Linear Equations in Two Unknowns

1
Linear Equations in Two Unknowns
7
Case Study
7.1 Linear Equations in Two Unknown and
Their Graphs
7.2 Solving Simultaneous Linear Equations in
Two Unknowns by the Graphical Method
7.3 Method of Substitution
7.4 Method of Elimination
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Chapter Summary
2
Case Study
Can you find the actual number of chicks and
rabbits there?
Suppose there are x chicks and y rabbits.
According to the above information, we can set up
2 equations as follows.
? x ? y ? 18 ............... (1) ? ? 2x ? 4y ?
46 ........... (2)
From (1), we have x ? 18 ? y.
......... (3)
Substituting (3) into (2), we have 2(18 ? y) ?
4y ? 46
Substituting y ? 5 into (3), we have x ? 18 ?
5 ? 13
36 ? 2y ? 4y ? 46
2y ? 10
y ? 5
? There are 13 chicks and 5 rabbits.
3
7.1 Linear Equations in Two Unknowns and
Their Graphs
In Book 1A, we have learnt about linear equations
in one unknown.
In this chapter, we are going to learn about
linear equations in two unknowns.
Examples of linear equations in two unknowns
? x ? y ? 18 ? 3x ? 5y ? ?7 ? 2y ? 4x ? ? 4
Each of the above equations consists of 2
unknowns with degree 1.
If one of the 2 variables is known, we can find
the other by the method of substitution.
4
7.1 Linear Equations in Two Unknowns and
Their Graphs
A. Solutions of Linear Equations in Two Unknowns
Consider the equation x ? y ? 10.
There are many pairs of values of x and y that
satisfy this equation
? x ? 0 and y ? 10
? x ? 1 and y ? 9
? x ? 2 and y ? 8
? x ? 3 and y ? 7 ?
Each of the above pairs of x and y is a solution
of the equation.
Besides the above 4 pairs, there are infinitely
many solutions for the linear equation in two
unknowns.
We can write the solutions in ordered pairs as
(0, 10), (1, 9), (2, 8), (3, 7), etc.,
or by using a table
x 0 1 2 3
y 10 9 8 7
5
7.1 Linear Equations in Two Unknowns and
Their Graphs
A. Solutions of Linear Equations in Two Unknowns
Example 7.1T
Determine whether the following pairs are
solutions of the equation 2x ? y ? 4. (a) (4,
0) (b) (3, ?2)
Solution
(a) Substitute (4, 0) into the equation.
L.H.S. ? 2(4) ? 0
? 8
? R.H.S.
? (4, 0) is not a solution of the equation.
(b) Substitute (3, ?2) into the equation.
L.H.S. ? 2(3) ? (?2)
? 4
? R.H.S.
? (3, ?2) is a solution of the equation.
6
7.1 Linear Equations in Two Unknowns and
Their Graphs
B. Graphs of Linear Equations in Two Unknowns
Consider the equation y ? x ? 2.
We write the solutions using a table
x ?2 ?1 0 1
y 0 1 2 3
For each pair of x and y, plot them onto the
rectangular coordinate plane and join them.
Consider some points from the line obtained such
as (?3, ?1).
Check if this point satisfies the equation y ? x
? 2
L.H.S. ? ?1
R.H.S. ? ?3 ? 2
? ?1
? L.H.S. ? R.H.S.
? (?3, ?1) satisfies the equation y ? x ? 2.
7
7.1 Linear Equations in Two Unknowns and
Their Graphs
B. Graphs of Linear Equations in Two Unknowns
In general, every point on the line satisfies the
equation y ? x ? 2.
(1.8, 3.8)
(0.7, 2.7)
The straight line obtained is called the graph of
the equation y ? x ? 2.
(?0.4, 1.4)
(?1.5, 0.5)
(?3.5, ?1.5)
Any point on the graph of a linear equation in
two unknowns is a solution of the equation. On
the other hand, if a point satisfies a linear
equation in two unknowns, the it lies on the
graph of the equation.
