic expressions Terms in the algebraic expression are said to be alike if they have the same SE variable, such as 4k, 8k, 5k and 7k. Like terms can be added to obtain a single algebraic term. In adding like terms, you add the coefficients and leave the variable unchanged. Terms which have different variables are called unlike U terms. For example; 3a, 4m, 6p and 7w. Unlike terms cannot be added to obtain a single algebraic term. E Example 1 N LI Simplify the following expressions: 1. (a) 4k + 8k = N (b) 6n + n + 2n = 2. (a) 2p + 4a + p + 2a = O (b) 4m + 3n + 4m + 3n = R Solution FO 1. (a) 4k + 8k = First, add the coefficients: 4 + 8 = 12. Therefore, 4k + 8k = 12k. 300 MATH std 5.indd 300 30/07/2021 14:50  FOR ONLINE USE ONLY DO NOT DUPLICATE (b) 6n + n + 2n = First, add the coefficients: 6 + 1 + 2 = 9. Therefore, 6n + n + 2n = 9n. 2. (a) 2p + 4a + p + 2a = First, rewrite the expression by collecting together the like terms as follows: LY 2p + p + 4a + 2a. Add the coefficients of like terms for the variable p: 2 + 1 = 3. N Add the coefficients of like terms for the variable a: 4 + 2 = 6. Therefore, 2p + p + 4a + 2a = 3p + 6a. O (b) 4m + 3n + 4m + 3n = Add the like terms: (4 + 4)m + (3 +3)n = 8m + 6n. SE Therefore, 4m + 3n + 4m + 3n = 8m + 6n. Exercise 2 U Simplify the following expressions: E 1. 3p + 4p + 5p = 2. 2m + 7m + m = N 3. m + n + m + n + m = 4. p+w+p+w+p+w+w= LI 5. 5n + k + 2k + 3n = 6. 6m + 2n + m + n = N 7. x + 3y + 4x + y + 2y = 8. x + 4w + 2w + 2w + x + 3x = O 9. 15m + 16p + 3t + t = 10. 5k + 2pq + k + pq = 11. 2p+ q + b + 3b + 2q = 12. 6ay + 4ab + 4ay + 8ab = R 13. 3a + 3b + 3c = 14. 2w + 5m + 5m + 8w = FO 15. 6gh + 6gh + 6gh = 301 MATH std 5.indd 301 30/07/2021 14:50  FOR ONLINE USE ONLY DO NOT DUPLICATE Subtraction of algebraic expressions Like terms in the algebraic expression can be subtracted to obtain a single algebraic expression. When subtracting like terms, subtract the coefficients leaving the variables unchanged. Unlike terms cannot be subtracted. Example Simplify the following expressions: LY (a) 4x – x = (b) 5ae – 2ab – 2ae = (c) 6k – 6n = N Solution (a) 4x – x = O Subtract coefficients: 4 – 1 = 3. Therefore, 4x – x = 3x. (b) 5ae – 2ab – 2ae = SE Collect together the like terms and then simplify. U 5ae – 2ae – 2ab = (5 – 2) ae – 2ab = 3ae – 2ab Therefore, 5ae – 2ab – 2ae = 3ae – 2ab. E N (c) 6k – 6n = This expression does not contain like terms. LI Thus, subtraction cannot be performed. Therefore, 6k – 6n = 6k – 6n. N O Exercise 3 Answer the following questions: R FO 1. 2m ‒ m = 2. 4m ‒ 2m = 3. 6k ‒ k ‒ 3k = 4. 10n ‒ 5n ‒ n = 302 MATH std 5.indd 302 30/07/2021 14:50  FOR ONLINE USE ONLY DO NOT DUPLICATE 5. 10x ‒ 4x ‒ x = 6. 5a ‒ 2a = 7. 4t ‒ t ‒ 2 = 8. 10a ‒ 2a ‒ 8a = 9. 8x ‒ 8 = 10. 4n ‒ 4k = 11. m ‒ m = 12. 7e ‒ 7e = 13. 7mk ‒ 4mk ‒ mk = 14. 8ad ‒ ad ‒ 2ad = 15. 16x ‒ x ‒ 14x = 16. 