Grade 9 Book 1 Unit 1 and 2

2. Use graphs to find the solutions to the following systems of linear equations in
two variables.
a. + = 8 and − = 2 Graph:
Method:

b. 2 + 3 = 8 and 4 + 5 = 12

Method: Graph:

c. − 3 = 6 and − 3 = 9 Graph:
Method:

Mathematics Book 1 41

2.1 Solving System of Linear Equation by Substitution Method

This method is used to solve the system of linear equation by solving one of
the equations (you choose which one) for one of the variables (you choose which
one), and then plugging this back into the other equation, "substituting" for the
chosen variable and solving for the other. Then you back-solve for the first variable.

Here are the steps of solving system of linear equations by substitution method:

Step 1: First solve one linear equation for y in terms of x.

Step 2: Then substitute that expression for y in the other linear equation.

Step 3: Solve this, and you have the x-coordinate of the intersection.

Step 4: Then plug in x to either equation to find the corresponding y-coordinate.

Example 10

Solve the system of linear equation by substitution method:
3 + 4 = 11 and 2 + = 9

Solution

Step 1: First solve one linear equation for y in terms of x.

let 3 + 4 = 11 ……………………………………… (1)

Solve for y 2 + = 9 ……………………………………….(2)

= 9 − 2 …….………………………… (3)

Step 2: Then substitute (3) into (1),

3 + 4(9 − 2 ) = 11

2 + 36 − 8 = 11

5 = 25

= 5

Step 3: Substitute = 5 into (3) (5,-1)
= 9 − 2(5)

= 9 − 10

= −1

42 Mathematics Book 1 Therefore, the solution is (5, -1)

Example 11

Solve:
+ = −1 and 3 − 4 = 4

Solution

+ = −1 …………………………………………………. (1)
3 − 4 = 4 ………………………………………………… (2)
From (1), = −1 − …………………………………………….. (3)
Substitute (3) into (2),

3(−1 − ) − 4 = 4
−3 − 3 − 4 = 4
−7 = 7
= −1

Substitute = −1 into (1),
+ (−1) = −1
− 1 = −1
= 0

Therefore, the solution is (0,-1)
Check:
Substitute = 0, = −1 into (1)

+ = −1
0 + (−1) = −1

−1 = −1

Mathematics Book 1 43

Exercise 2C

Solve the following equations by the substitution method:

1. 4 + = 3 and 6 − 5 = −2 Check:
Method:

2. 3 = 4 + 2 and 3 = 7 − 1 Check:
Method:

3. 2 − 3 = 4 and 2 − 5 = 8 Check:
Method:

4. 3 − 2 = 1 and 3 − 3 = 4
Method:

Check:

5. 4 − 3 = 2 and − 1 1 = −1
2

Method:

Check:

44 Mathematics Book 1

2.1 Solving System of Linear Equation by Elimination Method

This method is used to eliminate one of the unknowns by adding or subtracting
the equations.

Example 11

Solve the system of linear equation:
+ = 3 and 2 − = 0

Solution

+ = 3 ………………………………………………………(1)

2 − = 0 ……………………………………………………...(2)

(1) + (2) gives

+ + = 3 Check:
2 − = 0 Substitute = 1 and = 2 into (1)

3 = 3 + = 3
1+2=3
= 1
3=3
Substitute = 1 into (1) ∴ LHS = RHS

1 + = 3

= 2

Therefore, the solution is (1,2)

Example 12

Solve: 5 + 4 = 21 and 2 + 3 = 0

Solution

5 + 4 = 21 ………………………………………………………… (1)

2 + 3 = 0 ………………………………………………………… (2)

(1) x 3 15 + 12 = 63 ……………………………………………………….. (3)
(2) x 4 8 + 12 = 0 ……………………………………………………….. (4)
(3) – (4) gives

15 + 12 = 63
8 + 12 = 0
7 = 63

= 9 Mathematics Book 1 45

Substitute = 9 into (2) Check:
2(9) + 3 = 0 Substitute = 9 and = −6 into (1)
18 + 3 = 0
3 = −18 5(9) + 4(−6) = 21
= −6 45 − 24 = 21
21 = 21
Therefore, the solution is (9,-6). ∴ LHS = RHS

Exercise 2D

Solve the system of linear equation by elimination method.