8
7.1 Linear Equations in Two Unknowns and
Their Graphs
B. Graphs of Linear Equations in Two Unknowns
Example 7.2T
The figure shows the graph of 2x ? 4y ? 1. Find
the coordinates of C.
Solution
The y-coordinate of C is ?1.
Substitute y ? ?1 into the equation 2x ? 4y ? 1,
we have
2x ? 4(?1) ? 1
2x ? ?3
x ?
? The coordinates of C are .
9
7.1 Linear Equations in Two Unknowns and
Their Graphs
B. Graphs of Linear Equations in Two Unknowns
Example 7.3T
(a) Draw the graph of the equation 2y ? x ? 2.
(Take x from ?2 to 2.) (b) Using the graph in
(a), answer the following questions. (i) If
P(?3, y) lies on the graph of 2y ? x ? 2, find
the value of y. (ii) Is (3, 2.5) a solution of
the equation 2y ? x ? 2?
Solution
2y ? x ? 2
(a)
x ?2 0 2
y 0 1 2
(3, 2.5)
(b) (i)
? P(?3, ?0.5) lies on the graph of 2y ? x ? 2.
? y ? ?0.5
(?3, ?0.5)
(ii)
? (3, 2.5) lies on the graph of 2y ? x ? 2.
? (3, 2.5) is a solution of the equation 2y ? x
? 2.
10
7.1 Linear Equations in Two Unknowns and
Their Graphs
B. Graphs of Linear Equations in Two Unknowns
The equation of a straight line can be expressed
in different forms.
For example, y ? 4x ? 5 can be expressed as y ?
4x ? 5 or 4x ? y ? 5 ? 0.
However their graphs are the same.
Consider the graphs of equations in the form y ?
ax ? b, where a and b are constants.
(a) If a is fixed, consider (i) y ? x ?
3, (ii) y ? x and (iii) y ? x ? 3.
(b) If b is fixed, consider (i) y ? ?2x ?
1, (ii) y ? x ? 1 and (iii) y ? 3x ? 1.
The graphs are parallel lines.
The graphs pass through the same point on the
y-axis.
11
7.2 Solving Simultaneous Linear Equations in
Two Unknowns by the Graphical Method
The figure shows 2 linear equations y ? x ? 1 and
x ? y ? 3.
The coordinates of A, C and other points on the
graph of y ? x ? 1 satisfy the equation y ? x ? 1.
The coordinates of B, C and other points on the
graph of x ? y ? 3 satisfy the equation x ? y ? 3.
We observe that C(1, 2) satisfies both equations
y ? x ? 1 and x ? y ? 3 simultaneously.
In fact, C(1, 2) is the only point (point of
intersection) which lies on both graphs.
So, (1, 2) is the common solution of the 2
equations.
12
7.2 Solving Simultaneous Linear Equations in
Two Unknowns by the Graphical Method
We usually represent the pairs of equations
as ? y ? x ? 1 ? , ? x ? y ?
3 which is called a pair of simultaneous linear
equations in two unknowns.
Solving a pair of simultaneous linear equations
by finding the point of intersection of their
graphs is called the graphical method.
13
7.2 Solving Simultaneous Linear Equations in
Two Unknowns by the Graphical Method
Example 7.4T
The figure shows the graph of the equation x ? y
? 2. (a) Complete the following table for the
equation y ? 3x ? 4. (b) Draw the graph of y ?
3x ? 4 in the same coordinate plane. (c) Hence
solve graphically.
x 1 2 2.5
y
y ? 3x ? 4
Solution
(a)
x 1 2 2.5
y ?1 2 3.5
(1.5, 0.5)
(b) Refer to the graph.
(c) From to the graph, the solution is x ? 1.5, y
? 0.5.
14
7.2 Solving Simultaneous Linear Equations in
Two Unknowns by the Graphical Method
Example 7.5T
Solve the simultaneous equations
graphically.
Solution
y ? x ? 5 ? 0
x ?5 ?3 ?1
y
x ?5 ?3 ?1
y 0 2 4
(?4, ?1)
4y ? 3x ? 8
x 4 2 0
y
x 4 2 0
y 1 0.5 2
From to the graph, the solution is x ? ?4, y ?
