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Algebra
Collecting Terms
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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
Try these:
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Mxm
T+t+t
Rxrxrxr
G+g+g+g
Hxhxh
K+k
Jxjxjxjxjxjxj
Answers
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M2
3t
R4
4g
H3
2k
j7
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
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. 3 -1 -4
• This is +3 and -1 and -4
• 3 -1 = 2
• 2 - 4 = -2
Now try these;
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-5 +4 -3
3h + 2h – 6h
3s + 3s – 8s
7y – 5y – 2y
3d – d + 2d – 5d
7r – 4r – 5r -2r
5p – p + 3p – 2p
Answers
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-4
-1h or -h
-2s
0
-1d or –d
-4r
5p
Mixed terms
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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
Answers
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2j + 3k
-r + 5s
-y + 4
4t – 2s
6r + s
-2p – 3
4a - b
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
Multiplying
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2 x 4c
2xrxs
5 x 3c
2e x 6e
3a x 2a
(4k)2
4r x 2rs
Answers
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8c
2rs
15c
12e2
6a2
16k2
8r2s
Try these
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5ab – 3ab
6vw – 4w + 5wv
X + 2x – 3x2 + 5x2
8r + 6rs – 2sr – 3r
Xy + x2 – 3xy + 3x2
4y + 3y2 – 7y2 – 2y
Answers
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2ab
11 vw + 4w
3x + 2x2
5r – 4rs
-2xy + 4x2
2y – 4y2
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 12 + 4t
Try these
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6(1-s)
4(p + q)
3(10j-4k)
R2(3-2s)
2x3 (x-y)
5t2 (s + t)
3r (2r – 3s – t)
Answers
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6 -6s
4p + 4q
30j + 12k
3r2 + -2r2s
2x4 – 2x3y
5st2 + 5t3
6r2 – 9rs – 3rt
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
Try these
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3f + 3
15 + 20 t
18 + 6a
10j + 25
3r + 3s + 3t
Pq – q2
24p2 + 30pq
20ab2 + 36a2b2
Answers
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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)
Multiplying Brackets with •
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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
Try these
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-8(s-t)
-(r-5)
-(4r-3)
-9y(y-1)
-5s(s+4)
-3h(5-h)
- (x+y)
Answers
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-8s + 8t
-r + 5
-4r + 3
-9y2 + 9y
-5s2 + 20s
-15h + 3h2
-x - y
Multiplying sets of brackets
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If you are given 2(3+y) + 5(4+y)
First you must multiply out the brackets
6 + 2y + 20 + 5y
Secondly you collect terms
26 + 7y
Try these
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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)
Answers
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20 + 7d
28 + x
2 + 7e
5r + 5
3w + 5
2x2 + 7x + 8
20 + 11f + f2
Multiplying out brackets
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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
Try these
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(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)
Answers
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3s2+5s+2
2+9f+4f2
3d2-d-10
7+8k+k2
4a2+11a-3
Y2-4
A2-b2
(Brackets)2
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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)
Then we multiply this out
16g2 + 4gh
+4gh + h2
Then we collect terms
+4gh + 4gh = +8gh
So our equation is
16g2 + 8gh + h2
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
Try these
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x/5= 12
X-7=23
X + 12 = 45
0.5x=3
2x/3 = -6
x/3 + 2 =10
7x= x + 42
2.5x = 1.5x + 6
Answers
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X= 60
X= 30
X= 33
X= 6
X= -9
X= 24
X= 7
X= 6
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
Harder questions
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13x-15 = 12x+19
2.5x + 10 = 1.5x + 17
12x-1 = 7x+19
4x-3 = 12-x
2x-7 = 3 – 8x
3(2x+2) = 24
6(3x+1) = 3(4x+6)
10(3x-4) = 5(6-x)
Answers
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X= 34
X= 7
X= 4
X= 3
X= 1
X= 3
X= 1
X= 2
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
Try these
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T= sk - h
V=u + at
M=k + nk
T2 = 7r + x/4
H= y/4
T= s2 + 5
A=π r2
S= ½gt2
Answers
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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