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Paper 1
1. 2h = h + 3 2h – h = 3 h = 3
Answer: C
2. 3k = 12 – k 3k + k = 12 4k = 12
k = 12–––4
= 3
Answer: C
3. p––2
= 5––4
p = 5––4 × 2
= 5––2
Answer: B
4. x––3 – 1 = 4
x––3 = 4 + 1
x = 5 × 3 = 15
Answer: D
5. 2y–––5
+ 1 = 3
2y–––5
= 3 – 1
2y = 2 × 5
y = 10–––2
= 5
Answer: A
6. 6 – 1––2 x = x
6 = x + 1––2 x
6 = 3––2 x
6 × 2 = 3x 3x = 12
x = 12–––3
= 4
Answer: A
7. m – 1––––––2 = 5
m – 1 = 5 × 2 m – 1 = 10 m = 10 + 1 = 11
Answer: C
8. 2––n = 3––4
2 × 4 = 3 × n 8 = 3n
n = 8––3
Answer: B
Paper 2
1. 3x + y = 11 .....................1 x + y = 7 .......................21 – 2: 2x = 4
x = 4––2
= 2
Substituting x = 2 into 2, 2 + y = 7 y = 7 – 2 = 5
CHAPTER
4 Linear Equations I & IICHAPTER
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Mathematics SPM Chapter 4
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2. 2m + n = 15 .....................1 m – n = 3 .......................21 + 2: 3m = 18
m = 18–––3
= 6
Substituting m = 6 into 1, 2(6) + n = 15 12 + n = 15 n = 3
3. 5h – k = 19 .........................1 2h – k = 7 ...........................21 – 2: 3h = 12
h = 12–––3
= 4
Substituting h = 4 into 2, 2(4) – k = 7 8 – k = 7 8 – 7 = k k = 1
4. 3a + 2b = 5 ...........................1 3a – b = 11 .........................21 – 2: 3b = – 6
b = –6–––3
= –2
Substituting b = –2 into 1, 3a + 2(–2) = 5 3a – 4 = 5 3a = 5 + 4
a = 9––3
= 3
5. 4e + 3f = 18 ........................1 e – 3f = 12 ........................21 + 2: 5e = 30
e = 30–––5
= 6
Substituting e = 6 into 1, 4(6) + 3f = 18 24 + 3f = 18 3f = 18 – 24 = –6
f = –6–––3
= –2
Paper 1
1. x – 2 = 4(1 + 2x)3
3(x – 2) = 4(1 + 2x) 3x – 6 = 4 + 8x –6 – 4 = 8x – 3x –10 = 5x x = –2Answer: A
2. 7y – 23
= 4 – y
7y – 2 = 3(4 – y) 7y – 2 = 12 – 3y 7y + 3y = 12 + 2 10y = 14
y = 1410
= 75
Answer: C
3. p – 12
– p – 33
= 1
Multiply both sides by 6, 3(p – 1) – 2(p – 3) = 6 3p – 3 – (2p – 6) = 6 3p – 3 – 2p + 6 = 6 p + 3 = 6 p = 6 – 3 = 3
Answer: A
4. 4x – 15
= x + 13
3(4x – 1) = 5(x + 1) 12x – 3 = 5x + 5 12x – 5x = 5 + 3 7x = 8
x = 8––7
Answer: D
5. 12 – 3(1 – t) = 7t + 2 12 – (3 – 3t) = 7t + 2 12 – 3 + 3t = 7t + 2 9 – 2 = 7t – 3t 7 = 4t
t = 7––4
Answer: B
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6. 2 – 3(p – 1)2
= p3
Multiply both sides by 6, 12 – 9(p – 1) = 2p 12 – (9p – 9) = 2p 12 – 9p + 9 = 2p 21 = 2p + 9p 21 = 11p
p = 21–––11
Answer: D
7. 3––2 – m = –3(1 – m)
3––2 – m = –3 + 3m
3––2 + 3 = 3m + m
9––2 = 4m
m = 9–––––2 × 4
= 9––8
Answer: D
Paper 2
1. 3m – 4n = 13 .................... 1 6m + n = –1 .................... 22 × 4: 24m + 4n = –4 .................... 31 + 3: 27m = 9 m = 9–––
27
= 1—3
Substituting m = 1––3 into 2,
6 1––3 + n = –1
2 + n = –1 n = –1 – 2 = –3
2. 4v – 3k = 14 ...................... 1 2v + k = –3 ...................... 22 × 3: 6v + 3k = –9 ...................... 31 + 3: 10v = 5 v = 5–––
10
= 1––2
Substituting v = 1––2 into 2,
2 1––2 + k = –3
1 + k = –3 k = –3 – 1 = –4
3. 1––2 w – 3h = 5 ........................ 1
5w + 6h = 14 ...................... 21 × 2: w – 6h = 10 ...................... 32 + 3: 6w = 24
w = 24–––6
= 4
Substituting w = 4 into 2, 5(4) + 6h = 14 20 + 6h = 14 6h = 14 – 20
h = –6–––6
= –1
4. 3g – h = 10 ................... 1 g + 4h = –14 ................. 21 × 4: 12g – 4h = 40 ................... 32 + 3: 13g = 26 g = 2
