Book 2B: Chapter 8 – Linear Equations in Two Unknowns
Revision
1.
Simultaneous linear equations in two unknowns
A pair of linear equations with two common unknowns is called a pair of
x + y = 5
3 x − 2 y = 5
simultaneous linear equations in two unknowns, e.g. 
, 
,
2 y − x = 4  x − y = 1
2 x + 3 y = 1
.

 x − 2 y = −10
2.
Solving simultaneous linear equations in two unknowns
Algebraic methods:
(a) Method of substitution
This method involves substituting one of the equations into the other
equation in order to eliminate one of the unknowns.
(b) Method of elimination
This method involves adding or subtracting two linear equations in order to
eliminate one of the two unknowns.
3.
Solutions of simultaneous linear equations in two unknowns
A simultaneous linear equations in two unknowns may have
(i) exactly one solution,
(ii) no solutions,
(iii) an infinite number of solutions.
Example
1. Solve the followiong simultaneous equations by the method of substitution.
 x + 3 y = 10......(1)

 y − 2 x = 1......(2)
Solution:
From (1)
x = 10 − 3 y......(3)
Sub. (3) into (2)
y − 2(10 − 3 y ) = 1
y − 20 + 6 y = 1
7 y = 21
y =3
Sub. y = 3 into (3)
x = 10 − 3(3) = 1
Therefore, the solution is x = 1 and y = 3 .
1
Book 2B: Chapter 8 – Linear Equations in Two Unknowns
2.
Solve the following simultaneous equations by the method of substitution.
5 x − 4 y + 2 = 2 y − 3x = 0
Solution:
The given equations can be written as
5 x − 4 y + 2 = 0......(1)

2 y − 3 x = 0......(2)
From (2)
y=
3x
......(3)
2
Sub. (3) into (1)
3x
)+2 =0
2
5x − 6 x + 2 = 0
5 x − 4(
−x + 2 = 0
x=2
Sub. x = 2 into (3)
3(2)
=3
2
Therefore, the solution is x = 2 and y = 3 .
y=
3.
Solve the following simultaneous equations by the method of elimination.
5 x + 3 y = 1......(1)

5 x − 3 y = −11......(2)
Solution:
(1) + (2)
10 x = −10
x = −1
Sub. x = −1 into (1)
5(−1) + 3 y = 1
−5 + 3 y = 1
3y = 6
y=2
Therefore, the solution is x = −1 and y = 2 .
2
Book 2B: Chapter 8 – Linear Equations in Two Unknowns
4.
Solve the following simultaneous equations by the method of elimination.
5 x + 2 y = 12......(1)

2 x − y = 3......(2)
Solution:
(2) × 2 : 4 x − 2 y = 6......(3)
(1) + (3)
9 x = 18
x=2
Sub. x = 2 into (1)
5(2) + 2 y = 12
10 + 2 y = 12
2y = 2
y =1
Therefore, x = 2 and y = 1
5.
Solve the following simultaneous equations by the method of elimination.
2 x + 21 y = 15......(1)

3 x + 14 y = 5......(2)
Solution:
(1) × 3 : 6 x + 63 y = 45......(3)
(2) × 2 :
6 x + 28 y = 10......(4)
(3) − (4)
35 y = 35
y =1
Sub. y = 1 into (1)
2 x + 21(1) = 15
2 x = −6
x = −3
Therefore, x = −3 and y = 1
3
Book 2B: Chapter 8 – Linear Equations in Two Unknowns
Exercise
1
Solve the simultaneous equations by using the method of substitution:
2.
Solve the simultaneous equations by using the method of elimination:
3.
The weights of two goods are a kg and b kg respectively. The difference between
a and b is 2.6. a is 3 times as b. Find the weights of the two goods.
4.
Solve the simultaneous equations:
5.
Solve the simultaneous equations:
6.
The price of 5 cans of coke and 7 bottles of orange juice is $89, while the price
of 6 cans of coke and 8 bottles of orange juice is $103. Find the price of one can
of coke and the price of one bottle of orange juice.
7.
Solve the simultaneous equations: 10–x–2y = 2x + y + 7 = 5
8.
Solve the simultaneous equations:
9x–4y + 1 = 2y + 3x + 2 = 0
4
Book 2B: Chapter 8 – Linear Equations in Two Unknowns
9.
(a) Solve the simultaneous equations:
(b) Using the result in (a), solve the simultaneous equations:
10. Solve the simultaneous equations:
5 + 4p–2q = 9p + q + 3 = 22
11. A test has 10 questions. 20 points are given for each correct answer and 10 points
are deducted for each incorrect answer. If Sam’s score is 110, how many
questions does he answer correctly?
12. A bag contains balls of two colours, black and white. The number of the black
balls is 14 more than that of the white balls. If
1
1
of the black balls and
of
6
2
the white balls are taken away, 57 balls are left in the bag. Find the original
numbers of the black balls and the white balls.
2 x = 3 y
.
13. Solve 
4 x + 5 y − 11 = 0
3 x − 4 y + 1 = 0
14. Solve 
.
5 x + y − 6 = 0
5
Book 2B: Chapter 8 – Linear Equations in Two Unknowns
3( x + y ) = 2 y + 1
15. Solve 
.
3(2 x + 3 y ) = −33
 2( x + y ) = y − 1
16. Solve 
.
7(2 x + 3 y ) = −77
 x + y = 10