2p ‒ p= LY Addition and subtraction of algebraic terms N Addition and subtraction of algebraic terms may be used to simplify a long O expression. Like terms can be added and subtracted to obtain a simpler algebraic expression. Terms with different variables cannot be added or subtracted. SE U Example Simplify the following expressions: E 1. (a) 2a + 3b – a = (b) 4xy – xw + 3xw + 3xy = N LI Solution (a) 2a + 3b – a = N Collect together the like terms and simplify. In other words, arrange O together similar expressions and then, add or subtract their coefficients. 2a + 3b – a = 2a – a + 3b = a + 3b R Therefore, 2a + 3b – a = a + 3b. FO (b) 4xy – xw + 3xw + 3xy = 303 MATH std 5.indd 303 30/07/2021 14:50  FOR ONLINE USE ONLY DO NOT DUPLICATE Collect together the like terms and simplify. In other words, arrange similar expressions and then add or subtract their coefficients. 4xy – xw + 3xw + 3xy = 4xy + 3xy – xw + 3xw = 7xy + 2xw Therefore, 4xy – xw + 3xw + 3xy = 7xy + 2xw. LY 2. 3a – a + 4pq – pq = N Solution 3a – a + 4pq – pq = O Collect similar expressions together and then simplify: 3a – a + 4pq – pq = 2a + 3pq Therefore, 3a – a + 4pq – pq = 2a + 3pq. SE U Exercise 4 E Simplify the following expressions: N 1. 8a + 6a – a = 2. 7m – m + 2m = LI 3. 4b + 3b – 2b = 4. 8t + t – 8t = N 5. y + 4y – 3y = 6. 5t – 4t + t – t = O 7. 12e – 2e = 8. 9n – 8n + n = R 9. 15k – 4k + 3k – k = 10. 7w – 5w + 2w = FO 11. 10b + 2a – 5b = 12. 16pa + 4q – 6pa – 3q = 304 MATH std 5.indd 304 30/07/2021 14:50  FOR ONLINE USE ONLY DO NOT DUPLICATE 13. 16y – 10y + 2a + 11y = 14. 8wz + 8wy – wz – 2wy = 15. 6e – 4f + 6f – e = 16. 4w + 6z – 2w – 2z = 17. 4n + 2n + n = 18. 9m + 6n – m + n = 19. 10a – 10b – 10a + 10b = 20. xy + 3xy + 2xy – xy = LY N Multiplication of algebraic terms O Algebraic terms can be multiplied by a number or by another like or unlike terms. In multiplying terms by a number, multiply the coefficient by the number SE and maintain the variable. If the number is zero, 0, then the product is 0. In multiplying algebraic terms, first multiply the coefficients and then multiply U the variables. E Example N Simplify the following expressions: LI 1. (a) y × y = (b) 4a × 4 = 2. (a) 3k × 2k = (b) n × 3n = N O Solution 1. (a) y × y = R In this expression, y is the only variable involved, and both coefficients FO are 1. When a variable is multiplied by itself, the product is its square. Therefore, y × y = y2. 305 MATH std 5.indd 305 30/07/2021 14:50  FOR ONLINE USE ONLY DO NOT DUPLICATE (b) 4a × 4 = In this question, an expression is being multiplied by 4. Therefore, 4a × 4 = 16a. Solution 2. (a) 3k × 2k = This expression is a product of two like terms. The variable is k and the coefficients are 3 and 2. LY Multiply the coefficients: 3 × 2 = 6. Multiply the variables: k × k = k2. N Therefore, 3k × 2k = 6k2. O (b) n × 3n = SE This expression is a product of two like terms. The variable is n and the coefficients are 1 and 3. Multiply both the coefficients and the variables to get U n × 3n = (1× 3) × (n × n) = 3n2. Therefore, n × 3n = 3n2. E Exercise 5 N Simplify the following expressions: LI 1. b × b = 2. 2n × 2n = N 3. 3p × p = 4. 3y × 3y = O 5. t × k × r = 6. 4a × 3b × 2c = R 7. 