10. a. Copy and complete : No. of Rectangles (r) No. of Matches (m) 17 b. Find a formula for calculating m if you know r.

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1 Simplifying Exercise 1 Simplify the following: 1. 3x (-4x). -5x (+3x) 3. 10x + (-9x) 4. -6x (- 4x) 5. -1y + (-3y) 6. 16p (-1p) 7. -8p + (-1p) 8. 11z (-z) r + (-1r) c (+1c)11. -8b (-13b) x + (-9x) 13. p (-8p) 14. 7y + (-9y) 15. -p (-3p) 16. -y + (-3p) 4y s (-3s) + 1r 18. 5x + 7y (-x) 19. 7a (+5b) 3a 0. p (-3z ) + 4p 1. 8z (-y) + (-3y). 9q (+10r) + (-3q) 3. -1x + (-3y) + (-4y) 4. -4x (-6y) + (-5y) 5. 3p + (-1q) (-3q) 6. -1q + (-8p) (-9p) + (-3q) 7. -3x (-1y) + (-x) (-3y) 8. 5d (-1e) + (-3e) + (-d) 9. 8n + 5p (-p) + (-8n) 30. 1f + 6g (-y) + (+1f) 31. 8p + 1q + 10q 9p 3. 1p (-7q) + (-6p) (-9q) 33. 8p + 9q + 1r (-7r) + 8q (-p) 34. 3s (-1p) + (-5p) (-6s) 35. 4a (-b) + (-5b) (-a) 36. 5m (-n) + 5m (-3n) 37. 3p + (-q) (-3p) + 7q 38. 3j i + (-3i) + (-j) 39. 7z (-y) + 5z (-3y) 40. 1k (-n) + (-3m) + 45k Page 1 9. An electricity supplier charges its customers a Standing Charge of 1.50 plus 6p for each unit of electricity used. a. Find a formula for calculating c, the total cost in s, if you know n, the number of units used. b. Use your formula to calculate c if : i) n = 500 ii) n = 65 iii) n = 700 iv) n = 155 c. Use your formula to calculate n if : i) c = ii) c = iii) c = 33.6 iv) c = a. Copy and complete : No. of Rectangles (r) No. of Matches (m) 17 b. Find a formula for calculating m if you know r. c. Use the formula to find m if : i) r = 0 ii) r = 7 iii) r = 50 d. Use the formula to find r if : i) m = 80 ii) m = 164 iii) m = 59 Page 18

2 7. A car repair firm calculates its bills as follows. Cost of parts plus 18 per hour labour charges a. Calculate the cost of repairing a car that required 196 worth of parts and 5 hours of labour. b. If p = cost of parts in s, w = number of hours labour and c = total cost of the car repair in s, find a formula for calculating c if you know p and w. c. Use your formula to calculate c if : i) p = 36, w = ii) p = 70, w =.5 d. Use your formula to calculate p if : i) c = 49, w = 1.5 ii) c = 10, w = 3 e. Use your formula to calculate w if : i) c = 110, p = 56 ii) c = 405, p = A father is taking his 3 children to the cinema. The cost of a Child Ticket is 3 less than the cost of an Adult Ticket. a. If a stands for the cost of an Adult Ticket, find an expression in a for the cost of a Child Ticket. b. If the total cost of the 4 tickets is 7, form an equation in a and use it to find the cost of an Adult and a Child Ticket. Page 17 Exercise Multiply out the brackets (r + 3). -8 (-a 3) (g + ) 4. 6 (-c 10) 5. - (j + 3) 6. 3 (-p 5) (3t 5) (3y + 4) 9. 3 (-y 4) (-1t 4) 11. (-6p + 8) 1. 5 (-3r 1) (-p + y) (-w + 3f) (-6g 3r) (3t 4g) (-3w + x) (e + f + 3g) (-4h + 3c b) (4 d + 5f) Exercise 3 Multiply out the brackets and then simplify (k + 10). - 5(z 10) (h + ) (y 1) 5. 3w - 6 (w + 8) 6. 3p + 3 (9p 3) 7. g (3g 1) 8. d - 10 (3d + 1) 9. 4x - 7 (x 3) (5r ) 11. 3h + (6h + 7) (3t 4) (x + y) 14. c + 3d - 7 (c + 5d) (4g 3r) (3t 4d) (3w + q) 18. 5a - (e + f + 3g) 19. 5r - 3 (4r + 3x z) (4 d + 5f) Page

