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Mathematics

Stage 3

Algebra

S J Cooper

Algebra (1) Laws of indices

1. Simplify each of the following:

a) 34 xx

b) 27 pp

c) aa 3

d) ddd 35

e) 267 eee

f) 342 hhh

g) 23 mmm

h) 526 xxx

i) qqqq 463

j) 262 yyy


a) baba 463

b) 3752 nmnm

c) 3264 xyyx

d) 7435 jiji

e) 5739 qpqp

f) 4532 khkh

g) 328 dcdc

h) 72210 nmnm

i) 654 fefe

j) 3752 vuuu


a) 38 xx

b) 512 yy

c) 67 uu

d) 39 pp

e) 57 bb

f) 34 ee

g) 1320 hh

h) 1517 dd

i) 37 tt

j) 1313 xx


a) 23x

b) 42y

c) 35r

d) 254a

e) 233x

f) 562p

g) 442qp

h) 338ts

5. Simplify each of the following expressions:

a) 5

7

x

x

b) 8

12

t

t

c) c c

c

10 4

5

d) d d

d

6 4

8

e) x y

xy

3 5

2

f) p q p q6 2 5

g) 2 43 7 5 4r s r s

h) e f e

f e f

6 5 2

4 4


a) x5 4

b) a b2 6

c) c d4 7 3

d) a b

a b

3 2 7

8

e)

t t

t

6 2 4

3 4

f)

u v v

u v

5 4 3 6

2 4

Algebra (2) Continuing Sequences

1. Write down the next two terms for each of the following sequences and a rule in words.

a) 6, 9, 12, 15, 18

b) 1, 5, 9, 13, 17

c) 4, 13, 22, 31, 40

d) 8, 15, 22, 29, 36

e) 3, 8, 13, 18, 23

f) 6, 8, 10, 12, 14

g) 2, 10, 18, 26, 34

h) 9, 12, 15, 18, 21

i) 3, 10, 17, 24, 31

j) 6, 11, 16, 21, 26

2. Write down the next two terms for each of the following and give a rule for continuing the

sequence.

a) 61, 52, 43, 34,

b) 54, 48, 42, 36,

c) 70, 63, 56, 49,

d) 46, 41, 36, 31,

e) 20, 16, 12, 8,

3. Write down the first five terms for each of the following described sequences.

a) Add 2 to the previous term: 6, …….

b) Add five to the previous term: 2, …….

c) Subtract 1 from the last term: 20, ……

d) Subtract 8 from the previous term: 20, ……

e) Multiply the previous term by 2: 2, …..

f) Multiply the previous term by 3: 1, ……

g) Divide the last term by 2: 256, ……

h) Add the next even number each time: 2, 3, ……

i) Add a number that increases by 2 each time: 1, 4, …….

1 3 6 10

4. Fill in the missing gaps in the following sequences:

a) 7, ….. 19 25 ….. ….. 43

b) 5, ….. 9 ….. ….. 15 17

c) 13, ….. 25 ….. 37 43 …..

d) 4, ….. 20 28 ….. 44 …..

e) 8, ….. ….. ….. 28 33 …..

f) 1, 8 ….. 22 ….. 36 …..

g) 6, ….. ….. 18 22 ….. 30

5. Adding together the previous two terms generates the Fibonacci sequence. The first six terms

of the Fibonacci sequence starting with 1, 1 are 1, 1, 2, 3, 5, 8.

(a) Write down the next three terms in this sequence.

(b) Form a new Fibonacci sequence starting with 5, 10 and write down the next five terms.

6. Triangular numbers can be represented by dots arranged as triangles:

Draw and write down the next four in the sequence.

7.

Diagram 1 Diagram 2 Diagram 3

The diagrams above show a series of patterns made up by adding pencils to the previous

diagram.

(a) Draw the next diagram in the sequence.

