RD SHARMA Solutions for Class 10 Maths Chapter 3 - Pairs of Linear Equations in Two Variables
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Chapter 3 - Pairs of Linear Equations in Two Variables Exercise 3.114
Question 1
Solution 1
So, the correct option is (b).
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise 3.115
Question 1
Solution 1
Question 2
If am ≠ bl, then the system of equations
ax + by = c
lx + my = n
(a) has a unique solution
(b) has no solution
(c) has infinite many solution
(d) may or may not have a solution
Solution 2
So, the correct option is (a).
Question 3
Solution 3
So, the correct option is (b).
Question 4
Solution 4
So, the correct option is (a).
Question 5
If 2x-3y=7 and (a+b)x – (a+b-3)y = 4a+b represent coincident lines than a and b satisfy the equation
(a) a+5b=0 (b) 51+b=0 (c) a-5b=0 (d) 5a-b=0
Solution 5
So, the correct option is (c).
Question 6
If a pair of linear equations in two variables is consistent, then the lines represented by two equations are
(a) Intersecting (b) parallel
(c) always coincident (d) intersecting or coincident
Solution 6
Consistent solution means either linear equations have unique solutions or infinite solutions.
⇒ In case of unique solution; lines are intersecting
⇒ If solutions are infinite, lines are coincident.
So, lines are either intersecting or coincident
So, the correct option is (d).
Question 7
Solution 7
So, the correct option is (c).
Question 8
The area of the triangle formed by the lines
y=x, x=6, and y=0 is
(a) 36 sq. units
(b) 18 sq. units
(c) 9 sq. units
(d) 72 sq. units
Solution 8
So, the correct option is (b).
Question 9
If the system of equations 2x + 3y=5, 4x + ky =10 has infinitely many solutions, then k=
(a) 1
(b) 
(c) 3
(d) 6
Solution 9
So, the correct option is (d).
Question 10
If the system of equations kx - 5y = 2, 6x +2y=7 has no solution, then k=
(a) -10
(b) -5
(c) -6
(d)-15
Solution 10
So, the correct option is (d).
Question 11
The area of the triangle formed by the lines
x = 3, y = 4 and x = y is
(a) sq. unit
(b) 1 sq. unit
(c) 2 sq. unit
(d) None of these
Solution 11
So, the correct option is (a).
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise 3.116
Question 1
The area of the triangle formed by the lines
2x + 3y = 12, x – y – 1= 0 and x = 0
(a) 7 sq. units
(b) 7.5 sq. units
(c) 6.5 sq. units
(d) 6 sq. units
Solution 1
So, the correct option is (b).
Question 2
The sum of the digits of a two digit number is 9. If 27 is added to it, the digits of the number get reversed. The number is
- 25
- 72
- 63
- 36
Solution 2
Question 3
If x = a, y = b is the solution of the system of equations x - y = 2 and x + y = 4, then the values of a and b are, respectively
- 3 and 1
- 3 and 5
- 5 and 3
- -1 and -3
Solution 3
Since x = a and y = b is the solution of given system of equations x - y = 2 and x + y = 4, we have
a - b = 2 ….(i)
a + b = 4 ….(ii)
Adding (i) and (ii), we have
2a = 6 ⇒ a = 3
⇒ b = 4 - 3 = 1
Hence, correct option is (a).
Question 4
For what value k, do the equations 3x - y + 8 = 0 and 6x - ky + 16 = 0 represent coincident lines?
- 2
- -2
Solution 4
Question 5
Aruna has only Rs.1 and Rs.2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is Rs.75, then the number of Rs.1 and Rs.2 coins are, respectively
- 35 and 15
- 35 and 20
- 15 and 35
- 25 and 25
Solution 5
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.1
Question 1
Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheel and play Hoopla (a game in which you throw a rig on the items kept in the stall, and if the ring covers any object completely you get it). The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. Each ride costs Rs 3, and a game of Hoopla costs Rs 4. If she spent Rs 20 in the fair, represent this situation algebraically and graphically.
