1. Algebra

2. Collecting Terms This is a way of simplifying algebra
If you have b + b + b + b
This is the same as 4 b
The b could stand for boots
If you have p x p x p
This is p3
This happens when multiplying

3. Try these: M x m T + t + t R x r x r x r G + g + g + g H x h x h K + k
J x j x j x j x j x j x j

4. Answers M2 3t R4 4g H3 2k j7

5. Collecting terms Usually you have to collect terms from a mix
You cannot add 2t and t2, you can only add t2 and t2 together
E.g.
A2 + 3a + 3a + 2a2
3a + 3a = 6a
A2 + 2a2 = 3 a2
So the simplified equation is 6a + 3 a2

6. Collecting terms You may have to deal with minus numbers or terms
Always look at the sign before the term to see if it is positive (+) or negative (-)
E.g
This is +3 and -1 and -4
3 -1 = 2
2 - 4 = -2

7. Now try these; -5 +4 -3 3h + 2h – 6h 3s + 3s – 8s 7y – 5y – 2y
3d – d + 2d – 5d
7r – 4r – 5r -2r
5p – p + 3p – 2p

8. Answers -4
-1h or -h
-2s
-1d or –d
-4r 5p

9. Mixed terms J + j + k + k + k R – 2r +3s +2s 7y – 5y – 3y +4
3t + 6s – 8s + t
7r + 2s – r – s
5p + 6 – 7p – 9
3a + b + a - 2b

10. Answers
2j + 3k
-r + 5s
-y + 4
4t – 2s
6r + s
-2p – 3
4a - b

11. Multiplying Terms
When you multiply terms you multiply the numbers at the start of the term and then add together the number of letters you have
E.g.
2a x 3a
This is 2 x 3 = 6
And a x a = a2
This is 6a2

12. Multiplying
2 x 4c
2 x r x s
5 x 3c
2e x 6e
3a x 2a
(4k)2
4r x 2rs

13. Answers 8c
2rs
15c
12e2
6a2
16k2
8r2s

14. Try these 5ab – 3ab 6vw – 4w + 5wv X + 2x – 3x2 + 5x2
8r + 6rs – 2sr – 3r
Xy + x2 – 3xy + 3x2
4y + 3y2 – 7y2 – 2y

15. Answers
2ab
11 vw + 4w
3x + 2x2
5r – 4rs
-2xy + 4x2
2y – 4y2

16. Multiplying Out Brackets
Everything inside the brackets is multiplied by the term just outside it to the left
E.g.
4 (3 + t)
This is
4 x 3 = 12
And 4 x t= 4t
So it becomes t

17. Try these 6(1-s) 4(p + q) 3(10j-4k) R2(3-2s) 2x3 (x-y) 5t2 (s + t)
3r (2r – 3s – t)

18. Answers 6 -6s 4p + 4q 30j + 12k 3r2 + -2r2s 2x4 – 2x3y 5st2 + 5t3
6r2 – 9rs – 3rt

19. Factorising This is the opposite of multiplying out brackets
When you simplify you can place terms inside brackets
E.g. 4k + 2
Both can be divided by 2
So it becomes 2(2k + 1)
Everything inside the bracket is divided by 2

20. Try these 3f + 3 15 + 20 t 18 + 6a 10j + 25 3r + 3s + 3t Pq – q2
24p2 + 30pq
20ab2 + 36a2b2

21. Answers 3 (f + 1) 5 (3 + 4t) 6 (3 + a) 5 (2j + 5) 3 (r + s + t)
Q (p – q)
6p (4p + 5q)
4ab (5b + 9ab)

22. Multiplying Brackets with -
If there is a minus before the brackets
A minus x a plus = a minus
A minus x a minus = a plus
E.g.
-3(2r – r)
-3 x 2r= -6r
-3 x –r = +3r
So it becomes -6r + 3r

23. Try these
-8(s-t)
-(r-5)
-(4r-3)
-9y(y-1)
-5s(s+4)
-3h(5-h)
- (x+y)

24. Answers
-8s + 8t
-r + 5
-4r + 3
-9y2 + 9y
-5s2 + 20s
-15h + 3h2
-x - y

25. Multiplying sets of brackets
If you are given 2(3+y) + 5(4+y)
First you must multiply out the brackets
6 + 2y y
Secondly you collect terms
26 + 7y

26. Try these 3(4+d)+4(2+d) 6(3+x)+5(2-x) 2(10+5e)-3(6+e) 3(4r+1)-(7r-2)
4(w+1)-(w-1)
X(2x+1)+2(3x+4)
4(5+2f)+f(3+f)

27. Answers
20 + 7d
28 + x
2 + 7e
5r + 5
3w + 5
2x2 + 7x + 8
f + f2

28. Multiplying out brackets
If you are given (y+2)(y-4)
Then you really have two sums
Y(y-4)
+2 (y-4)
We have split the first bracket up to make sure we multiply everything together
So what do they workout as;
Y2 – 4y
+2y -8
We need to collect the terms
-4y + 2y = -2y
So the equation is y2 -2y -8

29. Try these (s+1)(3s+2) (2+f)(1+4f) (d-2)(3d+5) (7+k)(1+k) (a+3)(4a-1)
(y-2)(y+2)
(a+b)(a-b)

30. Answers
3s2+5s+2
2+9f+4f2
3d2-d-10
7+8k+k2
4a2+11a-3
Y2-4
A2-b2

31. (Brackets)2 If you have (4g+h)2 This means (4g+h) )(4g+h) First we …
Split up the first bracket
4g(4g+h)
+h(4g+h)
2. Then we multiply this out
16g2 + 4gh
+4gh + h2
3. Then we collect terms
+4gh + 4gh = +8gh
So our equation is
16g2 + 8gh + h2

32. Solving Equations
If you have a number and no x2 or x3 you can solve a linear equation
E.g.
4x=16
X = 16/4
X= 4
x/7=-2
Multiply both sides by 7
X = -14
What you do to one side of the equals sign you must do to the other to keep everything balanced

33. Try these x/5= 12 X-7=23 X + 12 = 45 0.5x=3 2x/3 = -6 x/3 + 2 =10

34. Answers
X= 60
X= 30
X= 33
X= 6
X= -9
X= 24
X= 7

35. Solving Equations
You should always try to keep x positive, so work from the side with the most x’s
E.g.
2x + 17= 4x
We need to take 2x away from both sides
17 = 2x
8.5 = x

36. Harder questions 13x-15 = 12x+19 2.5x + 10 = 1.5x + 17 12x-1 = 7x+19

37. Answers
X= 34
X= 7
X= 4
X= 3
X= 1
X= 2

38. Changing the Subject Sometimes you have to rearrange equations E.g.
C= a – b
I want to find out what a equals
If I add b to both sides I have a on it’s own
C + b = a

39. Try these T= sk - h V=u + at M=k + nk T2 = 7r + x/4 H= y/4 T= s2 + 5
A=π r2
S= ½gt2

40. Answers T/s + h/s = k v/a – u/a = t m/k– k/k = n T2/7- x/4/7 = r
4h = y
√t-5 = s
√a/ π = r
√2s/g= t