Chapter 3: Pair of Linear Equations in Two Variables — Class 10 Mathematics NCERT Solutions | CBSE
Chapter 3: Pair of Linear Equations in Two Variables – NCERT Class 10 Mathematics Solutions | CBSE
Chapter 3. Pair of Linear Equations in Two Variables
Chapter 3. Pair of Linear Equations in two variables |
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Exercise 3.1 complete solution Exercise 3.2 complete solution Exercise 3.3 complete solution Exercise 3.4 complete solution Exercise 3.5 complete solution Exercise 3.6 complete solution Exercise 3.7 (Optinal) complete solution |
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1. The general form for a pair of linear equations in two variables x and y is 2. Two lines in a plane , only one of the following three possibilities can happen : (i) The two lines will intersect at one point .
(ii) The two lines will not intersect, i.e., they are parallel .
(iii) The two lines will be coincident . 3. Graphical Method : |
and 
where,
are all real numbers and 
.
4.
|
Pair of lines |
Compare the ratios |
Graphical representation |
Algebraic interpretation |
|
1. |
|
Intersecting lines |
Exactly one solution (unique) |
|
2. |
|
Coincident lines |
Infinitely many solutions |
|
3. |
|
Parallel lines |
No solution |
EXERCISE 3.1
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 .
Solution: Let 
and 
be age of After and his daughter respectively .
Seven years ago, Aftab and daughter age will be 
and 
years respectively .
Again, three years from now , Aftab and daughter age will be 
and 
years respectively .
Algebraically :
A/Q, 
And 
Geometrically : We have ,
|
|
7 |
0 |
– 7 |
|
|
– 5 |
– 6 |
– 7 |
And 
|
|
0 |
3 |
3 |
|
|
– 2 |
– 1 |
– 3 |
Graph photo
2. The coach of a cricket team buys 3 bats and 6 balls for Rs 3900 . Later, she buys another bat and 3 more balls of the same kind for Rs 1300 . Represent this situation algebraically and geometrically .
Solution : Let 
and 
be the cost of one bats and one ball respectively .
Algebraically :
A/Q , 
and 
Geometrically : We have ,
|
|
300 |
100 |
– 100 |
|
|
500 |
600 |
700 |
And 
|
|
400 |
– 100 |
– 200 |
|
|
300 |
400 |
500 |
Photo of graph
3. The cost of 2 kg of apples and 1 kg of grapes on a day was found to be Rs 160 .After a month, the cost of 4 kg of apples and 2kg of grapes is Rs 300 . Represent the situation algebraically and geometrically .
Solution : Let and be the cost of 1 kg apple and 1 kg grape respectively .
Algebraically :
A/Q, 
and 
Geometrically :
We have , 
|
|
50 |
60 |
80 |
|
|
60 |
30 |
0 |
And 
|
|
40 |
50 |
60 |
|
|
70 |
50 |
30 |
Photo of graph
EXERCISE 3.2
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 .
Solution : Let and be number of girls and boys respectively .
A/Q , 
|
|
0 |
10 |
5 |
|
|
10 |
0 |
5 |
And 
|
|
4 |
0 |
5 |
|
|
0 |
– 4 |
1 |
Photo graph
(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 .
Solution: Let and be cost of one pencil and one pen respectively .
A/Q , 
|
|
10 |
3 |
– 4 |
|
|
0 |
5 |
10 |
And 
|
|
3 |
8 |
– 2 |
|
|
5 |
– 2 |
12 |
Photo graph
2. On comparing the ratios 
, find out whether the lines representing the following pairs of linear equations intersect at a point , are parallel or coincident :
(i) 
(ii) 
(iii) 
Solution : (i)
So , 
Therefore, the pair of linear equations are intersect at a point .
(ii)
So , 
Therefore, the pair of linear equations is coincident .
(iii)
So , 
Therefore, the pair of linear equations are parallel .
3. On comparing the ratios , find out whether the following pairs of linear equations are consistent or inconsistent .
(i) 
Solution : Here, 
So ,
Therefore , the pairs of linear equations are consistent .
(ii) 
Solution : Here, 
So,
Therefore , the pairs of linear equations are inconsistent .
(iii) 
Solution : We have , 
And 
Here , 
So ,
Therefore , the pairs of linear equations are consistent .
(iv) 
Solution : Here, 
So,
Therefore , the pairs of linear equations are consistent .
(v) 
Solution : We have , 
and 
Here,
So , 
Therefore , the pairs of linear equations are consistent .
4. Which of the following pairs of linear equations are consistent/inconsistent ? If consistent, obtain the solution graphically :
(i) 
Solution : Here, 
So ,
Therefore , the pairs of linear equations are consistent .
|
|
5 |
0 |
3 |
|
|
0 |
5 |
2 |
graph
(ii) 
Solution : Here, 
So
Therefore , the pairs of linear equations are inconsistent .