1. 2 + = −1 and 3 − 4 = 4

2. + 2 = 4 and 2 + 3 = 7

3. 5 − 3 = 85 and 12 + 5 = 21

4. 3 + 2 = 13 and 3 − 2 = 5

5. 7 + 3 = 10 and 35 − 6 = 1

46 Mathematics Book 1

Note:

It will be necessary to simplify the equations before applying any one of the
methods of solution.

Example 13

Solve the following equations.

5( + 2 ) − (3 + 11 ) = 14 ……………………………………… (1)

7 − 9 − 3( − 4 ) = 38 ……………………………………… (2

Solution

From (1), 5( + 2 ) − (3 + 11 ) = 14

2 − = 14 ………………………….. (3)

From (2), 7 − 9 − 3( − 4 ) = 38

4 + 3 = 38 …………………………. (4)

(3) × 3 , 6 − 3 = 42 …………………………. (5)

(4) + (5) gives

4 + 3 = 38

6 − 3 = 42

10 = 80

= 8

Substitute = 8 into (3), 2(8) − = 14

16 − = 14

= 2

Therefore, the solution is (8, 2).

Example 10

Solve 11 −5 = 3 + …………………………………………………… (1)

11 16

8 − 5 = 1 ………………………………………………………… (2)

Solution

From (1), multiply by 11 × 16,
176 − 80 = 33 + 11
143 − 91 = 0………………………………….. (3)

Mathematics Book 1 47

From (2), = 1 (1 + 5 ) ……………………………………………… (4)
8

Substitute x into (3),

143 (1 + 5 ) − 91 = 0 …………………………………… (5)

8

143 − 715 − 728 = 0

13 = 143

= 11

Substitute = 11 into (2),

8 − 55 = 1

8 = 56

= 7

Therefore, the solution is (7, 11).

Exercise 2E

Use a suitable method to solve the system of linear equation.

1. + = 3 and + 2 = 3
2 3 4 3

2. 3 − = 23 and + = 4
3 4

3. 5 − = 12 and 2 − 3 = −4
3 2

4. + 1 = 4 and 1 − 1 = 1
2 2 2 2

48 Mathematics Book 1

3. Problems Leading to System of Linear Equation

Many problems involving two unknowns can be used the method of solving
system of linear equations to find the required values of those quantities.

Before problems can be solved, we must be able to form equations from given
information. This means we have to identify the unknown quantities and then relate
them to the information given.

If there are two unknowns, there must be at least two pieces of information
leading to two equations before solving.

Example 11

The sum of two numbers is 56, and if the greater is subtracted from three times
the less, the remainder is 40. Find the numbers.

Solution

Let be the greater number, and the less.
From the first condition (the sum of two numbers is 56)

+ = 56 ………………………………………………………. (1)
From the second, (the greater is subtracted from three times the less, the remainder
is 40).

3 − = 40 ……………………………………………………… (2)
To eliminate , add (1) and (2)

4 = 96
= 96 ÷ 4
= 24
Substitute = 24 into (1)
+ 24 = 56
= 56 − 32
= 32
Therefore, the numbers are 32 and 24.

Mathematics Book 1 49

Example 12

The cost of one protractor and one set square is 60 baht where as the cost of
one protractor and two set squares is 80 baht. Find the cost of a protractor and that
of set square.

Solution

Let the cost of a protractor be baht and that of a set square be ℎ .
From the first condition,

+ = 60 …………………………………………………………….. (1)
From the second condition,

+ 2 = 80 …………………………………………………………….. (2)
To eliminate , (2) – (1);

= 20
Substitute = 20 into (1),

+ (20) = 60
= 40

Therefore, a protractor cost 40 baht and a set square is 20 baht.

Example 13

If 2 is subtracted from the numerator of a certain fraction, and three added to
the denominator, its remainder is 13, if 3 is added to the numerator, and the
denominator is multiplied by 2, its remainder is 2. What is the fraction?

