Transcript File

Brackets Factors and
Equations
Exercises 1 and 2
Oral Examples: Factorise
1. 2x + 6y = 2 (
2. 3x - 6y = 3 (
)
)
3. 8x + 2y = 2 (
4. 8x - 4y = 4 (
)
)
Oral Examples: Factorise
1. 2x + 6y = 2 ( x + 3y)
2. 3x - 6y = 3 ( x – 3y)
3. 8x + 2y = 2 (4x + y)
4. 8x - 4y = 4 ( 2x - y)
Oral Examples: Factorise
5. 3x + 12 = 3 (
6. 12x - 4 = 4 (
)
)
7. 30 + 5y = 5 (
8. 6 - 30y = 6 (
)
)
Oral Examples: Factorise
5. 3x + 12 = 3 (x + 4)
6. 12x - 4 = 4 ( 3x - 1)
7. 30 + 5y = 5 (6 + y)
8. 6 - 30y = 6 ( 1 – 5y)
Oral Examples: Factorise
9.
4x + 6y = 2 (
)
10. 6x - 9y = 3 (
)
11. 10x + 8 = 2 (
)
12. 15x - 12 = 3 ( )
Oral Examples: Factorise
9.
4x + 6y = 2 (2x + 3y)
10. 6x - 9y = 3 (2x – 3y)
11. 10x + 8 = 2 (5x + 4)
12. 15x - 12 = 3 (5x - 4)
Oral Examples: Factorise
13. 8 + 12y = 4 (
)
14. 20 - 24y = 4 (
)
15. 12x + 18y - 30z = 6 (
16. 18x - 24y + 30z = 6 (
)
)
Oral Examples: Factorise
13. 8 + 12y = 4 (2 + 3y)
14. 20 - 24y = 4 (5 – 6y)
15. 12x + 18y - 30z = 6 (2x + 3y – 5z)
16. 18x - 24y + 30z = 6 (3x – 4y + 5z)
Exercise 3: Factorise
1. 2x + 2y
2. 3x - 3y
3. 4x + 4
4. 5 - 5y
Exercise 3: Factorise
1. 2x + 2y = 2( x + y)
2. 3x - 3y = 3(x + y)
3. 4x + 4
= 4( x + 1)
4. 5 - 5y = 5( 1 – y)
Exercise 3: Expand
5. 2x + 10y
6. 10x - 5y
7. 12x + 2
8. 6 - 12y
Exercise 3: Expand
5. 2x + 10y
= 2(x + 5y)
6. 10x - 5y
= 5(2x – y)
7. 12x + 2
= 2(6x + 1)
8. 6 - 12y
= 6(1 – 2y)
Factorise
9.
3x + 15y
10. 15x - 5y
11. 18x + 6
12.
3 - 18y
Factorise
9.
3x + 15y = 3(x + 5y)
10. 15x - 5y
= 5(3x – y)
11. 18x + 6
= 6(3x + 1)
12. 3 - 18y
= 3(1 – 6y)
Exercise 3: Factorise
13. 4x + 20y
14. 20x - 5y
15. 24x + 6
16. 4 - 24y
Exercise 3: Factorise
13. 4x + 20y
= 4(x + 5y)
14. 20x - 5y
= 5(4x – y)
15. 24x + 6
= 6(4x + 1)
16. 4 - 24y
= 4(1 – 6y)
Exercise 3: Factorise
17.
2x + 4y
18. 9x - 3y
19. 16x + 4
20.
5 - 25y
Exercise 3: Factorise
17.
2x + 4y
= 2(x + 2y)
18. 9x - 3y
= 3(3x – y)
19. 16x + 4
= 4(4x + 1)
20.
= 5(1 – 5y)
5 - 25y
Exercise 3: Factorise
21. 6x + 10y
22. 10x - 4y
23. 6x + 8
24.
12 - 10y
Exercise 3: Factorise
21. 6x + 10y = 2(3x + 5y)
22. 10x - 4y = 2(5x – 2y)
23. 6x + 8
24.
= 2(3x +4)
12 - 10y = 2(6 – 5y)
Exercise 3: Factorise
25.
6x + 15y
26.
15x - 9y
27.
9x + 12
28.
18 - 15y
Exercise 3: Factorise
25.
6x + 15y
= 3(2x + 5y)
26.
15x - 9y
= 3(5x – 3y)
27.
9x + 12
= 3(3x + 4)
28.
18 - 15y
= 3(6x – 5y)
Exercise3: Factorise
29. 10x + 15y
30. 25x - 10y
31. 15x + 20
32.
25 - 15y
Exercise 3: Factorise
29. 10x + 15y = 5(2x + 3y)
30. 25x - 10y = 5(5x – 2y)
31. 15x + 20 = 5(3x + 4)
32.
25 - 15y = 5(5 – 3y)
Exercise 3: Factorise
33. 8x + 20y
34. 20x - 12y
35. 12x + 16
36.
