Contents
(Term II)
Preface
(v)
Latest Syllabus
(vi)
4.
8.
9.
10.
11.
13.
14.
15.
Linear Equations in Two Variables
Quadrilaterals
Areas of Parallelograms and Triangles
Circles
Constructions
Surface Areas and Volumes
Statistics
Probability
Value Based Questions
...
...
...
...
...
...
...
...
...
2
18
35
51
70
77
95
114
129
Practice Papers (I – III)
...
137
xv
Each chapter contains:
★ TIPS AND TRICKS
★ FORMATIVE ASSESSMENT
★ SUMMATIVE ASSESSMENT
●
CBSE and Other Important Questions
●
Objective Type Questions
●
Higher Order Thinking Skills (HOTS) Questions
●
NCERT Textbook Exercises
4
Linear Equations in Two Variables
CHAPTER
Tips and Tricks
• An equation of the form ax + by + c = 0, where a, b and
c are real numbers such that a ≠ 0, b ≠ 0, is called a linear
equation in two variables x and y. The process of finding
out the solution(s) is called solving an equation.
5(– 2) + 2p(– 1) = 2p
⇒
– 10 – 2p = 2p
⇒
4p = – 10
10
= – 2.5.
⇒
p= −
4
2. Check which of the following is (are) solution(s) of the
equation 3y – 2x = 1
• The solution of a linear equation is not affected when
(i) the same number is added to (or subtracted from) both
the sides of the equation.
(ii) both sides of the equation are multiplied or divided
by the same non-zero number.
• A linear equation in two variables has infinitely many
solutions.
• The graph of every linear equation in two variables is a
straight line and every point on the graph (straight line)
represents a solution of the linear equation. Also, every
solution of a linear equation can be represented by a
unique point on the graph of the equation.
• The graph of x = a is a straight line parallel to the y-axis.
• The graph of y = a is a straight line parallel to the x-axis.
(i) (4, 3)
(ii) (2 2 , 3 2 ) .
Sol. The given equation is
3y – 2x = 1
...(1)
For (i) (4, 3)
LHS = 3y – 2x
= 3(3) – 2(4)
=9–8
= 1 = RHS
Hence, (4, 3) is a solution of the equation 3y – 2x = 1.
For (ii) (2 2 , 3 2 )
ILLUSTRATIVE EXAMPLES
LHS = 3(3 3 ) − 2 (2 2 )
=9 2 −4 2
2 Marks, CBSE Exam.
=5 2
≠ RHS
1. Find a value of p for which x = – 2, y = – 1 is a solution
of the linear equation 5x + 2py = 2p.
Sol. The given linear equation is
5x + 2py = 2p
Hence, (2 2 , 3 2 ) is not a solution of the equation
3y – 2x = 1.
3. Give geometrical representation of equation 3x + 12 = 0
in (i) one variable (ii) two variables.
...(1)
If x = – 2, y = – 1 is a solution of equation (1), then these
values must satisfy the equation (1). So, putting x = – 2,
y = – 1 in equation (1), we get
Sol. (i) One variable
The given equation is
2
3
LINEAR EQUATIONS IN TWO VARIABLES
3x + 12 = 0
3x = – 12
⇒
Sol. The given linear equation is
12
⇒
x= −
=–4
3
The representation of 3x + 12 = 0 on the number line is
shown below, where 3x + 12 = 0 is treated as an equation
in one variable.
2x − 3 x + 3 2x + 3
+
=
5
4
4
⇒
4 (2 x − 3) + 5 ( x + 3)
2x + 3
=
20
4
⇒
8 x − 12 + 5 x + 15
2x + 3
=
20
4
⇒
13 x + 3 2 x + 3
=
20
4
x=–4
–5 –4 –3 –2–1
0
1
2
3
4
5
(ii) Two variables
⇒
The given equation is
⇒
13 x + 3 2 x + 3
=
5
1
13x + 3 = 5 (2x + 3)
3x + 12 = 0
⇒
13x + 3 = 10x + 15
3x + 0y + 12 = 0
⇒
⇒
It is a linear equation in two variables x and y. This is
represented by a line. All the values of y are permissible
because 0y is always 0. However, x must satisfy the
relation 3x + 12 = 0, i.e., x = – 4. Hence, two solutions of
the given equation are x = – 4, y = 0 and x = – 4, y = 2.