(iii) 
Solution : Here, 
So ,
Therefore , the pairs of linear equations are consistent .
|
|
3 |
0 |
2 |
|
|
0 |
6 |
2 |
graph
And 
|
|
0 |
1 |
2 |
|
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– 2 |
0 |
2 |
graph
(iv) 
Solution : Here, 
So ,
Therefore , the pairs of linear equations are inconsistent .
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 .
Solution : Let and be the length and width of the rectangular garden respectively .
A/Q , 
|
|
20 |
16 |
21 |
|
|
16 |
20 |
15 |
And 
|
|
0 |
– 4 |
2 |
|
|
– 4 |
0 |
– 2 |
Graph
6. Given the linear equation 
, 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
Solution : (i) intersecting lines :
Thus , the another linear equation in two variable is 
[ 
]
(ii) parallel lines :
Thus , the another linear equation in two variable is 
[ 
]
(iii) coincident lines :
Thus , the another linear equation in two variable is 
[ 
]
7. Draw the graphs of the equations 
and
. Determine the coordinates of the vertices of the triangle formed by these lines and the -axis , and shade the triangular region .
Solution : We have ,
|
|
0 |
– 4 |
2 |
|
|
– 4 |
0 |
– 2 |
and
|
|
4 |
0 |
2 |
|
|
0 |
6 |
3 |
Graph
EXERCISE 3.3
1. Solve the following pair of linear equations by the substitution method .
(i) 
Solution : We have , 
And 
[ From 
]
Putting the value of in eq.
, we get 
Therefore, the solution is 
and 
.
(ii) 
Solution : We have , 
And 
[ From ]
Putting the value of 
in equation , we get 
Therefore, the solution is 
and 
(iii) 
Solution : We have , 
And 
[ From ]
Which is a false statement . Therefore , the equation do not have a common solution . So, the two rails will not cross each other .
(iv) 
Solution : We have , 
And 
[ From ]
Putting the value of in equation , we get 
Therefore, the solution is 
and 
.
(v) 
Solution : We have, 
And 
[ From ]
Putting the value of in equation
, we have 
Therefore, the solution is 
and 
.
(vi) 
Solution : We have , 
And 
[ From (i) ] 
Putting the value of in equation , we have 
Therefore, the solution is 
and 
2. Solve 
and
and hence find the value of 
for which 
.
Solution : We have , 
and 
Putting in equation , we get 
Therefore, the value of m is – 1 .
3. From 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 .
Solution : Let and 
be the two numbers .
A/Q , 
And 
[From (i)]
Putting the value of in equation , we have 
Therefore , the two numbers are 39 and 13 .
(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees . Find them .
Solution : Let and be the two angles .
A/Q , 
And 
[ From ]
Putting the value of in equation , we get 
Therefore, the angles are 
and 
respectively .
(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 .
Solution : Let and be the cost of one bat and one ball respectively .
A/Q , 
and 
[ From
]
Putting the value of
in equation , we get 
Therefore , the cost of one bat and one ball are Rs 500 and Rs 50 respectively .
(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 ?
Solution : Let, and be the fixed charge and the charge per km respectively .
A/Q , 
And 
[ From ]
Putting 
in equation 
we get 
Therefore, a person have to pay for travelling a distance of 25 km 
Rs. ( 
) Rs.( 
) Rs.( 
) Rs. 
(v) 29. A fraction becomes 
, if 2 is added to both the numerator and the denominator . If 3 is added to both the numerator and the denominator it becomes
. Find the fraction .
Solution : Let, and are the numerator and the denominator of the fraction and the fraction is 
.
A/Q , 
And , 
Putting 
in equation , we ge 
Therefore , the fraction is 
.
(vi) Five years hence, the age of Jacob will be three times that of his son . Five years ago , Jocob’s age was seven times that of his son . What are their present ages ?
Solution : Let and be the present age of the Jacob and his son respectively .
Five years hence , the age of Jacob and his son will be 
and 
years respectively .
Five years ago , the age of Jacob and his son will be 
and 
years respectively .
A/Q , 
And 
[ From ]
Putting the value of in equation , we get 
Therefore , 40 yrs and 10 yrs are the present age of the Jacob and his son respectively .
EXERCISE 3.4
1. Solve the following pair of linear equations by the elimination method and the substitution method :
(i) 
Solution : Using elimination method :
And 
Putting the value of in equation , we get 
Therefore , the solution is 
and 
.
Using substitution method :
And 
[From (i)]
Putting the value of in equation , we ge 
Therefore , the solution is and .
(ii) 
Solution : Using elimination method :
And 
Putting the value of in equation , we have 
Therefore, the solution is and 
.