3

Solution

Let be the numerator of the fraction, and be the denominator of the
fraction; then the fraction is .

From the first condition,

−2 = 1 ………………………………………………………………….. (1)
+3 3

From the second,

+3 = 2 …………………………………………………………………. (2)
2 3

From (1), 3( − 2) = + 3

3 − = 9 …………………………………………………………………. (3)

50 Mathematics Book 1

From (2), 3( + 3) = 4

3 − 4 = −9 ……………………………………………………… (4)

(3) – (4), 3 = 18
= 6

Substitute = 6 into (2), = 5

Therefore, the fraction is .



Exercise 2F

Read and analyze the following problems.
1. Find the two numbers whose sum is 25, and whose difference is 1.

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2. The difference of two numbers is five-sixths of their sum, and the greater
exceeds ten times the less by 3; find the number.

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3. The sum of two numbers is 7. The difference between two times the larger
number and the smaller numbers is 5. Find the two numbers.

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4. One-seventh of the sum of two numbers is 6, and three times their difference
is 48; find them.

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_____________________________________________________M__a_t_h_e_m__a_t_ic_s_B__o_o_k__1____5_1___

5. Five years ago, Saman was three times as old as his daughter, Sunari. Five years
from now, Saman’s age will be twice as Sunari’s age. Find their present ages.

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6. The wages of 24 men and 10 women amount to 11,520 baht per day; half that
number of men, with 19 more women, would earn the same money. What are
the daily wages of each man and woman?

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7. A duck and a hen are together worth 145 baht, while 4 ducks and 7 hens cost
760 baht. Find the price of each animal.

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8. If either 9 tables and 7 chairs, or 10 tables and 2 chairs, can be bought 7,800
baht, what is the price of each?

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9. The length of a rectangle is 2 cm more than the width. If the perimeter of the
rectangle the rectangle is 56 cm. find the length and width.

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52 __M__a_t_h_e_m__a_t_ic_s__B_o_o_k__1___________________________________________________________

Summary

➢ Linear equations in two variables are equations which can be expressed
in general form as Ax + By + C = 0, where A, B and C are real numbers and
both A, and B are not zero.

➢ The equation is called linear if the variables are only to the first power, are
only in the numerator and there are no products of variables in any of the
equations.

➢ Slope intercept is a specific form of linear equation. It is in general form
as = + , where m and b are any two real numbers.

➢ Slope is the ratio of the change in y over the change in x between any two
points on the line. It can be denoted as

➢ A system of linear equations is a system made up of two linear equations.
To solve the system of equations, you need to find the exact values
of x and y that will satisfy both equations.

➢ To solve a system of linear equations graphically
we graph both equations in the same coordinate system. The solution to
the system will be in the point where the two lines intersect.

➢ A system of linear equation has one solution when the graphs are
intersecting each other.

➢ A system of linear equations has infinite solutions when the graphs are
the exact same line.

➢ A system of linear equations has no solution when the graphs are
parallel.

➢ Before you can graph a linear equation, you need to make sure that it is
written in slope-intercept form. The slope-intercept form of a linear
equation is y = mx + b.

➢ Substitution method is used to solve the system of linear equation by
solving one of the equations (you choose which one) for one of the
variables (you choose which one), and then plugging this back into the
other equation, "substituting" for the chosen variable and solving for the
other. Then you back-solve for the first variable.

➢ Elimination method is used to eliminate one of the unknowns by adding
or subtracting the equations.

➢ It will be necessary to simplify the equations before applying any one of
the methods of solution.

➢ Many problems involving two unknowns can be used the method of
solving system of linear equations to find the required values of those
quantities.

Mathematics Book 1 53

Revision Exercise

Part 1 (Skills 1-3)

1. Find the x-coordinate of points on the line = + 5 which have the y-
coordinate of:
a. 1 c. -1 e. 2

b. 3.5 d. −2 1

2

2. Find the y-coordinate of points on the line = − + 5 which have the x-

coordinate of:

a. 0 c. -2 e. 3 1

2

b. 4.5 d. -10

3. Solve the system of linear equation:

a. 7 + 3 = 10

35 + 6 = 1

b. 3 − 2 = 6
6 − 5 = 30

c. 7 + 10 = 4
14 = 46 − 18

d. 13 + 2 = 9
3 = 7

e. 5( + 2 ) − (3 + 11 ) = 14
7 − 9 − 3( − 4 ) = 38

54 Mathematics Book 1

Part 2. Concepts. Circle the letter of the correct answer.