?1.
15
7.2 Solving Simultaneous Linear Equations in
Two Unknowns by the Graphical Method
The figure shows the graphs of the linear
equations 7y ? 9x ? 16 and 6y ? 5x ? 1 ? 0.
From the graph, x ? 1.2 and y ? 0.8.
We do not know the exact solution by reading from
the graphs.
(1.2, 0.8)
We observe that the drawing of straight lines or
the point of intersection of the graphs may not
be accurate.
The reading of the solution may depend on the
scale of the grid lines.
Hence, the solutions obtained by the graphical
method are approximations only.
16
7.3 Method of Substitution
In the last section, we have learnt how to solve
simultaneous linear equations in two unknowns by
the graphical method.
However, as we know that there is a limitation to
use this method, we will explore other methods
(algebraic methods).
There are 2 algebraic methods
? Method of Substitution
? Method of Elimination
In this section, we are going to study the method
of substitution.
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7.3 Method of Substitution
Consider the following simultaneous
equations. ? x ? 3y ? 1 ..................
(1) ? ? 5x ? 6y ? ?1
.............. (2)
The following shows the steps to apply the method
of substitution.
Step 1 Make one of the unknowns, x or y, the
subject of the equation.
Step 2 Substitute the result (3) into equation
(2) to get a linear equation in one unknown. Then
solve this equation.
Step 3 Substitute the result (the value of y)
obtained into one of the above equation to find
the remaining unknown.
From (1), we have x ? 1 ? 3y ............ (3)
2 ? x ? 1 ? 3 ?
3
5(1 ? 3y) ? 6y ? ?1
5 ? 15y ? 6y ? ?1
? ?1
?9y ? ?6
2 y ? 3
2 ? The solution is x ? ?1, y ? . 3
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7.3 Method of Substitution
Example 7.6T
Solve the following simultaneous equations. ? y
? x ? 1 .................. (1) ?
? 4x ? y ? ?4 .............. (2)
Solution
Substitute (1) into (2)
4x ? (x ? 1) ? ?4
4x ? x ? 1 ? ?4
3x ? ?3
x ? ?1
Substitute x ? ?1 into (1)
y ? ?1 ? 1
? 0
? The solution is x ? ?1, y ? 0.
19
7.3 Method of Substitution
Example 7.7T
Solve the simultaneous equations 2x ? y ? 1 ? x ?
y ? 5.
Solution
? 2x ? y ? 1 ? 5 ......... (1) ?
? x ? y ? 5 ................. (2)
Rewriting the given equations, we have
From (2), we have x ? y ? 5 ............ (3)
Substitute (3) into (1)
2(y ? 5) ? y ? 1 ? 5
2y ? 10 ? y ? 1 ? 5
3y ? ?6
y ? ?2
Substitute y ? ?2 into (3)
x ? ?2 ? 5
? 3
? The solution is x ? 3, y ? ?2.
20
7.3 Method of Substitution
Example 7.8T
Solve the following simultaneous equations. ?
4x ? 3y ? 10 ............ (1) ?
? 6y ? 2x ? ?8 ............ (2)
Solution
From (2), we have 3y ? x ? ?4 x ? 3y ? 4
............ (3)
Substitute y ? into (3)
Substitute (3) into (1)
4(3y ? 4) ? 3y ? 10
12y ? 16 ? 3y ? 10
9y ? ?6
? 2
? The solution is x ? 2, y ? .
21
7.4 Method of Elimination
Besides the method of substitution, we can use
the method of elimination to solve simultaneous
linear equations in two unknowns.
The key idea of this method is to eliminate one
of the unknowns by adding or subtracting the
simultaneous equations.
Consider the following simultaneous equations. ?
x ? y ? 3 .............. (1) ?
? x ? y ? 1 .............. (2)
(1) ? (2)
x ? y ? 3 ?) x ? y ? 1
OR
(x ? y) ? (x ? y) ? 3 ? 1
x ? y ? x ? y ? 4
2x ? 4
2x ? 4
x ? 2
x ? 2
Substitute x ? 2 into (1), we have y ? 1.