Solution 1
Question 2
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' this interesting?) Represent this situation algebraically and graphically.
Solution 2
Let the present age of Aftab and his daughter be x and y respectively.
Seven years ago,
Age of Aftab = x - 7
Age of his daughter = y - 7
According to the given condition,
Three years hence,
Age of Aftab = x + 3
Age of his daughter = y + 3
According to the given condition,
Thus, the given conditions can be algebraically represented as:
x - 7y = -42
x - 3y = 6
Three solutions of this equation can be written in a table as follows:
Three solutions of this equation can be written in a table as follows:
The graphical representation is as follows:
Concept insight: In order to represent a given situation mathematically, first see what we need to find out in the problem. Here, Aftab and his daughter's present age needs to be found so, so the ages will be represented by variables x and y. The problem talks about their ages seven years ago and three years from now. Here, the words 'seven years ago' means we have to subtract 7 from their present ages, and 'three years from now' or 'three years hence' means we have to add 3 to their present ages. Remember in order to represent the algebraic equations graphically the solution set of equations must be taken as whole numbers only for the accuracy. Graph of the two linear equations will be represented by a straight line.
Seven years ago,
Age of Aftab = x - 7
Age of his daughter = y - 7
According to the given condition,
Three years hence,
Age of Aftab = x + 3
Age of his daughter = y + 3
According to the given condition,
Thus, the given conditions can be algebraically represented as:
x - 7y = -42
x - 3y = 6
Three solutions of this equation can be written in a table as follows:
| x | -7 | 0 | 7 |
| y | 5 | 6 | 7 |
Three solutions of this equation can be written in a table as follows:
| x | 6 | 3 | 0 |
| y | 0 | -1 | -2 |
The graphical representation is as follows:
Concept insight: In order to represent a given situation mathematically, first see what we need to find out in the problem. Here, Aftab and his daughter's present age needs to be found so, so the ages will be represented by variables x and y. The problem talks about their ages seven years ago and three years from now. Here, the words 'seven years ago' means we have to subtract 7 from their present ages, and 'three years from now' or 'three years hence' means we have to add 3 to their present ages. Remember in order to represent the algebraic equations graphically the solution set of equations must be taken as whole numbers only for the accuracy. Graph of the two linear equations will be represented by a straight line.
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
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 2 kg of grapes is Rs 300. Represent the situation algebraically and geometrically.
Solution 11
Let the cost of 1 kg of apples and 1 kg grapes be Rsx and Rsy.
The given conditions can be algebraically represented as:
Three solutions of this equation can be written in a table as follows:
Three solutions of this equation can be written in a table as follows:
The graphical representation is as follows:
Concept insight: cost of apples and grapes needs to be found so the cost of 1 kg apples and 1 kg grapes will be taken as the variables. From the given conditions of collective cost of apples and grapes, a pair of linear equations in two variables will be obtained. Then, in order to represent the obtained equations graphically, take the values of variables as whole numbers only. Since these values are large so take the suitable scale.
The given conditions can be algebraically represented as:
Three solutions of this equation can be written in a table as follows:
x |
50 | 60 | 70 |
| y | 60 | 40 | 20 |
Three solutions of this equation can be written in a table as follows:
x |
70 | 80 | 75 |
| y | 10 |
-10 |
0 |
The graphical representation is as follows:
Concept insight: cost of apples and grapes needs to be found so the cost of 1 kg apples and 1 kg grapes will be taken as the variables. From the given conditions of collective cost of apples and grapes, a pair of linear equations in two variables will be obtained. Then, in order to represent the obtained equations graphically, take the values of variables as whole numbers only. Since these values are large so take the suitable scale.
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.10
Question 1
Solution 1
Question 2
Solution 2
Question 3
The boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of stream and that of the boat in still water.
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
A person rowing at the rate of 5 km/h in still water, takes thrice as much time in going 40 km upstream as in going 40 km downstream. Find the speed of stream.