The graph AB is a line parallel to the y-axis at a distance
of 4 units to the left of it.
13x – 10x = 15 – 3
⇒
3x = 12
⇒
x=
12
=4
3
Hence, the required solution is x = 4.
5. Find ‘a’ if
a+3 a+2
=
, a ≠ 2, a ≠ – 7
a−2 a+7
y
Sol. We have,
4
4
(– 4, 2)
a+3 a+2
=
a−2 a+7
3
A
⇒
2
(– 4, 0)
–5 B–4 –3
–2
⇒ a2 + 7a + 3a + 21 = a2 – 4
1
x
0
–1
⇒
x
1
–1
2
3
–3
–4
10a + 21 = – 4
⇒
10a = – 4 – 21
⇒
10a = – 25
⇒
a=−
–2
x=–4
(a + 3) (a + 7) = (a + 2) (a – 2)
25
= – 2.5
10
Hence, the required value of a is – 2.5.
6. Draw the graph of the linear equation 2x – 3y + 7 = 0 and
hence find the coordinates of the point where the line
intersects x-axis.
y
4. Solve the linear equation for ‘x’.
2x − 3 x + 3 2x + 3
+
=
5
4
4
Sol. The given linear equation is
2x – 3y + 7 = 0
⇒
3y = 2x + 7
4
CCE MATHEMATICS–IX
y
2x + 7
y=
3
⇒
6
5
Table of solutions
x
1
3
4
2
y
3
–3
2x
4
5
( 7 , 0)
2
We plot the points (1, 3) and (4, 5) on a graph paper and
join by a ruler. The line obtained represents the graph of
the linear equation 2x – 3y + 7 = 0.
From graph, we see that the coordinates of the point where
F
H
the line 2x – 3y + 7 = 0 intersects x-axis are −
x
7
y+
=0
(4, 5)
(1, 3)
1
x
–3 –2 –1 0
–1
1
3
2
4
5
–2
–3
I
K
7
,0 .
2
–4
y
Formative Assessment
ORAL QUESTIONS (Conversation Type)
1. What is a linear equation in two variables?
2. How many solutions does a linear equation in two
variables have?
3. What is the form of any point on the x-axis?
4. What is the form of any point on the y-axis?
5. What is the equation of the x-axis?
TRUE OR FALSE
1. Infinitely many linear equations in two variables x and y
can be satisfied by x = 1 and y = 2.
2. If we add the same number to both sides of the linear
equation, then the solution of the linear equation remains
unchanged .
3. If we subtract the same number from both sides of the
linear equation, then the solution of the linear equation
remains unchanged.
4. If we multiply both the sides of a linear equation by a
non-zero number, then the solution of the equation
remains the same.
5. If we divide both the sides of a linear equation by a nonzero number, then the solution of the equation changes.
6. The graph of the linear equation y = x passes through the
point (1, 1).
7. x = 1, y = 2 is a solution of the linear equation
x + 5y = 11.
8. The equation 2x + 5y = 7 has a unique solution, if x, y are
natural numbers.
9. Any solution of the linear equation 3x + 0y + 5 = 0 in two
F
H
variables is of the form 0, –
10. Point (2, 0) lies on a y-axis.
I
K
5
.
3
5
LINEAR EQUATIONS IN TWO VARIABLES
Assignments
Name: ....................................... Class: ........
Section: .......
CLASS ASSIGNMENT 1
Roll No.: ...... Grade: ......
Teacher’s sign.: .....................
6. Does the graph of the linear equation x + y = 14 pass
through (7, 7)?
1. Compare the equation 2x – y + 1 = 0 with ax + by + c = 0
and find a, b and c.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
7. Does the graph of the linear equation 2x + 3y = 8 pass
through (4, 1)?
2. Does the line x + y = 0 pass through origin?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
3. Express y in terms of x when 5x + y = 4.
____________________________________________
____________________________________________
____________________________________________
4. Express x in terms of y when 4y – 3x – 23 = 0
____________________________________________
____________________________________________
8. Find a solution of the linear equation in two variables
y + x = 4.