? The solution is x ? 2, y ? 1.
22
7.4 Method of Elimination
Example 7.9T
Solve the following simultaneous equations by the
method of elimination. ? 4x ? y ? ?14
............ (1) ? ? 4x ?
5y ? ?2 ............ (2)
Solution
(1) ? (2)
(4x ? y) ? (4x ? 5y) ? ?14 ? (?2)
4x ? y ? ?14 ?) 4x ? 5y ? ?2
OR
4x ? y ? 4x ? 5y ? ?14 ? 2
6y ? ?12
6y ? ?12
y ? ?2
y ? ?2
Substitute y ? ?2 into (1)
4x ? (?2) ? ?14
4x ? ?12
x ? ?3
? The solution is x ? ?3, y ? ?2.
23
7.4 Method of Elimination
If the coefficients of x (or y) terms in a pair
of simultaneous equations are different, we
should multiply the equations by some constants
so that we can eliminate those terms by addition
or subtraction.
For example, consider the following simultaneous
equations ? x ? 2y ? 6 .............. (1) ?
? 3x ? y ? 11 ............
(2)
We can multiply equation (1) by 3 so that the
coefficients of x of the pair of simultaneous
equations are the same
(1) ? 3, 3(x ? 2y) ? 6 ? 3
3x ? 6y ? 18 ......... (3)
Hence the coefficients of x in equations (2) and
(3) are the same.
24
7.4 Method of Elimination
Example 7.10T
Solve the following simultaneous equations by the
method of elimination. ? 3y ? 2x ? 47
............ (1) ? ? 4y ?
x ? 48 .............. (2)
Solution
(2) ? 2 8y ? 2x ? 96 ...... (3)
(1) ? (3)
(3y ? 2x) ? (8y ? 2x) ? 47 ? 96
3y ? 2x ? 8y ? 2x ? 143
11y ? 143
y ? 13
Substitute y ? 13 into (2)
4(13) ? x ? 48
x ? 4
? The solution is x ? 4, y ? 13.
25
7.4 Method of Elimination
Example 7.11T
Solve the following simultaneous equations by the
method of elimination. ? 3x ? 2y ? 2
.............. (1) ? ? 4x
? 5y ? ?28 .......... (2)
Solution
(1) ? 4 12x ? 8y ? 8 ......... (3)
(2) ? 3 12x ? 15y ? ?84 ..... (4)
(3) ? (4)
(12x ? 8y) ? (12x ? 15y) ? 8 ? (?84)
23y ? 92
y ? 4
Substitute y ? 4 into (1)
3x ? 2(4) ? 2
x ? ?2
? The solution is x ? ?2, y ? 4.
26
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
In our daily lives, we often come across problems
that can be formulated as simultaneous equations.
We can solve these problems by applying the
technique of solving simultaneous equations.
Step 1 Identify the 2 unknowns and represent
them with letters such as x and y.
Step 2 Set up a pair of simultaneous linear
equations in 2 unknowns.
Step 3 Solve the simultaneous equations.
27
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Example 7.12T
The difference between 2 numbers is 8. If the
smaller number is doubled, the result is 2 more
than the larger number. Find the two numbers.
Solution
Let x and y be the smaller and the larger numbers
respectively.
From the problem, we have ? y ? x ? 8
............ (1) ? ? 2x ?
y ? 2 .......... (2)
(1) ? (2)
(y ? x) ? (2x ? y) ? 8 ? 2
x ? 10
Substitute x ? 10 into (1)
y ? 10 ? 8
y ? 18
? The two numbers are 10 and 18.
28
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Example 7.13T
Lily has some 3 and 5 stamps. If the total
number of stamps is 20 and their total value is
70, find the number of 3 and 5 stamps that she
has.
Solution
Let x and y be the numbers of 3 and 5 stamps
respectively.
? x ? y ? 20 .......... (1) ?
