school of mathematics · algebra for students preparing to start a level mathematics algebra is the...

Algebra for students preparing to start A Level Mathematics

Algebra is the backbone of mathematics. Nearly every problem you solve will require you to use some algebra. We have put this pack together to help you recap your GCSE algebra knowledge and fill in any gaps you may have. This way you will be ready to start when you come in September and can focus on learning the new and exciting things A level maths has to offer.

Work through the booklet, watching the suggested videos and try the exercises given. Clearly show all your working on lined paper. Answers are provided, so you can mark and correct you work. Please use a green pen for this. Highlight anything you need help with. There will be plenty of support available when you come to college in September.

If you need any help or clarification please e-mail Louisa Singleton at singleton.L@runshaw.ac.uk

Calculators Calculators are allowed in all the mathematics exams. We recommend you buy the Casio fx-991EX classwiz. It has all the features that you will need to see you through the next two years and beyond!

Name: __________________________________

Unit Questions I need help on

1. linear equations

2. simultaneous equations

3. factorising

4. transformation of formulae

School of Mathematics

Solving linear equations

Solve the following linear equations. Please check your answers and mark/correct them in green pen using this video: https://www.youtube.com/watch?v=-eNM4GV-X9s&feature=youtu.be

Solve 2+3x = 7

Solve 6x + 7 = 5 − 2x

Solve 2(3𝑥 − 2) = 20 − 3(𝑥 + 2)

Solve

𝑦−4

2=

𝑦

3+ 1

Further examples of fractions in equations can be seen here: https://www.youtube.com/watch?v=KBze9KjZzxQ https://www.youtube.com/watch?v=ofchn9z8_lU

Solve the following equations showing all the steps in your working on lined paper.

Check your answers and mark in green pen, completing any corrections as necessary.

1. 7x = 27 + 4x

2. 6x = 15 + 3x

3. 4x + 3 = 3x + 1

4. 3x - 6 = 2x - 7

5. 5x - 9 = 4x - 11

6. 6x + 7 = 4x + 6

7. 2(2x - 1) = 3(x + 3)

7. 2(2x - 1) = 3(x + 3)

8. 5(1 + x) = 3(2x + 3)

9. 4(3 - 2x) = 3(6 - 4x)

10. 7(x - 3) = 3(7 + x)

11. 6(2x + 7) = 7(x + 6)

12. 2(6x +7) = 6(7 - 6x) 13.

3

5 3

16

15

x x

14.

5

6

3

4 3

x x

15.

x x

2

8

9

2

30

16.

5

9

14

27

5

6

1

2x x x

17.

1

2 6

4

9 6

x

x

x

18.

x x

1

2

3 4

71

19.

3

1

2

2x x

(hint: cross multiply)

20.

4

55 2

2

32 3

7

106 5( ) ( ) ( )x x x

21.

x

x

x

x

2

1

3

4

Solve √7 − x + 3 = 5 check your answer here: https://youtu.be/-8q7Mx1-y-Y

Solve 3√x + 1 = 2√𝑥 + 6 check your answer here: https://www.youtube.com/watch?v=R7YBBzfKZh8

22. 2 8 2x

23. 2 5 3x x

24. 1 1

4x

25. 510

1x

ANSWERS

1. 9 11. 0 21. 1 14

2. 5 12. 712 22. 6

3. -2 13. 4 23. 4

4. -1 14. 1 12 24. 16

5. -2 15. 5 13 25. 2

5

6. 12 16. 13

27

7. 11 17. 89

8. -4 18. -13

9. 1 12 19. -4

10. 10 12 20. 2 1

46

Simultaneous equations – elimination

Access https://www.youtube.com/watch?v=phlus4x0UqM&feature=emb_logo

Complete the following exercise, filling in the gaps and completing the examples, whilst you watch

the video.

Make any extra notes you need to.

Solve simultaneously: 6𝑥 + 3𝑦 = 21 2𝑥 + 3𝑦 = 17

Solve simultaneously: 2𝑥 + 𝑦 = 7 3𝑥 − 𝑦 = 8

Solve simultaneously: 4𝑥 + 𝑦 = 43 4𝑥 − 2𝑦 = 34

Solve simultaneously: 7𝑝 − 2𝑞 = −1 21𝑝 + 𝑞 = 25

Solve simultaneously: 3𝑠 + 2𝑡 = 24 8𝑠 + 4𝑡 = 60

Solve simultaneously: 2𝑎 − 5𝑏 = 11 3𝑎 + 2𝑏 = 7

Solve simultaneously: 2𝑥 + 𝑦 = 5 2𝑥 = 15 − 3𝑦

Further worked examples and notes are available on mymaths:

https://app.mymaths.co.uk/196-lesson/simultaneous-equations-1

https://app.mymaths.co.uk/197-lesson/simultaneous-equations-2

https://app.mymaths.co.uk/198-lesson/simultaneous-equations-3

https://app.mymaths.co.uk/199-lesson/simultaneous-negatives

Further video examples with jack brown:

https://www.youtube.com/watch?v=Vw5_LV2n-GI&list=PLg2tfDG3Ww4vrstKAZ0dajHx_hq85P0G-&index=56&t=0s

Simultaneous equations – substitution Access https://www.youtube.com/watch?time_continue=7&v=FrECUQpaa80&feature=emb_logo

Complete the following exercise, filling in the gaps and completing the examples, whilst you watch

the video.