Substituting g = 2 into 2, 2 + 4h = –14 4h = –14 – 2
h = –16––––4
= –4
5. 2x – y = 10 ................. 1 5x + 2y = 7 ................... 21 × 2: 4x – 2y = 20 ................. 32 + 3: 9x = 27
x = 27–––9
= 3
Substituting x = 3 into 1, 2(3) – y = 10 6 – y = 10 6 – 10 = y y = –4
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Mathematics SPM Chapter 4
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6. 14
x + 2y = –3 ................... 1 x – 2y = 8 ..................... 21 × 4: x + 8y = –12 ................. 32 – 3: –2y – 8y = 8 – (–12) –10y = 20 y = –2Substituting y = –2 into 2, x – 2(–2) = 8 x + 4 = 8 x = 8 – 4 = 4
7. m + 4n = 4 ........................ 1
3––4 m – n = –5 ...................... 2
2 × 4: 3m – 4n = –20 .................... 31 + 3: 4m = –16
m = –16––––4
= –4
Substituting m = –4 into 1, –4 + 4n = 4 4n = 4 + 4
n = 8––4
= 2
Paper 1
1. 8 – 3(v + 2) = v 8 – (3v + 6) = v 8 – 3v – 6 = v 8 – 6 = v + 3v 2 = 4v
v = 2––4
= 1––2
Answer: D
2. 14 = 3p – 4(2p – 3) 14 = 3p – (8p – 12) 14 = 3p – 8p + 12 14 – 12 = –5p 2 = –5p
p = – 2––5
Answer: A
3. 5m + 4 = 18 – (2 – m) 5m + 4 = 18 – 2 + m 5m – m = 18 – 2 – 4 4m = 12
m = 12–––4
= 3
Answer: D
4. 12 = 2h – 3(2h – 2) 12 = 2h – (6h – 6) 12 = 2h – 6h + 6 12 – 6 = – 4h 6 = – 4h
h = – 6––4
= – 3––2
Answer: A
5. w + 2 = 5 – 2(w – 6) w + 2 = 5 – (2w – 12) w + 2 = 5 – 2w + 12 w + 2w = 5 + 12 – 2 3w = 15
w = 15–––3
= 5
Answer: D
6. 2x – 3 = 7 – 2(x + 1) 2x – 3 = 7 – (2x + 2) 2x – 3 = 7 – 2x – 2 2x + 2x = 7 – 2 + 3 4x = 8
x = 8––4
= 2
Answer: C
7. 3(2f – 5) – (3 – 2f) = 2 6f – 15 – 3 + 2f = 2 8f – 18 = 2 8f = 2 + 18
f = 20–––8
= 5––2
Answer: D
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8. 5(2q – 7) – (3 – 4q) = 4 10q – 35 – 3 + 4q = 4 14q – 38 = 4 14q = 4 + 38
q = 42–––14
= 3Answer: B
9. 1 – 2(p + 1) = –5(p – 1) 1 – (2p + 2) = (–5p + 5) 1 – 2p – 2 = –5p + 5 –2p + 5p = 5 – 1 + 2 3p = 6
p = 6––3
= 2Answer: C
10. 3h + 1 – 2h – 1 = 0 3h + 1 = 2h – 1 (3h + 1 )2 = (2h – 1 )2
9(h + 1) = 2h – 1 9h + 9 = 2h – 1 7h = –10
h = – 107
Answer: B
11. h + 1––––––2
= 3––5
5(h + 1) = 3 × 2 5h + 5 = 6 5h = 6 – 5
h = 1––5
Answer: C
12. 2h + 7––––––5 = h
2h + 7 = 5 × h 7 = 5h – 2h 7 = 3h
h = 7––3
Answer: A
13. 1 – k––2
= k – 3
1 + 3 = k + k––2
4 = 3k–––2
4 × 2 = 3k
k = 8––3
Answer: D
14. 5p2 – 1 + 2 = 2p
p + 1Multiply both sides by p2 – 1, 5 + 2(p2 – 1) = 2p(p – 1) 5 + 2p2 – 2 = 2p2 – 2p 2p = –3 p = – 3
2Answer: A
Paper 2
1. 2e – f = –8 ............................. 1 e + 3f = 10 ............................. 21 × 3: 6e – 3f = –24 ........................... 32 + 3: 7e = –14
e = –14––––7
= –2
Substituting e = –2 into 2, –2 + 3f = 10 3f = 10 + 2
f = 12–––3
= 4
2. a + 2b = 4 ........................ 1 3a – 4b = 7 ........................ 21 × 2: 2a + 4b = 8 ........................ 32 + 3: 5a = 15
a = 15–––5
= 3
Substituting a = 3 into 1, 3 + 2b = 4 2b = 4 – 3
b = 1––2
3. h + 1––2
k = 1 .......................... 1
2h – k = 10 ........................ 21 × 2: 2h + k = 2 .......................... 32 + 3: 4h = 12
h = 12–––4
= 3
Substituting h = 3 into 3, 2(3) + k = 2 6 + k = 2 k = 2 – 6 = – 4
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4. k – 2––3
w = 1 ......................... 1
3k + w = 9 ......................... 21 × 3: 3k – 2w = 3 ......................... 32 – 3: 3w = 6 w = 6––