? 3x ? 5y ? 70 ...... (2)
From the problem, we have
(1) ? 3 3x ? 3y ? 60 ..... (3)
(2) ? (3)
(3x ? 5y) ? (3x ? 3y) ? 70 ? 60
2y ? 10
y ? 5
Substitute y ? 5 into (1), we have x ? 5 ? 20
x ? 15
? Lily has 15 3 stamps and 5 5 stamps.
29
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Example 7.14T
A company produces washing powder in 2 kinds of
packages X and Y. The weight of one box of X is
50 g less than 2 boxes of Y. If the total weight
of 2 boxes of X and 1 box of Y is 2 kg, find the
weight of each package of washing powder.
Solution
Let x g and y g be the weights of each box of X
and Y respectively.
? 2y ? x ? 50 .......... (1) ?
? 2x ? y ? 2000 ...... (2)
From the problem, we have
(1) ? 2 4y ? 2x ? 100 ..... (3)
Substitute y ? 420 into (2) 2x ? 420 ? 2000
(2) ? (3)
(2x ? y) ? (4y ? 2x) ? 2000 ? 100
5y ? 2100
2x ? 1580
y ? 420
x ? 790
? The weights of each box of X and Y are 790 g
and 420 g respectively.
30
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Example 7.15T
Dickson is 3 years older than his sister Mary. 2
years ago, Dickson was twice as old as Mary. How
old is Dickson now?
Solution
Let x and y be the present ages of Dickson and
Mary respectively.
? x ? y ? 3 ................. (1) ?
? x ? 2 ? 2(y ? 2) ...... (2)
From the problem, we have
From (2), we have x ? 2 ? 2y ? 4 x ? 2y ? 2
........................... (3)
Substitute (3) into (1)
(2y ? 2) ? y ? 3
y ? 5
Substitute y ? 5 into (1), we have x ? 5 ? 3
x ? 8
? Dickson is 8 years old now.
31
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Example 7.16T
Fanny finished a test with 2 parts A and B. She
used of the time to finish part A. If she
used 20 minutes more to finish part A than part
B, (a) find the time she used to finish parts A
and B respectively, (b) find the time she used
to finish the whole test.
Solution
(a) Let x minutes and y minutes be the time she
used to finish parts A and B respectively.
? x ? y ? 20 .......... (1) ?
? x ? (x ? y) ....... (2)
From the problem, we have
From (1), we have x ? y ? 20
.................. (3)
Substitute (3) into (4)
From (2), we have 5x ? 3(x ? y) 2x ? 3y
...................... (4)
2(y ? 20) ? 3y y ? 40
Substitute y ? 40 into (3), x ? 60
? Part A 60 minutes Part B 40 minutes
32
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Example 7.16T
Fanny finished a test with 2 parts A and B. She
used of the time to finish part A. If she
used 20 minutes more to finish part A than part
B, (a) find the time she used to finish parts A
and B respectively, (b) find the time she used
to finish the whole test.
Solution
(b) Part A 60 minutes Part B 40 minutes
The time she used to finish the whole test ?
(60 ? 40) minutes
? 100 minutes
33
Chapter Summary
7.1 Linear Equations in Two Unknowns and Their
Graphs
1. An equation consisting of 2 unknowns with
degree 1 is called a linear equation in two
unknowns.
2. Any linear equation in two unknowns has
infinitely many solutions which can be
represented by ordered pairs.
3. The graph of a linear equation in two unknowns
is a straight line.
4. Any point on the graph of a linear equation in
two unknowns is a solution of the equation.
34
7.2 Solving Simultaneous Linear Equations in
Two Unknowns by the Graphical Method
Chapter Summary
1. The common solution of a pair of simultaneous
equations satisfies both equations.
2. Solving simultaneous equations by the
graphical method involves finding the coordinates
of the point of intersection of the graphs of the
2 equations.
3. Solutions obtained from solving simultaneous
equations by the graphical method are
approximations only.
35
7.3 Method of Substitution
Chapter Summary
The steps involved in solving simultaneous
equations by the method of substitution are as
follows.
1. Make one of the unknowns the subject of the
equation.
