NCERT Solutions Class 10 Mathematics Chapter 3 Pair Of Linear Equations In Two Variables Download In Pdf

chapter 3 Pair of Linear Equations In Two Variables Download in pdf

EXERCISE 3.1

**Question 1.** Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then.
Also, three years from now, I shall be three times as old as you will be.” (Isn’t this
interesting?) Represent this situation algebraically and graphically.

**Question 2.** The coach of a cricket team buys 3 bats and 6 balls for Rs 3900. Later, she buys another
bat and 2 more balls of the same kind for Rs 1300. Represent this situation algebraically
and geometrically.

**Question 3. **The cost of 2 kg of apples and 1kg of grapes on a day was found to be Rs 160. After a
month, the cost of 4 kg of apples and 2 kg of grapes is Rs 300. Represent the situation
algebraically and geometrically.

EXERCISE 3.2

** Question 1.** Form the pair of linear equations in the following problems, and find their solutions
graphically.

(i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4
more than the number of boys, find the number of boys and girls who took part in
the quiz.

(ii) 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together
cost Rs 46. Find the cost of one pencil and that of one pen.

**Question 2.** On comparing the ratios 1 1 1
2 2 2
, and
a b c
a b c
, find out whether the lines representing the
following pairs of linear equations intersect at a point, are parallel or coincident:
(i) 5x – 4y + 8 = 0

(ii) 9x + 3y + 12 = 0
7x + 6y – 9 = 0 18x + 6y + 24 = 0

(iii) 6x – 3y + 10 = 0
2x – y + 9 = 0

**Question 3. **On comparing the ratios 1 1
2 2
a , b
a b
and 1
2
c
c , find out whether the following pair of linear
equations are consistent, or inconsistent.

(i) 3x + 2y = 5 ; 2x – 3y = 7

(ii) 2x – 3y = 8 ; 4x – 6y = 9

(iii) 3 5 7
2 3
x + y = ; 9x – 10y = 14 (iv) 5x – 3y = 11 ; – 10x + 6y = –22

(v)
4 2 8
3
x + y = ; 2x + 3y = 12

**Question 4.** Which of the following pairs of linear equations are consistent/inconsistent? If
consistent, obtain the solution graphically:

(i) x + y = 5, 2x + 2y = 10

(ii) x – y = 8, 3x – 3y = 16

(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0

(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0

**Question 5.** Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is
36 m. Find the dimensions of the garden.

**Question 6.** Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables
such that the geometrical representation of the pair so formed is:

(i) intersecting lines

(ii) parallel lines

(iii) coincident lines

**Question 7.** Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the
coordinates of the vertices of the triangle formed by these lines and the x-axis, and
shade the triangular region.

EXERCISE 3.3

**Question 1.** Solve the following pair of linear equations by the substitution method.

**Question 2.** Solve 2x + 3y = 11 and 2x – 4y = – 24 and hence find the value of ‘m’ for which
y = mx + 3.

**Question 3.** Form the pair of linear equations for the following problems and find their solution by
substitution method.

(i) The difference between two numbers is 26 and one number is three times the other.
Find them.

(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find
them.

(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she buys 3
bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.

(iv) The taxi charges in a city consist of a fixed charge together with the charge for the
distance covered. For a distance of 10 km, the charge paid is Rs 105 and for a
journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the
charge per km? How much does a person have to pay for travelling a distance of
25 km?

(v) A fraction becomes 9
11
, if 2 is added to both the numerator and the denominator.
If, 3 is added to both the numerator and the denominator it becomes 5 6
. Find the
fraction.

(vi) Five years hence, the age of Jacob will be three times that of his son. Five years
ago, Jacob’s age was seven times that of his son. What are their present ages?

EXERCISE 3.4

**Question 1.** Solve the following pair of linear equations by the elimination method and the substitution
method :

(i) x + y = 5 and 2x – 3y = 4

(ii) 3x + 4y = 10 and 2x – 2y = 2

(iii) 3x – 5y – 4 = 0 and 9x = 2y + 7

**Question 2.** Form the pair of linear equations in the following problems, and find their solutions
(if they exist) by the elimination method :

(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces
to 1. It becomes
1
2
if we only add 1 to the denominator. What is the fraction?