17. Solve  3 x y .
 − =1
2 4
0.5 x + 0.25 y = 1.5
18. Solve 
.
1.5 x − 2.5 y = 4.5
25 y + 8 x = 40
19. Solve 
.
0.5 y + 0.4 x = 1.4
6 x + 5 y = 1

20. Solve 
4
8.
2 x − 3 y = − 3
x−4

6
y
−
= 10

3
21. Solve 
.
3
y
−
7
x +
=1

4
y−6

4 x − 3 = 18
22. Solve 
.
 1 y + 4 x − 12 = 8
2
 3
23. Solve 2(b – a) + (3a – 2b) = 4 – 3a = b – 2.
4 x + 3 y = 17
.
24. (a) Solve 
 x − y = −1
4 3
 x + y = 17

(b) Hence, solve 
.
 1 − 1 = −1
 x y
6
Book 2B: Chapter 8 – Linear Equations in Two Unknowns
x + 2 y = 2
25. (a) Solve 
.
3 x − 4 y = 1
1
 m +
(b) Hence, solve 
3 −
 m
2
=2
n
.
4
=1
n
x + 2 y = 3
26. (a) Using the method of substitution, solve 
.
6 x − 13 y = −32
(m + 1) + 2(4 − n) = 3
(b) Hence, solve 
.
6(m + 1) − 13(4 − n) = −32
27. If (x, y), where x = y, is the solution of the simultaneous equations
4 x + 3 y = 1
, find the value of k.

kx + (k − 1) y = 3
28. If the ordered pair (2, k) satisfies both equations 8x + 2y + b = 0 and
x – y + 2b – 1 = 0, find the values of b and k.
29. For a two-digit number, the sum of its digits is 11. If the digits are interchanged,
the number formed is smaller than the original one by 45. Find the original
number.
30. When both the numerator and the denominator of a fraction are increased by 1,
the fraction becomes
becomes
3
; when both of them are decreased by 5, the fraction
4
1
. Find the original fraction.
2
31. For a three-digit number, the sum of its hundreds digit and its units digit is 11. If
the hundreds digit is interchanged with the units digit, the number formed is
larger than the original one by 693. Given that its tens digit is 7, find the original
number.
7
Book 2B: Chapter 8 – Linear Equations in Two Unknowns
32. It is given that A and B are two alloys of silver and copper only, where alloy A
has a content of 25% silver and alloy B has a content of 37.5% silver. If a
craftsman has to prepare 100 kg of a new alloy with a content of 30% silver and
70% copper, how much alloy A and alloy B does he need?
33. Drink A contains 15% alcohol and drink B contains 5% alcohol. Steve wants to
blend a drink of 250 mL with 8% alcohol content by mixing drinks A and B, how
much drink A and drink B does he need?
34. There are 20 questions in a test. For each question answered correctly, m marks
will be given. For each unanswered question or wrongly answered question, n
marks will be deducted. If 10 questions are answered correctly, a student will get
20 marks; if 15 questions are answered correctly, a student will get 60 marks.
Find the values of m and n.
35. Car A and car B are 10 km apart. If they move towards each other, they will meet
in 8 minutes. If they move in the same direction, car A will catch up with car B in
40 minutes. Find the speeds of car A and car B. (Assume the speeds of both cars
do not change.)
36. After driving x hours at a speed of 40 km/h and y hours at a speed of 60 km/h,
Peter covers a total distance of 1800 km. If he drives back along the same road to
the starting point at an average speed of 50 km/h, the time required will be 2
hours shorter than the time taken in his forward journey. Find the values of x and
y.
8