2. Substitute the result obtained into the other
equation to get a linear equation in one unknown.
Then solve this equation.
3. Substitute the result obtained into one of the
given equations to find the remaining unknown.
36
7.4 Method of Elimination
Chapter Summary
1. Solving simultaneous equations by the method
of elimination involves eliminating one of the
unknowns by adding or subtracting the
simultaneous equations.
2. If the coefficients of the x (or y) terms in a
pair of simultaneous equations are different, we
should multiply one or both of the equations by
some constants so that we can eliminate those
terms by addition or subtraction.
37
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Chapter Summary
We can solve some daily-life problems by
formulating simultaneous equations and solving
them through the following steps.
1. Identify the 2 unknowns and represent them
with letters.
2. Set up a pair of simultaneous linear equations
in 2 unknowns.
3. Solve the simultaneous equations.
38
7.1 Linear Equations in Two Unknowns and
Their Graphs
A. Solutions of Linear Equations in Two Unknowns
Follow-up 7.1
Determine whether the following pairs are
solutions of the equation x ? 3y ? 1. (a) (4,
1) (b) (1, 4)
Solution
(a) Substitute (4, 1) into the equation.
L.H.S. ? 4 ? 3(1)
? 1
? R.H.S.
? (4, 1) is a solution of the equation.
(b) Substitute (1, 4) into the equation.
L.H.S. ? 1 ? 3(4)
? ?11
? R.H.S.
? (1, 4) is not a solution of the equation.
39
7.1 Linear Equations in Two Unknowns and
Their Graphs
B. Graphs of Linear Equations in Two Unknowns
Follow-up 7.2
The figure shows the graph of 9x ? 4y ? 36. Find
the coordinates of B.
Solution
The x-coordinate of B is 2.
Substitute x ? 2 into the equation 9x ? 4y ? 36,
we have
9(2) ? 4y ? 36
4y ? 18
y ? 4.5
? The coordinates of B are (2, 4.5).
40
7.1 Linear Equations in Two Unknowns and
Their Graphs
B. Graphs of Linear Equations in Two Unknowns
Follow-up 7.3
(a) Draw the graph of the equation y ? 1 ? x.
(Take x from ?2 to 2.) (b) (i) If P(x, ?0.5) lies
on the graph of y ? 1 ? x, find the value of
x. (i) If Q(?1.5, y) lies on the graph of y ? 1
? x, find the value of y. (ii) Is (2.5, ?1) a
solution of the equation y ? 1 ? x?
Solution
(a)
x ?2 0 2
y 3 1 ?1
(?1.5, 2.5)
(b) (i)
? P(1.5, ?0.5) lies on the graph of y ? 1 ? x.
? x ? 1.5
(1.5, ?0.5)
(2.5, ?1)
(ii)
From the graph, y ? 2.5.
y ? 1 ? x
(iii)
(2.5, ?1) is not a solution.
41
7.2 Solving Simultaneous Linear Equations in
Two Unknowns by the Graphical Method
Follow-up 7.4
The figure shows the graph of the equation 3x ?
2y ? ?12. (a) Complete the following table for
the equation 2y ? x ? 8. (b) Draw the graph of
2y ? x ? 8 in the same coordinate
plane. (c) Solve the simultaneous equations
graphically.
x ?5 ?4 ?1
y
(?2, 3)
2y ? x ? 8
Solution
(a)
x ?5 ?4 ?1
y 1.5 2 3.5
(b) Refer to the graph.
(c) From to the graph, the solution is x ? ?2, y
? 3.
42
7.2 Solving Simultaneous Linear Equations in
Two Unknowns by the Graphical Method
Follow-up 7.5
Solve the simultaneous equations
graphically.
Solution
y ? 2x ? 4
x ?6 ?4 ?2
y
x ?6 ?4 ?2
y 8 4 0
y ? x ? 1
(?5, ?6)
x ?6 ?4 ?2
y
x ?6 ?4 ?2
y 7 5 3
From to the graph, the solution is x ? ?5, y ?
?6.