____________________________________________
____________________________________________
____________________________________________
9. Where does the line 2x + 3y = 6 intersect the x-axis?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
10. Where does the line 3x + 4y = 12 intersect the y-axis?
5. Write the equation x = 1 as a linear equation in two
variables.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
6
CCE MATHEMATICS–IX
Name: ....................................... Class: ........
Section: .......
CLASS ASSIGNMENT 2
(From CBSE Examination Paper)
Roll No.: ...... Grade: ......
Teacher’s sign.: .....................
6. After 5 years, the age of father will be two times the age
of son. Write a linear equation in two variables to
represent this statement.
1. How many solution(s) of equation 2x + 1 = x – 3 are
there on the number line?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
7. The relation between the temperature in Celsius scale
5
(F – 30).
(°C) and in Fahrenheit scale (°F) is C =
9
Express F in terms of C.
____________________________________________
2. Find the point where the line 3x + y = 6 intersects x-axis.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
3. Express the equation 5x = – y in the general form and
indicate the values of a, b and c.
8. Find the value of k, if x = 2, y = 1 is a solution of
2x + 3y = k.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
4. Express y in terms of x from the equation 3x + 2y = 8.
9. Find the point where the line 2(x + 3) – 3(1 + y) = 0
intersects the y-axis.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
5. Check whether the point (4, – 2) lies on the line
3x + 2y = 8.
10. If the number of hours for which a labourer works is x
and y are his wages (in rupees) and y = 2x – 1, find the
wages of the labourer if he works for 6 hours.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
7
LINEAR EQUATIONS IN TWO VARIABLES
Name: ....................................... Class: ........
Section: .......
CLASS ASSIGNMENT 3
(From NCERT Exemplary Problems)
1. How many linear equations in x and y can be satisfied by
x = 1 and y = 2?
____________________________________________
____________________________________________
____________________________________________
2. Does the graph of the linear equation y = x pass through
the point (1, 1)?
____________________________________________
____________________________________________
____________________________________________
3. What is the equation of x-axis?
____________________________________________
____________________________________________
____________________________________________
Roll No.: ...... Grade: ......
Teacher’s sign.: .....................
6. At what point does the graph of the linear equation
2x + 3y = 6 cut the x-axis?
____________________________________________
____________________________________________
____________________________________________
7. Is it true to say that the graph of the equation y = mx + c
pass through the origin?
____________________________________________
____________________________________________
8. Is it false to say that a linear equation 2x + 3y = 5 has a
unique solution?
____________________________________________
____________________________________________
9. If the temperature of a liquid can be measured in Kelvin
units as x°K or in Fahrenheit units as y°F, the relation
between the two systems of measurement is given by the
linear equation
9
(x – 273) + 32
5
Express x in terms of y.
y=
____________________________________________
4. What is the equation of y-axis?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
10. The linear equation that converts Fahrenheit (F) to Celsius
5F − 160
. What is the
9
numerical value of the temperature which is same in both
the scales?
(C) is given by the relation C =
5. At what point does the graph of the linear equation
2x + 3y = 6 cut the y-axis?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
8
CCE MATHEMATICS–IX
Name: ....................................... Class: ........
Section: .......
HOME ASSIGNMENT 1
1. Where does the line 5x + 6y = 30 intersect the x-axis?
____________________________________________
____________________________________________
____________________________________________
2. Where does the line 4x + 5y = 20 intersect the y-axis?
____________________________________________
____________________________________________
____________________________________________
3. Find a solution of the linear equation x – y = 1.5.
____________________________________________
Roll No.: ...... Grade: ......
Teacher’s sign.: .....................
6. Write the equation x = 3y as a linear equation in two
variables.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
7. Express x in terms of y when 4x + 3y = 0.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
8. Express y in terms of x when 2x – y = 9.
____________________________________________
____________________________________________
____________________________________________
4. Does the graph of the linear equation – 5x + 6y = 9 pass
through (– 1, 0)?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
5. Does the graph of the linear equation 2x – 3y = 11 pass
through (– 4, – 1)?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
9. Prove that the line y = 2x pass through the origin.
____________________________________________
____________________________________________
____________________________________________
10. Compare the equation x = 7 with ax + by + c = 0 and find
the values of a, b and c.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________