3 Exercise 4 Multiply out the brackets then simplify (y - 3) + 4y (p ) + 4p (c + ) 7c (x 4) + 5x 5. - (-5w + ) 15w (p 3) 1p (-5r ) 18r (w + 1) + 5 7w (-z + 3) 15z (3m 3) (-5h 7) h 1. -(-g 1) 8g (a + b) a 5b (5f 3g) + 15g (k 3m) + 18m 16. 5(-h 4k) 7h + 5k (-6t 5y) + 0y 18. -r -(e r) 5e + 9r (-5t - w) (+10w) 0. -7(3y x) (-10x) (4w + 1) (-w) (w 3y) (+3y) (d e) (+3e) 4. 7a - (a + b) (+7b) (3x ) (-1) 6. -9d - 5(d + 1) (-3) 7. -4e + 3 (4e f) (-5f) 8. p - 3(3p ) (-7p) 9. -a + 7(-3a b) + (-9b) t - 5(3r t) (-0r) Exercise 5 Multyiply out the brackets. 1. -a (a + 3). -8a (-a 3) 3. -3c (c + ) 4. -c (-c 10) 5. -j (j + 1) 6. 3p (-p 4) 7. -3d (d 3) 8. -y (3y + 4) 9. x² (-x + 5) 10. 3s (-1ts 4s²) 11. p (-3p³ - ) 1. -8p (-3r 1) 13. -y (-5y² + y) w (-5w 3a) 15. -pq³ (-6q² 3p) 16. -gt²(3t 4g) 17. 4w² (-3w + y) 18. -ef (e + f + 3g) 19. -c³ (-4h + 3c a) 0. -d³ (4 d² + 5f) Page 3 5. The cost of hiring a van from a rental firm is calculated as follows : Basic Charge 5 Cost per Mile a. Calculate the cost of hiring a van if you travel : i) 10 miles ii) 100 miles iii) 50 miles b. Find a formula for calculating c, the cost of hiring a van in s, if you know m, the number of miles travelled. (Hint : costs in table above are in pounds and pence.) c. Use your formula to calculate c if : i) m = 0 ii) m = 75 iii) m = 450 iv) m = 300 v) m = 175 vi) m = 1000 d. Use your formula to calculate m if : i) c = 6.50 ii) c = 9 iii) c = 36 iv) c = v) c = 7.60 vi) c = Tim, Peter and Simon are brothers. Peter is 3 years older than Tim and Simon is 5 years younger than Tim. a. If t stands for Tim s age, find an expression in t for Peter s age and Simon s age. b. The combined total of their ages is 34 years. Form an equation in t and use it to find their ages. Page 16 5p