43
7.3 Method of Substitution
Follow-up 7.6
Solve the following simultaneous equations. ? y
? 5 ? x .................. (1) ?
? 2y ? 3x ? 0 .............. (2)
Solution
Substitute (1) into (2)
2(5 ? x) ? 3x ? 0
10 ? 2x ? 3x ? 0
5x ? 10
x ? 2
Substitute x ? 2 into (1)
y ? 5 ? 2
? 3
? The solution is x ? 2, y ? 3.
44
7.3 Method of Substitution
Follow-up 7.7
Solve the simultaneous equations y ? 2x ? 2y ? 7x
? 3.
Solution
? y ? 2x ? 3 ............ (1) ?
? 2y ? 7x ? 3 ........... (2)
Rewriting the given equations, we have
From (1), we have y ? 2x ? 3 ............ (3)
Substitute (3) into (2)
2(2x ? 3) ? 7x ? 3
4x ? 6 ? 7x ? 3
?3x ? ?3
x ? 1
Substitute x ? 1 into (3)
y ? 2(1) ? 3
? 5
? The solution is x ? 1, y ? 5.
45
7.3 Method of Substitution
Follow-up 7.8
Solve the following simultaneous equations. ?
6x ? 3y ? 1 .............. (1) ?
? 5x ? 6y ? 16 ............ (2)
Solution
From (1), we have 3y ? 1 ? 6x y ?
............... (3)
Substitute x ? ?2 into (3)
Substitute (3) into (2)
5x ? 6 ? 16
y
5x ? 2 ? 12x ? 16
?7x ? 14
x ? ?2
? The solution is x ? ?2, y ? .
46
7.3 Method of Substitution
Follow-up 7.8
Solve the following simultaneous equations. ?
6x ? 3y ? 1 .............. (1) ?
? 5x ? 6y ? 16 ............ (2)
Alternative Solution
From (1), we have 3y ? 1 ? 6x ............... (3)
From (2), we have 5x ? 2 ? 3y ? 16 ....... (4)
Substitute x ? ?2 into (3)
Substitute (3) into (4)
5x ? 2(1 ? 6x) ? 16
3y ? 1 ? 6(?2)
5x ? 2 ? 12x ? 16
? 13
?7x ? 14
x ? ?2
? The solution is x ? ?2, y ? .
47
7.4 Method of Elimination
Follow-up 7.9
Solve the following simultaneous equations by the
method of elimination. ? 2x ? 3y ? 5
.............. (1) ? ? 2x
? y ? 9 ................ (2)
Solution
(2) ? (1)
(2x ? y) ? (2x ? 3y) ? 9 ? 5
2x ? y ? 9 ?) 2x ? 3y ? 5
OR
2x ? y ? 2x ? 3y ? 4
4y ? 4
4y ? 4
y ? 1
y ? 1
Substitute y ? 1 into (1)
2x ? 3(1) ? 5
2x ? 8
x ? 4
? The solution is x ? 4, y ? 1.
48
7.4 Method of Elimination
Follow-up 7.10
Solve the following simultaneous equations by the
method of elimination. ? 4x ? 3y ? 7
.............. (1) ? ? 2x
? 5y ? ?3 ............ (2)
Solution
(2) ? 2 4x ? 10y ? ?6 ..... (3)
(1) ? (3)
(4x ? 3y) ? (4x ? 10y) ? 7 ? (?6)
4x ? 3y ? 4x ? 10y ? 7 ? 6
13y ? 13
y ? 1
Substitute y ? 1 into (1)
4x ? 3(1) ? 7
x ? 1
? The solution is x ? 1, y ? 1.
49
7.4 Method of Elimination
Follow-up 7.11
Solve the following simultaneous equations by the
method of elimination. ? 5x ? 4y ? 2
.............. (1) ? ? 8x
? 3y ? 27 ............ (2)
Solution
(1) ? 3 15x ? 12y ? 6 ......... (3)
(2) ? 4 32x ? 12y ? 108 ..... (4)
(4) ? (3)
(32x ? 12y) ? (15x ? 12y) ? 108 ? 6
17x ? 102
x ? 6
Substitute x ? 6 into (1)
5(6) ? 4y ? 2
y ? 7
? The solution is x ? 6, y ? 7.