3 = 2
Substituting w = 2 into 2, 3k + 2 = 9 3k = 9 – 2 3k = 7
k = 7––3
5. 2p + q = 3 ......................... 1 4p – 3q = 11 ....................... 21 × 3: 6p + 3q = 9 ......................... 32 + 3: 10p = 20
p = 20–––10
= 2
Substituting p = 2 into 1, 2(2) + q = 3 4 + q = 3 q = 3 – 4 = –1
6. 5p + 3n = 4 ........................ 1 2p – n = –5 ...................... 22 × 3: 6p – 3n = –15 .................... 31 + 3: 11p = –11
p = –11––––11
= –1
Substituting p = –1 into 1, 5(–1) + 3n = 4 –5 + 3n = 4 3n = 4 + 5
n = 9––3
= 3
7. f – 2g = 8 .......................... 1 3f + 4g = 4 .......................... 21 × 2: 2f – 4g = 16 ........................ 32 + 3: 5f = 20
f = 20–––5
= 4
Substituting f = 4 into 1,4 – 2g = 8 4 – 8 = 2g 2g = – 4
g = – 4–––2
= –2
8. e – 1––4
h = 6 ............................ 1
3e + h = 11 .......................... 21 × 4: 4e – h = 24 .......................... 32 + 3: 7e = 35
e = 35–––7
= 5
Substituting e = 5 into 2, 3(5) + h = 11 15 + h = 11 h = 11 – 15 = – 4
9. 2x – 3y = 12 ....................... 1 4x + y = 10 ....................... 22 × 3: 12x + 3y = 30 ....................... 31 + 3: 14x = 42
x = 42–––14
= 3
Substituting x = 3 into 2, 4(3) + y = 10 12 + y = 10 y = 10 – 12 = –2
10. e – 4w = 16 ....................... 1 3e + 2w = 6 ......................... 22 × 2: 6e + 4w = 12 ....................... 31 + 3: 7e = 28
e = 28–––7
= 4
Substituting e = 4 into 2, 3(4) + 2w = 6 12 + 2w = 6 2w = 6 – 12
w = – 6–––2
= –3
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11. 2h + 6k = 9 ........................... 1
1––3
h – 2k = 3 ........................... 2
2 × 3: h – 6k = 9 ........................... 31 + 3: 3h = 18
h = 18–––3
= 6
Substituting h = 6 into 1, 2(6) + 6k = 9 12 + 6k = 9 6k = 9 – 12
k = –3–––6
= – 1––2
12. h + 12k = 1 ...................... 1 4h – 3k = 21 .................... 22 × 4: 16h – 12k = 84 .................... 31 + 3: 17h = 85
h = 85–––17
= 5
Substituting h = 5 into 1, 5 + 12k = 1 12k = 1 – 5
k = – 4–––12
= – 1––3
13. 1––3
m + n = –4 ..................... 1
m – 2n = 23 ..................... 21 × 3: m + 3n = –12 ................... 32 – 3: –2n – 3n = 23 – (–12) –5n = 35
n = 35––––5
= –7
Substituting n = –7 into 2, m – 2(–7) = 23 m + 14 = 23 m = 23 – 14 = 9
14. 1––4
g + 3k = 10 .................. 1
g – 7k = –36 ................ 21 × 4: g + 12k = 40 .................. 33 – 2: 12k – (–7k) = 40 – (–36) 19k = 76
k = 76–––19
= 4
Substituting k = 4 into 2, g – 7(4) = –36 g – 28 = –36 g = –36 + 28 = –8
15. 7v – 2w = 24 ....................... 1 3v – 4w = 26 ....................... 21 × 2: 14v – 4w = 48 ....................... 33 – 2: 11v = 22
v = 22–––11
= 2
Substituting v = 2 into 1, 7(2) – 2w = 24 14 – 2w = 24 –2w = 24 – 14
w = 10––––2
= –5
16. 3m + 2n = 9 ......................... 1 2m – 3n = 19 ....................... 21 × 3: 9m + 6n = 27 ....................... 32 × 2: 4m – 6n = 38 ....................... 43 + 4: 13m = 65
m = 65–––13
= 5
Substituting m = 5 into 1, 3(5) + 2n = 9 15 + 2n = 9 2n = 9 – 15
n = –6–––2
= –3
17. 3t – 4w = 23 ...................... 1 2t + 3w = 4 ........................ 21 × 3: 9t – 12w = 69 ...................... 32 × 4: 8t + 12w = 16 ...................... 43 + 4: 17t = 85
t = 85–––17
= 5
Substituting t = 5 into 2, 2(5) + 3w = 4 10 + 3w = 4 3w = 4 – 10
w = –6–––3
= –2