(b) Copy and complete the table below

Diagram number Number of pencils

1 3

2 5

3

4

5

(c) How many pencils will there be in diagram

(i) 7 (ii) 9 (iii) 15

Algebra (3) Collection of like terms

Simplify each of the following terms:

1. mmm 410

2. dddd 63

3. eee 736

4. ff 1512

5. xxx 61321

6. yyy 96

7. ppp 342

8. aaa 947

9. bbb 34

10. ccc 735

11. baba 238

12. dcdc 345

13. pqpq 437

14. yxyx 8429

15. tsts 355

16. fefe 627

17. hghg 2543

18. yxyy 72

19. nmnm 10668

20. jiji 3643

21. yxyx 10364

22. pkpk 475

23. cbcb 3879

24. yxyx 61288

25. baba 7723

26. vuvu 66

27. wwzz 426

28. mnnnm 34235

29. cbacba 28537

30. gfegfe 413538

31. jihjih 3679

32. pnmpnm 644823

33. tsrsrt 9779

34. zyxzyx 3485

35. gfegfe 913201715

36. bababa 5232

37. yxyxyx 4223

38. rqprqp 7251058

39. wvuwvu 1310877

40. xxyx 437

41. utut 363

42. baba 89312

43. yxyx 33

44. hghg 257

45. kjkj 1284

46. qpqp 1292

47. dcdc 356

48. vxvw 7238

49. zxzx 8659

50. cbacba 124332

Algebra (4) Simplifying Expressions

Simplify each of the following terms:

1. x7

2. ba 43

3. yx 35

4. dc 26

5. qp47

6. xy 82

7. nm 75

8. de 54

9. af 98

10. rs3

11. ba 43

12. nm 67

13. pq 412

14. jk 9

15. xy 37

16. vu 115

17. xx

18. pp

19. aa 92

20. rs 65

21. rr4

22. ih 77

23. rqp 32

24. cba 326

25. gfe 74

26. zyx 64

27. tt 154

28. rpq 36

29. yy 85

30. nm 27

Algebra (5) Brackets

1. Remove the brackets for each of the following:

(a) 34 x

(b) 75 x

(c) 13 x

(d) 52 x

(e) 322 x

(f) 123 x

(g) 435 x

(h) 2xx

(i) 52 yy

(j) x327

(k) x414

(l) 732 aa

(m) 52 x

(n) 323 x

(o) x328

(p) 654 t

(q) 732 b

(r) xx 232

(s) b293

(t) 23 mm

(u) m36

(v) p326

(w) ba 327

2. Rewrite these expressions using brackets. {Look for common factors}

(a) 82 x

(b) 123 a

(c) 243 p

(d) x48

(e) y530

(f) 64 m

(g) n912

(h) q2128

(i) xx 32

(j) yy 23 2

(k) bab 3

(l) eef 612

(m) cdc 129

(n) yxy 1221

(o) xx 164 2

3. Remove the brackets and simplify each of the following:

(a) 532 x

(b) 7423 x

(c) 324 x

(d) xx 224

(e) 1233 xx

(f) 4332 xx

(g) 522123 xx

(h) xx 312134

(i) 23548 xx

(j) xxxx 3212

Algebra (6) Substituting into formulae

1. If a p 4 find a when 8p

2. If m n 7 find m when 6n

3. Given st 26 work out t when s 9

4. Given bca what is the value of a when 24b and c 13?

5. If zyx what is the value of x when y 34and z 19?

6. If 32 xy work out the value of y given x 5 .

7. If 54 xy what is the value of y when 3x ?

8. Given ed 26 work out the value of d when e 3 .

9. If 73 pq what is the value of q when 11p ?

10. If g h 8 4 find the value of g when 7h .

11. If pnrm work out the value of m when n 8, r 2 and 5p .

12. Given y mx c what is the value of y when 3m , 5x and 2c ?

13. Given v u at find v when u 7 , a 3 and t 10.

14.Evaluate UTS 10 when 5.2T and U 5.

15. If VM

D work out V when 80M and 5D .

16. Given SD

T work out S for 350D and 20T .

17. If h i 6 3 work out h for 6i .

18. Given m n 4 2 work out the value of m when 8n .

19. Evaluate P q r 5 when 4q and 3r .

20.Given y x 2 what is the value of y when 8x ?

21. Given 72 xy what is the value of y when 4x ?

22.Given xxy 22 what is the value of y when 2x ?

23.Given xxy 32 what is the value of y when 5x ?