50
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Follow-up 7.12
The sum of 2 numbers is 41 and the difference
between the 2 numbers is 5. Find the 2 numbers.
Solution
Let x and y be the 2 numbers respectively.
? x ? y ? 41 ............ (1) ?
? x ? y ? 5 .............. (2)
From the problem, we have
(1) ? (2)
(x ? y) ? (x ? y) ? 41 ? 5
2x ? 46
x ? 23
Substitute x ? 23 into (1)
23 ? y ? 41
y ? 18
? The 2 numbers are 18 and 23.
51
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Follow-up 7.13
John has some 10 and 20 notes. If the total
number of notes is 18 and their total value is
300, find the number of 10 and 20 notes that
he has.
Solution
Let x and y be the numbers of 10 and 20 notes
respectively.
? x ? y ? 18 ................ (1) ?
? 10x ? 20y ? 300 ...... (2)
From the problem, we have
(1) ? 10 10x ? 10y ? 180 ..... (3)
(2) ? (3)
(10x ? 20y) ? (10x ? 10y) ? 300 ? 180
10y ? 120
y ? 12
Substitute y ? 12 into (1), we have x ? 12 ? 18
x ? 6
? John has 6 10 notes and 12 20 notes.
52
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Follow-up 7.14
Mrs. Lee bought some bottles of drink X and Y.
The total volume of 2 bottles of X and 1 bottles
of Y is 4 L. If the volume of 2 bottles of X is 1
L more than 2 bottles of Y, find the volume of
each bottle of X and Y respectively.
Solution
Let x L and y L be the volumes of each bottle of
X and Y respectively.
? 2x ? y ? 4 .......... (1) ?
? 2x ? 2y ? 1 ........ (2)
From the problem, we have
(1) ? (2)
(2x ? y) ? (2x ? 2y) ? 4 ? 1
Substitute y ? 1 into (1) 2x ? 1 ? 4
3y ? 3
x ? 1.5
y ? 1
? The volumes of each bottle of X and Y are 1.5 L
and 1 L respectively.
53
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Follow-up 7.15
Mrs. Wan is 30 years older than her daughter. 4
years ago, the total age of Mrs. Wan and her
daughter was 34. How old is her daughter now?
Solution
Let x and y be the ages of Mrs. Wan and her
daughter now respectively.
? x ? y ? 30 ........................ (1) ?
? (x ? 4) ? (y ? 4) ? 34
...... (2)
From the problem, we have
From (2), we have x ? y ? 42 .....................
..................... (3)
(1) ? (3)
(x ? y) ? (x ? y) ? 30 ? 42
2x ? 72
x ? 36
Substitute x ? 36 into (1), we have 36 ? y ? 30
y ? 6
? Her daughter is 6 years old now.
54
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Follow-up 7.16
A box of sweets is divided between Jerry and
Gloria in the ratio 5 2. If Jerry gets 60
sweets more than Gloria, find (a) the number of
sweets each of them gets, (b) the total number
of sweets in the box.
Solution
(a) Let x and y be the numbers of sweets that
Jerry and Gloria get respectively.
? x ? y ? 60 .......... (1) ?
? ? .................. (2)
From the problem, we have
From (1), we have x ? y ? 60
.................. (3)
Substitute (3) into (4)
2(y ? 60) ? 5y y ? 40
From (2), we have 2x ? 5y .....................
. (4)
Substitute y ? 40 into (3), x ? 100
? Jerry 100 sweets Gloria 40 sweets
55
7.5 Applications of Simultaneous Linear
Equations in Two Unknowns
Follow-up 7.16
A box of sweets is divided between Jerry and
Gloria in the ratio 5 2. If Jerry gets 60
sweets more than Gloria, find (a) the number of
sweets each of them gets, (b) the total number
of sweets in the box.
Solution
(b) Jerry 100 sweets Gloria 40 sweets
Total number of sweets in the box ? 100 ? 40
? 140