Make any extra notes you need to.

Solve simultaneously: 2𝑥 − 𝑦 = 2 4𝑥 + 2𝑦 = 36

Solve simultaneously: 3𝑦 − 𝑥 = 7 10𝑦 + 3𝑥 = −2

Further video examples with jack brown:

https://www.youtube.com/watch?v=o2D1CrZ1Kjs&list=PLg2tfDG3Ww4vrstKAZ0dajHx_hq85P0G-&index=57&t=0s

Now try the following questions practising both methods.

1. x + 5y = 8

2x - 5y = 1

2. 7x - 3y = 14

5x + 4y = 53

3. 800 tickets were sold for a pop concert, some costing £9 and some costing £12. The total cash

received was £8550. Find the number of each ticket sold.

4. The sum of 2 numbers is 42, and one is 6 times the other. Calling the numbers x and y, write down

a pair of simultaneous equations in x and y and solve.

Answers: 1) x = 3, y = 1 2) x = 5, y = 7 3) 350 and 450 4) x + y = 42, x = 6y; x = 36, y = 6

Simultaneous equations – solving graphically

Access https://www.youtube.com/watch?time_continue=1&v=20R39KaGwmc&feature=emb_logo

Complete the following exercise, filling in the gaps and completing the examples, whilst you watch

the video.

Make any extra notes you need to.

Solve simultaneously: 𝑦 = 2𝑥 𝑦 = 𝑥 + 1

Solve simultaneously: 2𝑥 + 𝑦 = 4 4𝑥 − 3𝑦 = 3

Further worked examples and notes are available on mymaths:

https://app.mymaths.co.uk/200-lesson/solving-sim-equations-graphically

Factorising

Access https://www.youtube.com/watch?v=6_4AfAK4x3s

Complete the following exercise, filling in the gaps and completing the examples, whilst you watch the

video. Make any extra notes you need to.

Factorise 6𝑥2 − 2𝑥𝑦

Simplify and factorise 3𝑥(2𝑥 − 1) − 4(2𝑥 − 1)

Factorise 𝑥2 − 9𝑥 − 10

Factorise 𝑥2 − 9

Factorise 5𝑥2 + 13𝑥 − 6

Now try some GCSE exam questions on factorising:

Factorise the following, thinking carefully about each step in your working.

1. 4(3𝑥 + 2𝑦) + 8(𝑥 − 3𝑦)

2. 𝑥(𝑥 − 2) − 𝑥(𝑥 − 8) + 6

3. 𝑥2 + 5𝑥 + 6

4. 𝑥2 + 10𝑥 + 24

5. 𝑥2 − 16𝑥 + 48

6. 2𝑥2 + 5𝑥 + 2

7. 15𝑥2 + 2𝑝 − 1

8. 9𝑥2 − 30𝑥 + 25

9. 2𝑥2 − 3𝑥 − 27

10. 4𝑥2 + 4𝑥 − 15

Answers

1. 4(5𝑥 − 4𝑦) 2. 6(𝑥 + 1)

3. (𝑥 + 3)(𝑥 + 2) 4. (𝑥 + 6)(𝑥 + 4)

5. (𝑥 − 12)(𝑥 − 4) 6. (2𝑥 + 1)(𝑥 + 2)

7. (5𝑥 − 1)(3𝑥 + 1) 8. (3𝑥 − 5)(3𝑥 − 5)

9. (2𝑥 − 9)(𝑥 + 3) 10. (2𝑥 + 5)(2𝑥 − 3)

Transformation of Formulae

Access https://www.youtube.com/watch?v=qb7qSdmJwT8

Complete the following exercise, filling in the gaps and completing the examples, whilst you watch the

video. Make any extra notes you need to.

Transform 𝑦 = 4𝑥 + 3 for x

𝐶 =5(𝐹−32)

9 for F

𝑥2 + 𝑦2 = 𝑎2 for x

𝑎 − 𝑥𝑡 = 𝑏 + 𝑦𝑡 for t

𝑇 − 𝑊 =𝑊

2𝑎 for W

Now try some GCSE exam questions on transforming formulae:

Transpose the following, thinking carefully about each step in your working.

1. yx

5

17 for x

2. D = B - 1.28d for d

3. Sv u t

( )

2 for u

4. TD d

L

12( ) for d

5. PS C F

C

( ) for C

6. yt

t

1

1 for t

7. 54

3

ta for t

8. g

Lt 2 for L

9. P mgmv

r

2

for m

10. D

d

f p

f p

for f

Answers

1. x = 5y - 85 = 5(y -17) 2. dB D

1 28.

3. us vt

t

s

tv

2 2 4. d= = D -

5. CSF

S P

6. t

y

y

1

1

7. ta

a

3 5

4 8. L

t g

2

24

9. mv gr

Pr2

10. f= =

12D - LT

12

LT

12

D2p + d2p

D2 - d2

p(D2 + d2)

D2 - d2

5