24.Given 32 xxy what is the value of y when 1x ?

25.Given 722 xxy what is the value of y when 3x ?

Algebra (7) Fill in the missing gaps

In each of the following questions fill in the missing gaps:

1. What number should be written in the circles in the following:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

7 +3

7 –4

7 X 4

7 + 𝑥

7 –𝑦

𝑎 + 𝑏

𝑚 X 𝑛

𝑦 + 2𝑥

5d –2d

4h X 3i

2. What should be written in the rectangles in the following?

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

–2

X 5

X 2

+ a

– b

X f

+ 2 x

–5d

X 3a

+ 3

11

6

30

2a

2 + a

x - b

fg

8x

4e –5d

12ab

3. What should be written next to the sign in the following:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

4 +

12 –

3 +

p X

5s +

b X

2y –

9d –

5h ×

4 x

12

12

4

3 + c

8p

5s + r

4bc

2y – 3x

2d

30hg

4. What should be written in the circles or rectangles for each of the following:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

5. Here is a flow diagram

(a) Work out the output value when the input value is 4

(b) Work out the input value when the output value was 32.

5 Double, and add 3

Double, and subtract 1

11

3 Multiply by 3 and add x

a Multiply by 3 and add x

2 Multiply by a and subtract b

d Double, and subtract 1

Double, and add 5

2x + 5

Multiply by 4 and subtract 2

4a –2

3p Multiply by 4 and subtract 3q

Double, and add 2k

8 + 2k

Algebra(8) Sequences

1. For each of the following nth terms write down the first five terms

a) 𝑈𝑛 = 5𝑛 + 1

b) 𝑈𝑛 = 4𝑛 − 2

c) 𝑈𝑛 = 6𝑛 + 4

d) 𝑈𝑛 = 2𝑛 − 1

e) 𝑈𝑛 = 2𝑛 + 3

f) 𝑈𝑛 = 7𝑛 − 5

g) 𝑈𝑛 = 6𝑛 + 3

h) 𝑈𝑛 = 8𝑛 − 5

i) 𝑈𝑛 = 4𝑛 + 1

j) 𝑈𝑛 = 9𝑛 − 3

2. A sequence has nth term given by 3𝑛 + 6

(a) Write down the first two terms of the sequence

(b) Write down the tenth term of the sequence.

3. Using the flow diagram below write down an expression for the output value when the input

value is n


value is n


value is n

6. Find the nth term for each of the following sequences

7. Find the nth term for the sequence below

20, 17, 14, 11, 8,

8.

Diagram 1 Diagram 2 Diagram 3 Diagram 4

The diagrams above show a series of patterns made up by adding pencils to the previous

diagram.

(a) Copy and complete the table below

Diagram number Number of pencils

1 3

2 5

3

4

5

(b) Work out an expression for the Number pf pencils (P) required for diagram D.

(c) How many pencils will be required for diagram 20?