(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as
old as Sonu. How old are Nuri and Sonu?

(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is
twice the number obtained by reversing the order of the digits. Find the number.

(iv) Meena went to a bank to withdraw Rs 2000. She asked the cashier to give her
Rs 50 and Rs 100 notes only. Meena got 25 notes in all. Find how many notes of
Rs 50 and Rs 100 she received.

(v) A lending library has a fixed charge for the first three days and an additional charge
for each day thereafter. Saritha paid Rs 27 for a book kept for seven days, while
Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the
charge for each extra day.

EXERCISE 3.5

**Question 1.** Which of the following pairs of linear equations has unique solution, no solution, or
infinitely many solutions. In case there is a unique solution, find it by using cross
multiplication method.

(i) x – 3y – 3 = 0

(ii) 2x + y = 5
3x – 9y – 2 = 0 3x + 2y = 8

(iii) 3x – 5y = 20

(iv) x – 3y – 7 = 0
6x – 10y = 40 3x – 3y – 15 = 0

**Question 2.** (i) For which values of a and b does the following pair of linear equations have an
infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2

(ii) For which value of k will the following pair of linear equations have no solution?
3x + y = 1
(2k – 1) x + (k – 1) y = 2k + 1

**Question 3.** Solve the following pair of linear equations by the substitution and cross-multiplication
methods :
8x + 5y = 9
3x + 2y = 4

**Question 4.** Form the pair of linear equations in the following problems and find their solutions (if
they exist) by any algebraic method :

(i) A part of monthly hostel charges is fixed and the remaining depends on the
number of days one has taken food in the mess. When a student A takes food for
20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes
food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charges and the
cost of food per day.

(ii) A fraction becomes
1
3 when 1 is subtracted from the numerator and it becomes
1
4
when 8 is added to its denominator. Find the fraction.

(iii) Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1
mark for each wrong answer. Had 4 marks been awarded for each correct answer
and 2 marks been deducted for each incorrect answer, then Yash would have
scored 50 marks. How many questions were there in the test?

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another
from B at the same time. If the cars travel in the same direction at different speeds,
they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What
are the speeds of the two cars?

(v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by
5 units and breadth is increased by 3 units. If we increase the length by 3 units and
the breadth by 2 units, the area increases by 67 square units. Find the dimensions
of the rectangle.

EXERCISE 3.6

**Question 1.** Solve the following pairs of equations by reducing them to a pair of linear equations:

**Question 2.** Formulate the following problems as a pair of equations, and hence find their solutions:

(i) Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her
speed of rowing in still water and the speed of the current.

(ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3
women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to
finish the work, and also that taken by 1 man alone

(iii) Roohi travels 300 km to her home partly by train and partly by bus. She takes 4
hours if she travels 60 km by train and the remaining by bus. If she travels 100 km
by train and the remaining by bus, she takes 10 minutes longer. Find the speed of
the train and the bus separately.

EXERCISE 3.7 (Optional)*

**Question 1.** The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old
as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ
by 30 years. Find the ages of Ani and Biju.

**Question 2.** One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The
other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the
amount of their (respective) capital? [From the Bijaganita of Bhaskara II]
[Hint : x + 100 = 2(y – 100), y + 10 = 6(x – 10)].

**Question 3.** A train covered a certain distance at a uniform speed. If the train would have been
10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train
were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find
the distance covered by the train.

**Question 4.** The students of a class are made to stand in rows. If 3 students are extra in a row, there
would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the
number of students in the class.

**Question 5.** In a Δ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

**Question 6.** Draw the graphs of the equations 5x – y = 5 and 3x – y = 3. Determine the co-ordinates of
the vertices of the triangle formed by these lines and the y axis.

**Question 7.** Solve the following pair of linear equations:

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- Chapter 3 Pair of Linear Equations In Two Variables
- Chapter 4 Quadratic Equations
- Chapter 7 Coordinate Geometry
- Chapter 8 Introduction To Trigonometry
- Chapter 11 Constructions
- Chapter 9 Some Applications of Trigonometry
- chapter 10 Circles
- Chapter 12 Areas Related To Circles
- Chapter 13 Surface Areas and Volumes
- Chapter 14 Statistics
- Chapter 15 Probability
- Chapter 5 Arithmetic Progressions

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