Using substitution method :
And 
Putting the value of in equation , we get 
Therefore, the solution is and .
(iii) 
Solution : Using elimination method :
We have ,
And 
Putting the value of in equation 
, we get 
Therefore, the solution is 
and 
Using substitution method : We have ,
And
Putting the value of in equation we get 
Therefore, the solution is 
and .
(iv) 
Solution : Using elimination method :
We have , 
and 
Putting 
in equation , we have 
Therefore, the solution is 
and 
Using substitution method : We have
We have ,
and
[ From (i)]
Putting the value of in equation , we get 
Therefore, the solution is and
2. Form the pair of linear equations in the following problem , 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 
if we only add 1 to the denominator . What is the fraction ?
Solution : Let and be the numerator and denominator of the fraction .
The fraction is .
A/Q , 
And 
Putting the value of in equation , we get 
Therefore , the fraction is 
.
(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 ?
Solution : Let and be the present age of the Nuri and Sonu respectively .
Five years ago , the age of Nuri and Sonu will be and years respectively .
Five years later , the age of Nuri and Sonu will be and years respectively .
A/Q , 
And 
Putting the value of in equation , we get
Therefore , 35 yrs and 15 yrs be the present age of the Nuri and Sonu respectively .
(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 .
Solution : Let and be the ten’s and the unit’s digit of the number respectively .
The number is 
.
When the digits are reversed , then the
number is 
.
A/Q , 
Putting the value of in equation , we get 
Therefore , the number is 18 .
[ 
]
(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 .
Solution : Let and be the number of notes of Rs 50 and Rs 100 respectively .
A/Q , 
Putting the value of in equation , we get 
Therefore , 10 and 15 be the number of notes of Rs 50 and Rs 100 respectively .
(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 .
Solution : let, and be the fixed charge and the charges for each extra day .
A/Q , 
And 
Putting the value of in equation , we get 
Therefore, the fixed charge is Rs 13 and the charge for each extra day is Rs 3 .
EXERCISE 3.5
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 it by using cross multiplication method .
(i) 
Solution : Here, 
So ,
Therefore, the pair of linear equations has no solution .
(ii) 
Solution: Here, 
So,
Therefore, the pair of linear equations has unique solution .
Using cross-multiplication method :
We know that , 
and 
Therefore, the solution is and .
(iii) 
Solution : We have , 
Here, 
So ,
Therefore, the pair of linear equations has infinitely many solutions .
(iv) 
Solution : We have, 
and 
Here, 
So ,
Therefore, the pair of linear equations has unique solution .
Using cross-multiplication method :
We know that ,
and 
Therefore, the solution is 
and .
2. (i) For which values of 
and 
does the following pair of linear equations have an infinite number of solutions ?
Solution : We have, 
Here, 
Since, the pair of linear equation have an infinitely number of solutions .
...........(i)
and 
[ From (i)]
Putting 
in Eq. (i) we get, 
Therefore, the value of is 
and the value of is 
.
(ii) For which value of 
will the following pair of linear equations have no solution ?
Solution : We have , 
and 
Here, 
, 
,
,
,
,
A/Q , 
Hence, for k 
, the given equation has no solution .
3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :
Solution : Using substitution method : We have ,
and 
[ From ]
Putting the value of 
in equation , we get 
Therefore, the solution is 
and 
.
Using cross-multiplication method : We have ,
We know that , 
and 
Therefore, the solution is 
and .
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 , pay Rs 1180 as hostel charges . Find the fixed charges and the cost of food per day .
Solution : Let and be the fixed charges and the cost of food per day .
For student A : 
For student B : 
Putting the value of in equation , we get 
Therefore, Rs 400 and Rs 30 be the fixed charges and the cost of food per day respectively .
(ii) A fraction becomes 
when 1 is subtracted from the numerator and it becomes 
when 8 is added to its denominator . Find the fraction .
Solution : Let and be the numerator and denominator of the fraction respectively .
The fraction is .
A/Q , 
And 
Putting the value of in equation , we get 
Therefore, the fraction is 
.
(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 ?
Solution : let and are the number of right answers and wrong answers respectively .
A/Q, 
and 
Putting 
in equation , we get 
Therefore, the number of question in the test is 20 (15+5 = 20) .
(iv) Places A and B are 100km 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 ?
Solution : let and (in km/h) be the speed of two cars start from A and B respectively .
A/Q , when the two cars travel in the same direction at different speed , then
Again , when the two cars travel towards each other , they meet in 1 hour , then
Thus, the speeds of the two cars are 60 km/h and 40 km/h respectively .
(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 .
Solution : let, and are the length and breadth of the rectangle respectively.
Therefore , the area of rectangle 
.