16 - 20y
Exercise 3: Factorise
33. 8x + 20y = 4(2x +5y)
34. 20x - 12y = 4(5x – 3y)
35. 12x + 16 = 4(3x + 4)
36.
16 - 20y = 4(4 – 5y)
Exercise 3: Factorise
37. 12x + 18y
38. 30x - 12y
39. 18x + 24
40.
30 - 18y
Exercise 3: Factorise
37. 12x + 18y
= 6(2x + 3y)
38. 30x - 12y
= 6(5x – 2y)
39. 18x + 24
= 6(3x + 4)
40.
= 6(5 – 3y)
30 - 18y
Exercise 3: Factorise
41. 25x + 20y
42. 25x - 30y
43. 30x + 24
44.
30 - 36y
Exercise 3: Factorise
41. 25x + 20y = 5(5x + 4y)
42. 25x - 30y = 5(5x – 6y)
43. 30x + 24 = 6(5x + 4)
44.
30 - 36y = 6(5 – 6y)
Exercise 3: Factorise
45. 4x + 6y + 10z
46. 6x - 8y + 10z
47. 6x + 9y - 15z
48. 9x - 12y - 15z
Exercise 3: Factorise
45. 4x + 6y + 10z
= 2(2x + 3y + 5z)
46. 6x - 8y + 10z
= 2(3x – 4y + 5z)
47. 6x + 9y - 15z
= 3(2x + 3y – 5z)
48. 9x - 12y - 15z = 3(3x – 4y – 5z)
Exercise 3: Factorise
49. 10x + 15y + 25z
50. 15x - 20y + 25z
51. 8x + 12y - 20z
52. 12x - 16y - 20z
Exercise 3: Factorise
49. 10x + 15y + 25z = 5(2x + 3y + 5z)
50. 15x - 20y + 25z = 5(3x – 4y +5z)
51. 8x + 12y - 20z
= 4(2x +3y – 5z)
52. 12x - 16y - 20z = 4(3x – 4y – 5z)
Exercise 4
1.
xy + xz
2.
xy - xz
3. yz + yx
4.
zx - zy
Exercise 4
1.
xy + xz
= x(y + z)
2.
xy - xz
= x(y – z)
3. yz + yx
= y(z + x)
4. zx - zy
= z(x – y)
Exercise 4
5. x + xy
2
2
6. xy- y
7. xy+ x
8. y- xy
Exercise 4
5. x + xy = x ( x+ y)
2
6. xy- y = y( x- y)
2
7. xy+ x = x ( y+1)
8. y- xy = y(1- x)
Exercise 4
1. xy+ xz
2. xy- xz
3. yz+ yx
4. zx - zy
Exercise 4
1. xy+ xz = x ( y+ z)
2. xy- xz = x ( y- z)
3. yz+ yx = y( z+ x)
4. zx - zy = z( x - y)
Exercise 4
5. x + xy
2
2
6. xy- y
7.
xy+ x
8.
y- xy
Exercise 4
5. x + xy = x ( x+ y)
2
2
6. xy- y
= y ( x - y)
7.
xy+ x
= x ( y+1)
8.
y- xy
= y (1- x)
Exercise 4
9.
10.
x +x
2
y- y
2
2
2
11. x y+ xy
2 2
12. xy- x y
Exercise 4
9.
10.
x +x
= x ( x+1)
2
= y (1- y)
2
y- y
2
2
= xy( x+ y)
2 2
= xy (1- xy)
11. x y+ xy
12. xy- x y
Exercise 4
13. pq+ pr
14. pq - pr
15. qr + qp
16.
rp- rq
Exercise 4
pq+ pr
= p ( q+ r )
14. pq - pr
= p( q - r )
15.
qr + qp
= q ( r + p)
16.
rp- rq
= r ( p- q)
13.
Exercise 4
17.
p + pq
18.
pq - q
19.
pq+ p
20.
q- pq
2
2
Exercise 4
17.
p + pq = p ( p+ q)
2
18.
2
pq - q
= q ( p- q)
19.
pq+ p
= p ( q+1)
20.
q- pq
= q (1- p)
Exercise 4
21.
22.
p +p
2
2
q- q
2
2
23. p q+ pq
24. pq - p q
2
2
Exercise 4
21.
22.
