Chapter 14
Algebra
Exercise 14.1
1.
Determine the values of the letters in these formulae :a
A = m + n when m = 19 and n = 18
b
B = 2k + 4p when k = 5 and p = 6
c
C = 50 – 3a when a = 14
d
D=
e
E=
f
F = pq – r when p = 7 q = 4 and r = 6
g
G = w 2 when w = 20
h
H = x when x = 144
i
I = 5y 2 − 3 when y = 4
j
J = (a − b )2 when a = 3 and b = 2
l
⎛ x ⎞2
L = ⎜⎜ ⎟⎟ when x = 14 and y = 2.
⎝y ⎠
k
2.
K=
65
c + 1
v
w
when c = 9
when v = 72 and w = 2
36
g
when g = 9
The cost of hiring a wheelbarrow is a one-off payment of £12 plus £5·50 per day.
The cost £C for n days is given by the formula C = 12 + 5·5n .
Find the cost of hiring a wheelbarrow for 6 days.
Exercise 14.2
1.
2.
Multiply out the brackets :a
4(x + 5)
b
3(x – 9)
c
8(1 – x)
d
2(x + y)
e
15(x – y)
f
20(y – x)
g
3(2x + 1)
h
7(3x + 4)
i
5(2x – 1)
j
9(1 – 2x)
k
10(3 – 4x)
l
3(2x + 3y)
m
8(3y – x)
n
7(5p + 4q)
o
6(6k – 3mp)
p
20(5ab + 2c)
q
–2(a + 5)
r
–3(a + 7)
s
–4(6a + 1)
t
–8(2 – 3a)
u
–2(v – 1)
v
–5(v – 3)
w
–7(v – 2w)
x
–6(10v – w).
Rewrite the following without brackets :a
2(a + 2b + c)
b
3(a + 4b + 5c)
c
4(a – b + 6c)
d
5(3a – 2b + c)
e
10(5a – 4b – 3c)
f
20(a – b – 1)
g
6(ab + bc – cd)
h
9(abc + d – 4)
i
–2(a + b + c)
j
–3(2a + 5b + 7c)
k –4(3a – 2b + 5c)
l
–5(4a – 3b – 6c).
©TeeJay Publishers 2007
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Algebra
Exercise 14.3
Multiply out the brackets and collect like terms :1.
2.
a
2(x + 3) + 1
b
3(p + 4) + 2
c
2(d + 5) – 9
d
8(n + 1) – 2
e
5(x + 1) + 3x
f
9(x + 3) + x
g
7(t + 1) – 2t
h
10(p + 4) – 6p
i
7x + 7(x + 1)
j
9w + 5(w – 1)
k
4k + 2(5k + 1)
l
6k + 2(2k – 5)
m
4x + 2(4x – 3y)
n
4(4p – 2q) – 10p
o
9 – 2(p – 4)
a
2(x + 4) + 4(x + 1)
b
3(a + 4) + 6(a + 2)
c
4(m – 2) + 5(m + 3)
d
3(p – 2) + 8(p + 1)
e
4(2 + 3x) + 9(1 – x)
f
7(1 – x) + 10(1 + x)
g
12(2m + 2) + 2(2m – 8)
h
9(3 – 2g) + 4(1 + 5g)
i
2(5d – 3) + 3(2d + 4)
j
2(5v + w) + 2(3w – 2v)
k
4 + 2(x – 5)
l
4 – 2(x – 5)
(careful)
(careful)
Exercise 14.4
1.
2.
3.
Factorise each of the following by taking out a common factor :a
3x + 12
b
6x + 24
c
8x + 40
d
2x – 10
e
3x – 15
f
6x – 30
g
9x + 72
h
8x – 56
i
5x + 45
j
7x + 14y
k
4a + 12b
l
5p + 20q
m
2v – 18w
n
10m – 90n
o
110k + 11h.
Factorise each of the following by taking out the highest common factor each time :a
4x + 10
b
8x + 20
c
9x + 15
d
4x – 14
e
10x – 25
f
15x – 35
g
27x + 18
h
20x + 24
i
21x + 35
j
6p + 9q
k
20p + 30q
l
12x + 18
m
6a – 14b
n
35a – 15b
o
14a – 35b.
Factorise each of the following by taking out a common factor or letter :a
5a + 5b + 5c
b
2a + 4b + 12c
c
25a + 15b – 45c
d
14a – 21b – 56c
e
pr – pq – pm
f
ab – av – a.