A/Q , 
and 
Fromi , we get 
Required the length and breadth are 17 unit and 9 unit respectively.
EXERCISE 3.6
1. Solve the following pairs of equations by reducing them to a pair of linear equations :
(i) 
Solution : Let 
and 
We have, 
and 
Putting the value of 
in equation , we get
and 
Therefore, the solution is 
and 
.
(ii) 
, 
Solution : Let 
and 
Therefore , 
and 
From we get 
and 
Therefore, the solution is and 
(iii) 
Solution : Let
and 
Putting 
in equation , we get 
Hence, 
and 
are the required solution of the given pair of equations .
(iv) 
Solution: Let, 
and 
We have, 
and 
Putting 
in equation , we get 
Therefore, 
and 
Hence, 
and are the required solution of the given pair of equations .
(v) 
Solution: We have , 
and 
Let, 
and 
So, 
and 
Putting in equation , we have 
and 
Hence, 
and are the required solution of the given pair of equations .
(vi) 
Solution : We have , 
and 
Let, 
and
Now , 
and 
From (i) , we get , 
[ Putting 
]
and 
(vii) 
Solution : We have , 
and 
Let, 
and 
and 
[ From ]
Putting 
in , we get 
and 
Hence, 
and 
are the required solution of the given pair of equations .
(viii) 
, 
Solution : We have , and
Let, 
and 
and 
Putting 
in 
equation , we get 
and 
Hence, and are the required solution of the given pair of equations .
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 .
Solution: Let and (in km/h) are the speed of rowing and current water respectively .
Then the speed of downstream of water 
km/h and upstream of water 
km/h
A/Q , 
and 
Therefore, 6 km/h and 4 km/h are the speed of rowing and current water respectively .
(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 women alone to finish the work and also that taken by 1 man alone .
Solution : let time taken by 1 woman and 1 man to finish the work and days respectively .
A/Q , 
and 
Let, an
and 
and 
Putting the value of 
in equation , we get 
and 
Thus time taken by 1 woman and 1 man to finish the work 18 days and 36 days respectively .
(iii) Roohi travels 300km to her home party by train and party 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 .
Solution : Let and (in km/h) be the speed of the train and the bus respectively .
A/Q , 
and 
Let, and
Putting the value of
in equation , we get 
and 
Therefore, 60 km/h and 80 km/h be the speed of the train and the bus respectively .
EXERCISE 3.7 (Optional)*
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.
Solution: Let 
and 
(in years)be the ages of Ani and Biju respectively .
A/Q, 
Again, Ani’s father Dharam age will be 
and the age of Cathy will be 
From (i) , we get 
From (ii) , we get 
Therefore, the age of Ani is 19 or 21 years and the age of Biju is 16 or 24 years .
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)].
Solution: Let, first friend and second friend has Rs and Rs respectively .
A/Q, 
and 
[from (i)]
From (i) , we get 
Therefore, first friend and second friend capital has Rs 40 and Rs 170 respectively .
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.
Solution: Let 
(in km/h) be the speed of train and 
(in hours) be the time taken distance covered by the train .
Therefore, the distance covered by the train is 
(in km) .
A/Q, 
and 
From (i) , we get 
Therefore, the distance covered by the train is 600 km 
.
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.
Solution: Let and be the number of student in row and the number of row respectively .
Therefore, the number of students in the class is .
A/Q, 
and 
[from (i)]
From (i) , we get 
Therefore, the number of students in the class is 
.
5. In a 
. Find the three angles.
Solution: Given, 
In 
, we have 
, 
and 
6. Draw the graphs of the equations 
and 
. Determine the co-ordinates of the vertices of the triangle formed by these lines and the y axis.
Solution: We have, 
|
x |
0 |
1 |
2 |
|
y |
– 5 |
0 |
5 |
and 
|
|
0 |
1 |
2 |
|
|
– 3 |
0 |
3 |
7. Solve the following pair of linear equations:
(i) 
; 
(ii) 
; 
(iii) 
; 
(iv) 
; 
(v) 
; 
Solution: (i) 
; 
We have, 
and 
From (i) , we get 
Therefore, the solutions are 
and 
(ii) 
; 
We have, 
From (i) , we get 
Therefore, the solutions are 
and 
(iii) We have, 
From (i) , we get 
(iv) We have, 
………. (i)
……….. (ii)
From (ii) , we get 
(v) We have, 
and 
From (iii) , we get 
Therefore, the solutions are 
and 
8. ABCD is a cyclic quadrilateral (see Fig. 3.7). Find the angles of the cyclic quadrilateral
Solution: We know that , the sum of either pair of opposite angles of a cyclic quadrilateral is 180° .
and 
From (i) and (ii) , we get 
From (ii) , we get 
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