1. Which graph has no solution? 6. Which method is used to eliminate
a. c.
one of the unknowns by adding or

subtracting the equations?

a. substitution

b. elimination

b. d. c. graphing

d. equating

2. Which one is true about the values 7. What is the solution to the system

of x and y we recovered from of equations, 5 − 8 = −2 and

solving system of linear equation? 7 − 6 = 18?

a. will be equal a. (6,3) c. (6,4)

b. will be bigger than 1 b. (4,6) d. (7,2)

c. will satisfy both equations 8. In graphing, what does it mean

d. will be numerically equal by when the lines overlap at every

multiplying point?

3. Which pair of equations is formed a. It has exact solution.

from the statement below? b. It has infinitely many

“Two numbers whose sum is m and solutions.

the larger number is twice the c. It has no solution.

smaller number”. d. It is parallel.

a. = = 9. What is the slope intercept form of

= a linear equation?

b. = , 2 a. = +

c. = 2 , = b. + = 4
2
c. = +
d. + = , = 2
d. = +
4. When can you say that the graph
10. Which equation has the slope of 3?
of a system of linear equation has
a. = 4 + 4
infinite solution?
b. + 2 = 5
a. When two graphs are parallel
c. 5 − 2 = 6
b. When two graphs intersect
d. − 3 = 7

c. When two graphs are on the 11.What does it mean when the

exact same line graph to the system of the

d. When two graphs intersect equation is parallel?

perpendicularly a. It has one solution.

5. What is the slope of the line from b. It has infinitely many

the equation = 3 + 4? solutions

a. 3 c. 2 c. It has no solution

b. 4 d. 1 d. It intersects

Mathematics Book 1 55

Part 3. Word problems. Circle the letter of the correct answer.

12. I spend 450 baht in buying eggs at 16. Kanyawat sells ticket for
2 for 5 baht and apples at 3 for 10 baht. admission to your school play and
If I were to sell them all alike at the rate collects a total of 3,600 baht.
of 10 for 40 baht, I should gain 150 Admission prices are 36 baht for
baht. How of each did I buy? adults and 27 baht for children. She
sold 21 tickets total. Write a system
a. 30 eggs and 60 apples to represent this situation assuming
b. 60 eggs and 90 apples “x” represents the number tickets
c. 100 eggs and 50 apples and “y” represents the number of
d. 90 eggs and 60 apples children’s ticket.
e. None of these a. + = 21
36 + 27 = 3,600
13. One-fourth of the sum of two b. 36 + = 21
angles is 13°, and one-sixths of their + 27 = 3,600
difference is 3°. What are the two c. + 27 = 21
angles? + 27 = 3,600
d. 36 + = 3,600
a. 17°, 18° + 27 = 21
b. 18°, 35° e. None of these
c. 35°, 17°
d. 70°, 17° 17. A rectangle is 5 cm longer than it is
e. None of these wide and its area is 66 cm2. What are
the dimensions of the rectangle?
14. What are the two numbers whose a. breadth 5cm, length 11 cm
sum is 61, and whose difference is 15? b. breadth 6 cm, length 11 cm
c. breadth 6 cm, length 11.5 cm
a. 15, 61 d. breadth 6.5 cm, length 11.5 cm
b. 19, 42 e. None of these
c. 23, 38
d. 51, 10 18. A number of two digits is equal to
e. None of these eight times the sum of its digits; if 45
is subtracted from the number will
15. If two notebooks and four pencils be reversed; what is the number?
cost 90 baht, two pencils and two ball a. 19
pens cost 45 baht, one notebook and b. 36
three ball pens cost 75 baht; what is c. 45
the price of each notebook? d. 72
e. None of these
a. 15 baht
b. 20 baht
c. 25 baht
d. 30 baht
e. None of these

56 Mathematics Book 1