= p ( p+1)
p +p
2
= q (1- q)
2
q- q
2
2
23. p q+ pq
24. pq - p q
2
2
= pq ( p+ q)
= pq (1- pq)
Exercise 4
25. x + 2x
2
26. x - 3x
2
27. 4 y+ y
2
2
28. 5y- y
Exercise 4
25. x + 2x = x ( x+ 2 )
2
26. x - 3x = x ( x - 3)
2
2
= y ( 4 + y)
2
= y( 5 - y)
27. 4y+ y
28. 5y- y
Exercise 4
29. x + 6x
2
30. x - 7x
2
31. 8y+ y
2
2
32. 9y- y
Exercise 4
29. x + 6x = x ( x+ 6 )
2
30. x - 7x = x ( x - 7 )
2
2
= y( 8 + y)
2
= y ( 9 - y)
31. 8y+ y
32. 9y- y
Exercise 4
33. x + x
2
34. x - x
2
35. y+ y
2
2
36. y- y
Exercise 4
33. x + x = x ( x+1)
2
34. x - x = x ( x -1)
2
2
= y(1+ y)
2
= y (1- y)
35. y+ y
36. y- y
Exercise 4
37. 2x + 3x
2
38. 2x - 7x
2
39. 5y+ 2y
2
2
40. 9y- 2y
Exercise 4
37. 2x + 3x = x ( 2x+ 3)
2
38. 2x - 7x = x ( 2x - 7 )
2
2
= y( 5 + 2y)
2
= y( 9 - 2y)
39. 5y+ 2y
40. 9y- 2y
Exercise 4
41. 3x + 2x
2
42. 3x - 5x
2
43. 4y+ 3y
2
2
44. 7y- 5y
Exercise 4
41. 3x + 2x = x ( 3x+ 2 )
2
42. 3x - 5x = x ( 3x - 5 )
2
2
= y( 4 + 3y)
2
= y ( 7 - 5y)
43. 4y+ 3y
44. 7y- 5y
Exercise 4
45. 2x + x
2
46. 2x - x
2
47. y+ 3y
2
2
48. y- 3y
Exercise 4
45. 2x + x = x ( 2x+1)
2
46. 2x - x = x ( 2x -1)
2
2
= y (1+ 3y)
2
= y (1- 3y)
47. y+ 3y
48. y- 3y
Exercise 4
49.
50.
51.
2x + 4x
2
2x - 8x
2
6y+ 2y
2
2
52. 10y- 2y
Exercise 4
49.
2x + 4x
= 2x ( x+ 2 )
50.
2x - 8x
= 2x ( x - 4 )
2
= 2y ( 3+ y)
51.
2
2
6y+ 2y
2
52. 10y- 2y
= 2y ( 5 - y)
Exercise 4
53. 3x + 6x
2
54. 3x -12x
2
55. 9y+ 3y
2
2
56. 15y- 3y
Exercise 4
53.
3x + 6x
2
= 3x ( x+ 2 )
54. 3x -12x = 3x ( x - 4 )
2
55.
2
9y+ 3y
2
56. 15y- 3y
= 3y ( 3+ y)
= 3y ( 5 - y)
Exercise 4
57. 4x + 8x
2
58. 4x -16x
2
59. 12y+ 4y
2
2
60. 20y- 4y
Exercise 4
57.
4x + 8x
2
= 4x ( x+ 2 )
58. 4x -16x = 4x ( x - 4 )
2
2
= 4y ( 3+ y)
2
= 4y ( 5 - y)
59. 12y+ 4y
60. 20y- 4y
Exercise 4
61. 5x +10x
2
62. 5x - 20x
2
63. 15y+ 5y
2
2
64. 25y- 5y
Exercise 4
61. 5x +10x = 5x ( x+ 2 )
2
62. 5x - 20x = 5x ( x - 4 )
2
2
= 5y( 3+ y)
2
= 5y ( 5 - y)
63. 15y+ 5y
64. 25y- 5y
Exercise 4
65. 2x + 2x
2
66. 3x - 3x
2
67. 4y+ 4 y
2
2
68. 5y- 5y
Exercise 4
65. 2x + 2x = 2x ( x+1)
2
66. 3x - 3x = 3x ( x -1)
2
2
= 4y (1+ y)
2
= 5y (1- y)
67. 4y+ 4y
68. 5y- 5y
Exercise 4
69.
70.
71.
4x + 6x
2
6x - 9x
2
6y+ 9y
2
2
72. 8y-12y
Exercise 4
69.
4x + 6x
= 2x ( 2x+ 3)
70.
6x - 9x
= 3x ( 2x - 3)
2
= 3y ( 2 + 3y)
71.
2
2
6y+ 9y
2
72. 8y-12y
= 4y ( 2 - 3y)
Exercise 4
73.
x + xy+ xz
74.
xy+ y - yz
2
2
2
75.
xz- yz+ z
2
2
2
76. x yz- xy z- xyz
Exercise 4
73.
x + xy+ xz
= x ( x+ y+ z)
74.
xy+ y - yz
= y( x+ y- z)
2
2
= z( x - y+ z)
2
75.
xz- yz+ z
2
2
2
76. x yz- xy z- xyz
= xyz( x - y- z)
Exercise 4
77.
6x + 8xy+10xz
78.
12xy+15y2 - 6yz
79.
20xz- 8yz+12z2
2
2
2
2
80. 10x yz-15xy z- 20xyz
Exercise 4
77.
6x2 + 8xy+10xz
= 2x ( x+ 4y+ 5z)
78.
12xy+15y2 - 6yz
= 3y( 4x+ 5y- 2z)
79.
2
20xz- 8yz+12z
80. 10x2 yz-15xy2 z- 20xyz2
= 4z( 5x - 2y+ 3z)
= 5xyz( 2x - 3y- 4z)