3a × 3b × a = 8. m×m= FO 9. 4k × 4t = 10. 3a × 2e × 4 = 11. 2p × t × 3p = 12. y × w × 2y × 3w = 306 MATH std 5.indd 306 30/07/2021 14:50  FOR ONLINE USE ONLY DO NOT DUPLICATE 13. 4a × b × a × b = 14. 10t × 10p = 15. 6p × q × 6p = 16. 3b × 3 × 4b = 17. 3na × 4 × 2n = 18. p × q × 5q × 5w = 19. 3tp × 0 = 20. 7vfk × kp = LY Division of algebraic terms In dividing algebraic terms, divide the coefficients and the same variables. N Example O Simplify the following expressions: 1. (a) 4a ÷ 2a = 2. (a) 4k ÷ 2k = SE (b) 4b ÷ 4b = (b) 6ab ÷ 3ab = U Solution 1. (a) 4a ÷ 2a = E Identify the coefficients and the variables. Then, divide coefficient by coefficient and variable by variable: N 4a ÷ 2a = ​​   ​​ = ( ​​ ​   ​)​ × ​(​   ​)​​ = 2 × 1 = 2 4a _ 4 _ a _ 2a 2 a LI Therefore, 4a ÷ 2a = 2. N O (b) 4b ÷ 4b = Identify the coefficients and the variables. Then divide coefficient by coefficient and variable by variable: R 4b ÷ 4b = ​​   ​​ = ( ​​ ​   ​)​ × ​(​   ​)​​ = 1 × 1 = 1 4b _ 4 _ _ b 4b 4 b FO Therefore, 4b ÷ 4b = 1. 307 MATH std 5.indd 307 30/07/2021 14:50  FOR ONLINE USE ONLY DO NOT DUPLICATE 2. (a) 4k ÷ 2k = ​ ​  ​​ = ( ​​ ​   ​)​ × ​(​   ​)​​= ​(​  _ ​)​× ​(_ )​ _ 4 ​k​​ 2​ _ 4 ​k​​  2​ _ 4 ​k × k ​​​ 2 ​   ​ 2k 2 k 2 k =2×k = 2k Therefore, 4k2 ÷ 2k = 2k. ​   ​)​ × ​(_  ​​ = ​​(_ ​  a ​)​ × ​(_ ​   ​)​​ 6 ​a​​  2​  b _ 6 ​a​​  2​ b (b) 6a2 b ÷ 3ab = ​​  3ab 3 b LY 6 a× a b =​​(​  _ ​)​ × ​(​  _ ​​ × ​ ​  _ ​ ​​ a ) ( ) 3 b N =2×a×1 = 2a O Therefore, 6a2b ÷ 3ab = 2a. Exercise 6 SE Simplify the following expressions: U 1. 6b ÷ 2 = 2. 3b ÷ b = 3. 4c ÷ 4c = 4. 8b ÷ 2b = E N 5. 6b ÷ 3b = 6. 10y ÷ y = LI 7. 15a ÷ 5a = 8. 5a2 ÷ a = 9. 8k2 ÷ 8 = 10. 10p2 ÷ 2p = N 11. t2 ÷ t = 12. 4m2 ÷ 2m2 = O 13. 6k2 p ÷ 2p = 14. 10n2 ÷ 10n = R 15. 18x2 ÷ 9x = 16. 10y2p ÷ 10yp = FO 17. 12nm2 ÷ 12m = 18. 10a2b2 ÷ 10ab = 19. 12m2n ÷ 3m2 = 20. 8y2w2 ÷ 4yw2 = 308 MATH std 5.indd 308 30/07/2021 14:50  FOR ONLINE USE ONLY DO NOT DUPLICATE Exercise 7 Simplify the following expressions: 1. 3n + n = 2. 7n – p = 3. 8k – 4k = 4. 3a × b × b = 5. 4y + 2y = 6. 8n – 6n = 7. 8p2 ÷ p2 = 8. 3n × n = LY 9. 4k2 ÷ 2k2 = 10. 6e ÷ e = N 11. 4p2 ÷ 4 = 12. 6k2 ÷ 6k = O 13. 16m ÷ 4m = 14. 10n2 ÷ 10n = 15. 18x2 ÷ 9x = 16. pt2 ÷ pt = 17. 5ya2 ÷ 5ya = SE18. 12t2 p ÷ 4tp = 19. 10ab2 c ÷ 5ab = 20. a2 b2 e ÷ abe = U 21. 10 + 2y – 10 = 22. 5a + 3b – 2a – 2b = E 23. 9t – t + z – z = 24. 6y2 – 2x2 – 2y2 + 3x2 = N Summary LI 1. Algebraic terms are made up of coefficients and variables. 2. Algebraic expressions are constructed using the terms. N 3. Like terms have the same variables. 4. To add like terms, add the coefficients. The variable remains the same. O 5. To subtract like terms, subtract the coefficients. The variable remains the same. R 6. Unlike terms cannot be combined to obtain a single expression, but they may be simplified. FO 7. To multiply algebraic terms, multiply the coefficients, as well as the variables. 8. To divide algebraic terms, divide the coefficients, as well as the variables. 309 MATH std 5.indd 309 30/07/2021 14:50