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
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 12
Question 13
A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find the speed of the boat in still water and the speed of the stream.
Solution 13
Question 14
Solution 14
Question 15
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 15
Let the speed of the train be x km/h and the time taken by train to travel the given distance be t hours and the distance to travel be d km.
Or, d = xt ... (1)
According to the question,
By using equation (1), we obtain:
3x - 10t = 30 ... (3)
Adding equations (2) and (3), we obtain:
x = 50
Substituting the value of x in equation (2), we obtain:
(-2) x (50) + 10t = 20
-100 + 10t = 20
10t = 120
t = 12
From equation (1), we obtain:
d = xt = 50 x 12 = 600
Thus, the distance covered by the train is 600 km.
Concept insight: To solve this problem, it is very important to remember the relation
. Now, all these three quantities are unknown. So, we will represent these
by three different
variables. By using the given conditions, a pair of equations will be obtained. Mind one thing that the equations obtained will not be linear. But they can be reduced to linear form by using the fact that. Then two linear equations can be formed which can
be solved easily by elimination method.
According to the question,
By using equation (1), we obtain:
3x - 10t = 30 ... (3)
Adding equations (2) and (3), we obtain:
x = 50
Substituting the value of x in equation (2), we obtain:
(-2) x (50) + 10t = 20
-100 + 10t = 20
10t = 120
t = 12
From equation (1), we obtain:
d = xt = 50 x 12 = 600
Thus, the distance covered by the train is 600 km.
Concept insight: To solve this problem, it is very important to remember the relation
. Now, all these three quantities are unknown. So, we will represent these by three different
variables. By using the given conditions, a pair of equations will be obtained. Mind one thing that the equations obtained will not be linear. But they can be reduced to linear form by using the fact that
. Then two linear equations can be formed which can be solved easily by elimination method.
Question 16
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 hours. What are the speeds of two cars?
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.11
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
ABCD is a cyclic quadrilateral such that 
A = (4y + 20)o, B = (3y - 5)o, C = (-4x)o and D = (7x + 5)o. Find the four angles.
Solution 6
We know that the sum of the measures of opposite angles in a cyclic quadrilateral is 180°. 
A + C = 180
4y + 20 - 4x = 180
-4x + 4y = 160
x - y = -40 ... (1)
Also, B + D = 180
3y - 5 - 7x + 5 = 180
-7x + 3y = 180 ... (2)
Multiplying equation (1) by 3, we obtain:
3x - 3y = -120 ... (3)
Adding equations (2) and (3), we obtain:
-4x = 60
x = -15
Substituting the value of x in equation (1), we obtain:
-15 - y = -40
y = -15 + 40 = 25 A = 4y + 20 = 4(25) + 20 = 120oB = 3y - 5 = 3(25) - 5 = 70oC = -4x = -4(-15) = 60oD = -7x + 5 = -7(-15) + 5 = 110o
Concept insight: The most important idea to solve this problem is by using the fact that the sum of the measures of opposite angles in a cyclic quadrilateral is 180o. By using this relation, two linear equations can be obtained which can be solved easily by eliminating a suitable variable.
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
The car hire charges in a city comprise of fixed charges together with the charge for the distance covered. For a journey of 12 km, the charge paid is Rs 89 and for a journey of 20 km, the charge paid is Rs. 145. What will a person have to pay for travelling a distance of 30 km?
Solution 12
Question 13
A part of monthly hostel charges in a college are 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, he 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 charge and the cost of food per day.
Solution 13
Question 14
Solution 14
Question 15
The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
Solution 15
Question 16
2 Women and 5 men can together finish a piece of embroidery 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 embroideery, and that taken by 1 man alone.
Solution 16
Question 17
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 Rs 50 and Rs 100 she received.
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
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 number of students in the class.
Solution 21
Question 22
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?
Solution 22
Let the money with the first person and second person be Rs x and Rs y respectively.