a) 8, 11, 14, 17, 20

b) 3, 10, 17, 24, 31

c) 7, 11, 15, 19, 23

d) 1, 9, 17, 25, 33

e) 5, 11, 17, 23, 29

f) 4, 13, 22, 31, 40

g) 6, 9, 12, 15, 18

h) 5, 12, 19, 26, 33

i) 9, 11, 13, 15, 17

j) 12, 18, 24, 30, 36

Algebra(9) Solving Equations

Exercise 1

Solve each of the following equations

1. 𝑥 + 5 = 12

2. 𝑥 + 7 = 16

3. 𝑥 − 4 = 7

4. 𝑥 − 9 = 3

5. 𝑥 + 6 = 4

6. 𝑥 + 7 = 21

7. 𝑥 − 5 = 12

8. 𝑥 − 1 = 3

9. 𝑦 + 7 = 11

10. 𝑎 + 3 = 28

Exercise 2


1. 3𝑥 = 27

2. 4𝑥 = 36

3. 5𝑥 = 14

4. 2𝑥 = 54

5. 6𝑦 = 48

6. 4𝑚 = 60

7. 9𝑎 = 54

8. 12𝑔 = 48

9. 3𝑡 = 36

10. 11𝑟 = 77

Exercise 3


1. 2𝑥 − 3 = 7

2. 3𝑥 + 1 = 16

3. 2𝑥 + 6 = 14

4. 4𝑥 − 5 = 19

5. 7𝑦 + 2 = 23

6. 4𝑒 − 7 = 5

7. 2𝑚 + 8 = 9

8. 5𝑎 − 6 = 39

9. 8𝑥 − 1 = 23

10. 6𝑛 + 5 = 36

11. 7𝑝 − 4 = 31

12. 2𝑒 − 8 = 7

13. 3𝑟 + 4 = 22

14. 9𝑖 − 3 = 51

15. 4𝑥 − 1 = 31

16. 8𝑤 + 6 = 46

17. 6ℎ + 9 = 27

18. 10𝑦 − 4 = 46

19. 5𝑏 + 8 = 53

20. 9𝑥 − 5 = 31

Exercise 4


1. 3𝑥 + 1 = 2𝑥 + 8

2. 4𝑥 − 3 = 2𝑥 + 7

3. 5𝑥 + 7 = 3𝑥 + 3

4. 7𝑥 − 4 = 3𝑥 + 12

5. 7𝑚 + 3 = 2𝑚 + 28

6. 6𝑝 − 9 = 2𝑝 + 11

7. 8𝑥 + 1 = 𝑥 + 22

8. 9𝑦 − 5 = 3𝑦 + 22

9. 6ℎ + 2 = 4ℎ + 16

10. 7𝑏 − 5 = 4𝑏 + 4

11. 5𝑥 − 9 = 3𝑥 + 7

12. 9𝑑 + 6 = 5𝑑 + 30

13. 3𝑐 − 7 = 2𝑐 − 5

14. 4𝑥 − 5 = 2𝑥 − 9

15. 5𝑤 + 12 = 2𝑤 + 3

16. 4𝑎 + 11 = 𝑎 + 8

17. 8𝑦 − 3 = 3𝑦 + 32

18. 10𝑥 + 6 = 5𝑥 − 9

19. 5𝑦 + 8 = 3𝑦 − 2

20. 12𝑥 + 1 = 4𝑥 − 23

Exercise 5


1. 2(𝑥 + 7) = 16

2. 3(𝑥 − 5) = 6

3. 2(2𝑥 − 1) = 10

4. 3(𝑥 + 1) = 12

5. 5(𝑎 + 4) = 30

6. 4(𝑦 − 3) = 16

7. 2(2𝑥 + 3) = 18

8. 3(2𝑥 − 5) = 9

9. 8(𝑏 + 2) = 24

10. 9(𝑐 − 5) = 36

11. 7(𝑦 + 5) = 42

12. 3(3𝑦 + 1) = 21

13. 4(2𝑥 + 3) = 52

14. 2(2𝑦 − 7) = 6

Exercise 6


1. 63

x

2. 35

y

3. 204

p

4. 62

5

x

5. 52

3

x

6. 26

1

y

7. 123

3

m

8. 72

8

x

9. 73

12

x

10. 35

32

x

S J Cooper

Algebra(10) Constructing equations

1. I think of a number, double it and then add 5. The answer is 15. What was my number?

2. I think of a number, double it and subtract 4. The answer is 10. What was my number?

3. I think of a number, treble it and add 1. The answer is 25. What was my number?

4. I think of a number, multiply it by 3 and subtract 4. The answer is 17. What was my number?

5. I think of a number, multiply it by 4 and subtract 2. The answer is 10. What was my number?

6. I think of a number, multiply it by 6 and add 5. The answer is 29. What was my number?

7. The rectangle drawn opposite has dimensions 𝑥 by

𝑥 + 3.

(a) Write down an expression for the perimeter of the

rectangle.

(b) Given that the perimeter is 30, find the value of 𝑥.

8. Given the perimeter of the rectangle opposite is

40cm find its length and width

9. If the perimeter of the triangle is 35 cm find the sides of the triangle.

10. (a) Find an expression for the perimeter of the triangle drawn below

(b) Given that the perimeter of the triangle above is 48cm, find the value of y.