©TeeJay Publishers 2007
page 58
Algebra
Exercise 14.5
Solve these equations :1.
x + 4 = 11
2.
2x = 16
3.
2x + 7 = 17
4.
x – 3 = 11
5.
2x = x + 7
6.
6x = 2x + 20
7.
x – 11 = 12
8.
3x – 1 = 26
9.
2(x + 1) = 10
10.
4 x – 3 = 3x + 7
11.
7x = 56
12. 8x = 5x + 21
13.
7 x + 2 = 5x + 9
14.
3(x – 4) = 15
15. 6x + 2 = 50
16.
4x = 40
17.
5(x – 1) = 40
18. x – 0·5 = 7·5
19.
18x = 15x + 21
20.
3x – 8 = x + 1
21. x + 1·5 = 7·5
22.
11x = 2x + 72
23.
21x = 4x + 17
24. 5x – 6 = 44
25.
4x = 10
26.
11x – 12 = 5x
27. 40 = 4(x + 2)
28.
4x = 27 + x
29.
3x + 19 = 19
30. 6x = 33
31.
7(x + 1) = 56
32.
7 + 5x = 27
33. 210x = 0
34.
6x – 24 = x + 1
35.
60 = x + 51
36. 77 – 2x = 9x
37.
1
2
38.
1
2
39.
x=9
x=0
1
4
x = 4·5.
Exercise 14.6
1.
2.
3.
Write True OR False ?
a
3 > 7
b
–5 > –2
c
– 4 < –3
d
2 > –2
e
–21 < –20
f
5 > –1
g
–19 < –18
h
–50 > –49
i
20 < –20.
COPY each statement and put < or > between each pair to make the statement true :a
–3 ... 1
b
–4 ... –5
c
–9 ... –8
d
–1 ... 3
e
–14 ... –15
f
4 ... –3
g
–7 ... –6
h
–17 ... –18
i
–24 ... –26.
Solve the inequalities, leaving your answer in the form e.g. “x > 3”.
a
x + 3 > 11
b
x–7<3
c
7x > 21
d
7x – 4 ≥ 31
e
4x + 5 < 21
f
5x – 2 < 2x + 13
g
2(x – 3) ≤ 8
h
3x – 1 > 19 – x
i
10(x + 1) < 10.
©TeeJay Publishers 2007
page 59
Algebra
Revision Exercise
1.
2.
3.
4.
5.
6.
a
P = 7Q + R
Find P when Q = 6 and R = 8.
b
H = vw – xy
Find H if v = 5, w = 9, x = 11 and y = 4.
c
M = n 2k
Find M when n = 8 and k = 0·5.
d
T = 75
Find T when g = 24.
g + 1
Multiply out the brackets :a
3(c + 6)
b
11(a – 3d)
c
v(w – 8)
d
x(x – 1)
e
5(3n – m)
f
–2(b + 4)
g
–6(c – y)
h
2(2a + 3b – 5c)
i
2(k – 3gy – 5h).
Remove the brackets and simplify :a
4(x + 2) + 1
b
9(x + 2) – 15
c
3(x + 7) + 7x
d
8(x + 3) – 7x
e
4x + 5(x + 2y)
f
10 + 4(x – 1)
g
4(x + 2) + 7(x + 1)
h
3(x – 2) + 5(x + 2)
i
8(x – 1) – 6(x – 2).
Factorise each of the following fully :a
2x – 18
b
20x + 4
c
8x – 20
d
9x + 30
e
4x + 6 y
f
10x – 45w
g
0·8v + 0·8w
h
40p + 50q – 60r
i
8x – 12y – 16z
j
ab + ac
k
m – mn
l
2pq – 4aq.
Find the value of x by solving these equations :a
x + 11 = 17
b
x – 9 = 11
c
12 – x = 0
d
17 + x = 40
e
6x = 42
f
2x = 11
g
3x + 4 = 28
h
6x – 1 = 41
i
7x – 7 = 0
j
5(x + 1) = 30
k
12(x – 1) = 0
l
5x – 4 = 3x + 3.
Solve, leaving your answer in the form “x > 5”, etc.
a
x+6>9
b
x–7<8
c
x + 14 ≤ 14
d
x–9≥9
e
8x ≤ 32
f
6x ≥ 54
g
10x < 200
h
2x > 17
i
2x – 12 > 2
j
7x + 1 ≥ 71
k
4x – 10 ≤ 0
l
8x + 10 < 14.
©TeeJay Publishers 2007
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Algebra