According to the question,
x + 100 = 2(y - 100)
x + 100 = 2y - 200
x - 2y = -300 ... (1)
6(x - 10) = (y + 10)
6x - 60 = y + 10
6x - y = 70 ... (2)
Multiplying equation (2) by 2, we obtain:
12x - 2y = 140 ... (3)
Subtracting equation (1) from equation (3), we obtain:
11x = 140 + 300
11x = 440
x = 40
Putting the value of x in equation (1), we obtain:
40 - 2y = -300
40 + 300 = 2y
2y = 340
y = 170
Thus, the two friends had Rs 40 and Rs 170 with them.
Concept insight: This problem talks about the amount of capital with two friends. So, we will represent them by variables x and y respectively. Now, using the given conditions, a pair of linear equations can be formed which can then be solved easily using elimination method.
Question 23
A shopkeeper sells a saree at 8% profit and a sweater at 10% discount, thereby getting a sum of Rs.1008. If she had sold the saree at 10% profit and sweater at 8% discount, she would have got Rs.1028. Find the cost price of the saree and the list price (price before discount) of the sweater.
Solution 23
Question 24
In a competitive examination, one mark is awarded for each correct answer while ½ mark is deducted for every wrong answer. Jayanti answered 120 questions and got 90 marks. How many questions did she answer correctly?
Solution 24
Question 25
A shopkeeper gives book on rent for reading. She takes a fixed charge for the first two days, and an additional charge for each day thereafter. Latika paid Rs.22 for a book kept for 6 days, while Rs.16 for the book kept for four days. Find the fixed charges and charge for each extraday.
Solution 25
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.2
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Since, the graph of the two lines coincide, the given system of equations have infinitely many solutions.
Question 13
Solution 13
Question 14
Solution 14
Question 15
Show graphically that each one of the following systems of equations is in-coinsistent (i.e. has no solution):
3x - 5y = 20
6x - 10y = - 40
3x - 5y = 20
6x - 10y = - 40
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solve graphically each of the following systems of linear equations. Also find the coordinates of the points where the lines meet axis of y.
2x + y - 11 = 0
x - y - 1 = 0
2x + y - 11 = 0
x - y - 1 = 0
Solution 26
Question 27
Solve graphically each of the following systems of linear equations. Also find the coordinates of the points where the lines meet axis of y.
x + 2y - 7 = 0
2x - y - 4 = 0
x + 2y - 7 = 0
2x - y - 4 = 0
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31
Question 32
Solution 32
Question 33
Solution 33
Question 34
Solution 34
Question 35
Solution 35
Question 36
Solution 36
Question 37
Solution 37
Question 38
Solution 38
Question 39
Solution 39
Question 40
Solution 40
Question 41
Solution 41
Question 42
Solution 42
Question 43
Solution 43
Question 44
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. Calculate the area of the triangle so formed.
Solution 44
Three solutions of this equation can be written in a table as follows:
| x | 0 | 1 | 2 |
| y | -5 | 0 | 5 |
| x | 0 | 1 | 2 |
| y | -3 | 0 | 3 |
The graphical representation of the two lines will be as follows:
It can be observed that the required triangle is 
ABC.
The coordinates of its vertices are A (1, 0), B (0, -3), C (0, -5).
Concept insight: In order to find the coordinates of the vertices of the triangle so formed, find the points where the two lines intersects the y-axis and also where the two lines intersect each other. Here, note that the coordinates of the intersection of lines with y-axis is taken and not with x-axis, this is because the question says to find the triangle formed by the two lines and the y-axis.
Question 45
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.
(iii) Champa went to a 'Sale' to purchase some pants and skirts. When her friends asked her how many of each she had bought, she answered, "The number of skirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased". Help her friends to find how many pants and skirts Champa a bought.
Solution 45
(i) Let the number of girls and boys in the class be x and y respectively.