𝑥 + 3

𝑥

y

2y 2y

2𝑥 − 1

𝑥

𝑥 + 5

𝑥 − 3

3𝑥 + 1

S J Cooper

Algebra (11) Inequalities 1. Represent each of the following inequalities on a number line

a) 1x

b) 3x

c) 2x

d) 5x

e) 3x

f) 1x

g) 7x

h) 5x

2. State what inequality is represented by each of the following diagrams

a)

b)

c)

d)

e)

f)

g)

3. Solve each of the following inequalities

a) 2𝑥 − 1 < 7

b) 3𝑥 + 4 > 10

c) 6𝑥 − 3 ≤ 15

d) 2𝑥 + 6 < 2

e) 4𝑥 − 1 ≤ 11

f) 5𝑥 + 8 ≥ 18

g) 3𝑥 − 7 > 2

h) 2𝑥 + 9 ≤ 3

i) 4𝑥 − 5 ≤ 19

j) 2𝑥 + 3 > 9

k) 9𝑥 − 12 ≥ 24

l) 7𝑥 + 6 > 34

m) 3𝑥 + 4 < 1

n) 5𝑥 − 6 ≤ 19

o) 6𝑥 + 8 ≥ 32

p) 10𝑥 − 9 ≥ 61

q) 2𝑥 + 11 < 5

r) 3𝑥 + 20 > 5

s) 4𝑥 + 1 ≤ 29

t) 3𝑥 − 8 > 19

– 2 –1 0 1 2 3 4

– 2 –1 0 1 2 3 4

– 4 –3 –2 –1 0 1 2

– 5 –4 –3 –2 –1 0 1

– 2 –1 0 1 2 3 4

– 5 –4 –3 –2 –1 0 1

– 2 –1 0 1 2 3 4

S J Cooper

Algebra(12) Straight line graphs

1. a) Using the formula 𝑦 = 2𝑥 + 3 complete the table below for different values of x.

x 0 1 2 3 4

y

b) Plot each of the points found above on a set of axes labelling the 𝑥 axis from 0 to 5 and

the y axis from 0 to 15

2. a) Using the formula 𝑦 = 2𝑥 − 5 complete the table below for different values of x.

x 0 1 2 3 4

y


the y axis from −6 to 6

3. a) Using the formula 𝑦 = 3𝑥 + 1 complete the table below for different values of 𝑥.

x 0 1 2 3 4

y


the y axis from 0 to 18

4. By completing the table below, plot the graph of 𝑦 = 3𝑥 − 2

x 0 1 2 3

y


x 0 1 2 3

y

6. By completing the table below, plot the graph of 𝑦 = 2𝑥 + 4

x 0 1 2 3

y


x 0 1 2 3

y

S J Cooper

8. By completing the table below, plot the graph of 𝑥 + 𝑦 = 6

x 0 1 2 3

y

9. By completing the table below, plot the graph of 𝑥 + 𝑦 = 3

x 0 1 2 3

y

10. By completing the table below, plot the graph of 2𝑥 + 𝑦 = 10

x 0 1 2 3

y

11. Using your own table, draw each of the following graphs.

a) 𝑦 = 𝑥 − 3

b) 𝑦 = 3𝑥 + 3

c) 𝑦 = 4𝑥 − 2

d) 𝑥 + 𝑦 = 12

e) 𝑦 = 8 − 2𝑥

12. a) Draw on the same set of axes the graphs of 𝑦 = 2𝑥 − 1 and 𝑦 = 3𝑥 − 5

b) Hence solve the simultaneous equations 𝑦 = 2𝑥 − 1 and 𝑦 = 3𝑥 − 5

13. a) Draw on the same set of axes the graphs of 𝑦 = 2𝑥 + 5 and 𝑥 + 𝑦 = 11

b) Hence solve the simultaneous equations 𝑦 = 2𝑥 + 5 and 𝑥 + 𝑦 = 11

14. a) Draw on the same set of axes the graphs of 𝑦 = 3𝑥 − 7 and 𝑦 = 2𝑥 − 8

b) Hence solve the simultaneous equations 𝑦 = 3𝑥 − 7 and 𝑦 = 2𝑥 − 8

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