4 b. The combined total of their three ages is 51 years. Form an equation in b, and use it to find their ages. 4. a. Draw the next two patterns in the sequence. b. Copy and complete the table. No. of Black Tiles (b) No. of White Tiles (w) 8 c. Find a formula for finding w, the number of white tiles if you know b, the number of black tiles. d. Find the number of white tiles w, if : i) b = 10 ii) b = 0 iii) b = 13 iv) b = 15 v) b = 100 vi) b = 500 e. Find the number of black tiles b, if : i) w = 34 ii) w = 5 iii) w = 15 iv) w = 7 v) w = 86 vi) w = 60 Page 15 Exercise 6 Multiply out the brackets and simplify. 1. -a (ab - 1) + (-5a). -x (3x + 5) (-8x) 3. -y (y + x) + (-3y²) 4. x (-x² - 4) + (-7x) 5. a² (-a - 3b) (-5a²b) 6. -6a (4a 1) (-10a) 7. -k (k² 5) (-4k³) 8. -p (-5p + p²) + (-4p²) 9. -3q² (-q² + 3) + (-6q²) 10. -ab (-b b²) (+ab²) 11. -x (-6xy + 5y) + (-x²y) 1. 5f (-3ef f) + (+3f²) 13. -ab (-a + b) (+5a²b) 14. -xy (-3x + 3) (-10xy) 15. t² (-t 3r) + (-4t²r) w (4w 5) (-3w) 17. -k (k² 5) (-4k³) 18. -a (-a + a²) - (-3a²) q² (-q² + 3) + (-6q²) 0. -xy (-y y²) (+xy²) Exercise 7 Multiply out the brackets and simplify. 1. -p -p(pq - ) + (-3p). ab -b(b +3a) + (-b²) 3. x³ - x²(-x 3y) (-5x²y) 4. 1a -6a(4a 1) (-10a) 5. 3m³- m(m² 1) (-m³) 6. 5d³ +3d² -d²(-3d + 1) + (-6d²) 7. 1ab² -5b(-3ab + 7b) + (-a²b) 8. 1gh² -gh(-4g + 3h) (-g²h) 9. -3e³ - e²(-f 3e) + (-4e²f) b³ - b(7b² 1) (-3b³) 11. 3p³ -p(-p + 3p²) (-4p²) 1. -3w³ + (-5w²) -w²(-3w + 1) + (-5w²) Page 4

5 Substitution Exercise 1 If a =, b = -3, c = -1, x = - and y = -4 calculate : 1. a + c. x - y 3. b + c 4. a + b 5. c - x 6. a x - y 7. b + c - a 8. x - b - c 9. y + b - a 10. b + y - c 11. x - y - b 1. x - c + y 13. y b + a 14. x - c + y 15. a - b - c + x Exercise If p = -, q = 3, r = -3, s = -5 and t = -4 calculate : 1. -s. -q 3. -t 4. -p 5. -r 6. r 7. -p 8. 3q 9. -4t 10. 4q r 1. 3s s 14. 7t r Exercise 3 If f = -, g = 3, h = -, j = -1 and k = -3 calculate : 1. f k. j + 4f 3. g 3h 4. -g + k 5. -f j 6. k - 3j 7. -k + 5f 8. j + h 9. -3f + k 10. -g + j 11. j 3g 1. -h + 3g 13. j + 4h 14. j - g 15. 6f k. a. Draw the next two patterns in the sequence. b. Copy and complete the table. No. of Black Tiles (b) No. of White Tiles (w) 10 c. Find a formula for finding w, the number of white tiles if you know b, the number of black tiles. d. Find the number of white tiles w, if : i) b = 10 ii) b = 14 iii) b = 0 iv) b = 5 v) b = 300 vi) b = 99 e. Find the number of black tiles b, if : i) w = 30 ii) w = 50 iii) w = 06 iv) w = 506 v) w = 48 vi) w = Bens' mother is three times his age. His sister is four years younger than him. Let b stand for Ben s age. a) Find an expression in b for the age of : i) Bens mother ii) Bens sister Page 5 Page 14

6 Finding and Using Formulae 1. A fence is constructed using posts and chains as shown below. posts 3 posts 4 posts 3 chains 6 chains 9 chains a. Draw the next two patterns in the sequence. b. Copy and complete the table. Number of Posts (p) Number of Chains (c) 9 c. Find a formula for finding c, the number of chains if you know p, the number of posts. d. Find the number of chains c, if : i) p = 7 ii) p = 10 iii) p = 8 iv) p = 5 v) p = 100 vi) p = 50 e. Find the number of posts p, if : i) c = 30 ii) c = 57 iii) c = 153 iv) c = 51 v) c = 81 vi) c = 1497 Page 13 Exercise 4 If d = -, e = -3, f =, k = 5 and m = -4 calculate : 1. -3d - k. e 3m 3. k + e d 4. -4m e 5. 3k - m 6. d + k 4m 7. e - 3f 8. 4m 5e 9. -3m d + 3k 10. 6e + 6m 11. -d + 3f 1. 3k e + 3m 13. -f + e 14. 3d + 4m m d 3e Exercise 5 If x = -, y = -3 and z = 4 calculate : 1. -z - x. y - x 3. -z + y 4. -x + y 5. z + x 6. -3y - x 7. -4z + 3y 8. 3y z 9. -5x + z 10. -y x z 8x 1. -3y + 3x x 5y 14. 7z 8y z + 10x Exercise 6 If r = -4, g = -3, h = - and m = 3 calculate : 1. -g - h. -r - h 3. -m - g 4. m - h 5. g + r 6. -m + r 7. -3h - r 8. 3g - m 9. -4m - 3g r m g + r 1. -7h - m 13. 4r + g 14. 3h - m 15. -m - 5h 16. 4r - g - m 17. 5m - r + 4g g + 5h - r 19. 3g + 8r - m 0. -8h - g + r 1. -m + 5r 6h Page 6

7 Exercise 7 If a = -1, f = -, r =, h = 3 and p = -3 calculate : 1. 3af. 4ahp 3. arh 4. af + h 5. ap + 3fr 6. fh r 7. 1p rh 8. 4hp 5ar 9. 5p 10. 3h p + a 1. h 3 + a 13. 8fr a 14. p rh 15. 5a 3f 16. 9ap hf 17. 6ah afp 18. afr h 19. f + ap 0. 4h 3 p 1. p. 3a 4 f 3. h 4 p 4. 5afrh 4ap 5. 7a 3 5frh 6. 6f p 3 5frh 8. 3a + p²h 9. 4a³ - r² 30. h³ + f²r Exercise 8 If x = -, y = 3 and z = -4 calculate : 1. ( 4x ). ( y ) 3 3. ( 4x) 4 4. ( 6xy ) 5. ( 6x² ) 6. ( z² ) 7. ( xz ) 8. ( 3y ) 4 9. ( 5x²y) 3 Page 7 Exercise 7 Solve these equations using cross multiplication. x 3 x 1 1 7x x 1 5 3x 9 6 5x x 1 7x 9 3x 1 x 5 3 4x 7 7 5x x 4 x 4 x 4 4 x Exercise 8 Solve these inequations. 1. x - 8 > 6x h + 10 < 3h x + 1 < 6x e - 3 > 7e t + 3 > 8t d 3 > 8d a - 5 < 5a w + 5 < 6w n +6 > 3n r + 9 > r b - 4 > 3b m - 10 > 8m r + 10 < 3r z + 5 > -7-3z 15. h - 3 < 3h e + 7 > e a > a 18. 3g + < -14 5g g < g 0. 3n + > -10 n Page

8 Exercise 5 Solve the following equations. Simplify fraction answers where appropriate. 1. 3x + 6 = (x 9). 5x 3 = (x + 4) 3. 4x 7 = 5(x ) 4. 7( x) = 3(x ) 5. 5(x 3) = 7x (x + 4) = 5x (x + 1) = 3(3x + 7) 8. 5(x + 1) = (x 7) 9. 3(x 3) = 5(x 1) 10. 6(x 3) = (4x + 3) 11. 5(x 3) = 8(x ) 1. 7(x + 3) = 4(3x 1) 13. 4(x + 5) = 5(3x + 4) 14. 3(x 1) = 1(x + 1) 15. 7(x 1) = 7(3x + 4) 16. (x 1) = (-8 x) 17. 5(x 7) = (3x + 1) 18. 3(x + 4) = 4(x 1) 19. 7(3x 4) = 5(4x + 1) 0. 1(3x 4) = 9(x 5) Exercise 6 Solve these equations using cross multiplication. x 6 x 3x x Exercise 9 If p = -, r = -3 and t = calculate: 1. (p + 5). 5 (r 3) 3. 5 (p + r) 4. 6 (t p) 5. (p + 3t) 6. 3 (9p 3r) 7. 8 (3t r) (3p + t²) 9. 3 (p³ t) (5t p²) 11. (6p + r²) 1. 5 (r² p³) 13. (p² + p³) (p + 5r²) (4pr 3t) (3t² 4p³) (3prt + r²) 18. (p + p² + p³) (4r + 3p p²) 0. 7 (4t p³ + 5r²) Exercise 10 If e = -3, f = - and g = 6 calculate: 1. e f g. e f 3. (g 3e) f 4. 3e 7 f x 10 x 6x 6 3x 5. g e 6. g ef 7. ( ef ) g 8. (e e ) 5g 15 x 6x 1 x d x 3x x x (e 3 f ) g 10. e f g 11. f 8 f 1. g ef Page 11 Page 8

9 Solving Equations and Inequations Exercise 1 Solve the following equations : 1. 3x - 8 = 6x +1. 4r 10 = 9r x + = x d + 5 = 1d x + 0 = x x 3 = 4x c + 1 = c x +1 = x n +5 = 3n x + 8 = x b - 4 = 5b x - 10 = x d + 5 = 15d z - 9 = -7-3z 15. 5h + 6 = h n + 8 = n g = 0 + 4g t - 4 = -16-3t g = 15-15g 0. 14e +5 = e Exercise Solve the following equations : 1. x 1 = 5 + 3x. 4r + 10 = 4 + 6r 3. 3x + 3 = -1 x 4. -5d 4 = 1 + 3d 5. 4x + 1 = -6 3x 6. 5x + 10 = 4 + 3x c = 4 4c 8. 5x + 8 = -3-15x 9. 8n + 9 = 5n 10. 6x 1 = 3 + 3x 11. -b + 15 = 3 4b p = 4 p t = 10-1t x = x d = d g = g n = -13 n h = -1 - h g 3 = 1-5g 0. 5m + 4 = m - 5 Exercise 3 Solve the following equations : (x 3) = (x + ) = (x + 1) = (3x 4) = (3h 1) = (d + ) = (3w ) = (3s 3) = (3w + ) = (5b + ) = (d + 4) = (6t 3) = (3g 10) = (w 8) = (a + 1) = (3r + ) = (g 3) = (3k 1) = (x 1) = (4d ) = 5 Exercise 4 Solve the following equations. Simplify fraction answers. 1. -x 3 = = 4a = 6p = 7 3z = 6a = 4p = 6a a + 4 = z = 3z q = - 18 q a = -35 3a 1. -9a = 5a p = 8p a + 1 = a 15. 5q + 15 = -q 16. 0q + 7 = q z 9 = 41z q = p = x = 1 Page 9 Page 10

10 Formulae and Graphs 1. The relationship between the roasting time, t hours and the weight w kg of a joint of pork can be expressed as : t = ½(3w + 1) a. Draw the graph for 0 < w < 4. b. From the graph find the weight of a joint of pork which needs 5.9 hours roasting time. c. From the graph, find the roasting time for a joint that weighs 5.5 kg. Alva Academy Maths Department Algebra Booklet Three. The employees in a factory earn a basic pay of 0 per week plus 8 for each hour of overtime. a. Find a formula for calculating p, the total pay in s, if you know n, the number of hours of overtime worked. b. Draw the graph for 0 < n < 10. (Hint: Begin the vertical axis (p) at 00.) c. From the graph, find the number of hours of overtime worked if the total pay is 34. Content: Page 1 Page 5 Page 9 Page 13 Page 19 Simplifying Substitution Solving Equations and Inequations Finding and using Formulae Formulae and Graphs Page 19

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