According to the given conditions, we have:
x + y = 10
x - y = 4
x + y = 10 
x = 10 - y
Three solutions of this equation can be written in a table as follows:
| x | 4 | 5 | 6 |
| y | 6 | 5 | 4 |
x - y = 4 x = 4 + y
Three solutions of this equation can be written in a table as follows:
| x | 5 | 4 | 3 |
| y | 1 | 0 | -1 |
The graphical representation is as follows: 
From the graph, it can be observed that the two lines intersect each other at the point (7, 3).
So, x = 7 and y = 3.
Thus, the number of girls and boys in the class are 7 and 3 respectively.
(ii) Let the cost of one pencil and one pen be Rs x and Rs y respectively.
According to the given conditions, we have:
5x + 7y = 50
7x + 5y = 46
Three solutions of this equation can be written in a table as follows:
| x | 3 | 10 | -4 |
| y | 5 | 0 | 10 |
Three solutions of this equation can be written in a table as follows:
| x | 8 | 3 | -2 |
| y | -2 | 5 | 12 |
The graphical representation is as follows: 
From the graph, it can be observed that the two lines intersect each other at the point (3, 5).
So, x = 3 and y = 5.
Therefore, the cost of one pencil and one pen are Rs 3 and Rs 5 respectively.
(iii)
Let us denote the number of pants by x and the number of skirts by y. Then the equations formed are:
y = 2x - 2 ...(1)
and y = 4x - 4 ...(2)
Let us draw the graphs of Equations (1) and (2) by finding two solutions for each of the equations.
They are given in Table
| x | 2 | 0 |
| y = 2x - 2 | 2 | -2 |
| x | 0 | 1 |
| y = 4x - 4 | -4 | 0 |
Plot the points and draw the lines passing through them to represent the equations, as shown in fig.,
The two lines intersect at the point (1,0). So, x = 1, y = 0 is the required solution of the pair of linear equations, i.e., the number of pants she purchased is 1 and she did not buy any skirt.
Concept insight: Read the question carefully and examine what are the unknowns. Represent the given conditions with the help of equations by taking the unknowns quantities as variables. Also carefully state the variables as whole solution is based on it. On the graph paper, mark the points accurately and neatly using a sharp pencil. Also, take at least three points satisfying the two equations in order to obtain the correct straight line of the equation. Since joining any two points gives a straight line and if one of the points is computed incorrect will give a wrong line and taking third point will give a correct line. The point where the two straight lines will intersect will give the values of the two variables, i.e., the solution of the two linear equations. State the solution point.
Question 46
Solution 46
Question 47
Solution 47
Question 48
Solution 48
Question 49
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
Solution 49
(i) For the two lines a1x + b1x + c1 = 0 and a2x + b2x + c2 = 0, to be intersecting, we must have 
So, the other linear equation can be 5x + 6y - 16 = 0 
(ii) For the two lines a1x + b1x + c1 = 0 and a2x + b2x + c2 = 0, to be parallel, we must have 
So, the other linear equation can be 6x + 9y + 24 = 0, 
(iii) For the two lines a1x + b1x + c1 = 0 and a2x + b2x + c2 = 0 to be coincident, we must have 
So, the other linear equation can be 8x + 12y - 32 = 0, 
Concept insight: In order to answer such type of problems, just remember the conditions for two lines to be intersecting, parallel, and coincident. This problem will have multiple answers as their can be many equations satisfying the required conditions.
Question 50
Solution 50
Question 51
Solution 51
Question 52
Graphically, solve the following pair of equations:
2x + y = 6
2x - y + 2 = 0
Find the ratio of the areas of the two triangles formed by the lines representing these equations with the x-axis and the lines with the y-axis.
Solution 52
The lines AB and CD intersect at point R(1, 4). Hence, the solution of the given pair of linear equations is x = 1, y = 4.
From R, draw RM ⊥ X-axis and RN ⊥ Y-axis.
Then, from graph, we have
RM = 4 units, RN = 1 unit, AP = 4 units, BQ = 4 units
Question 53
Determine, graphically, the vertices of the triangle formed by the lines y = x, 3y = x, x + y = 8.
Solution 53
From the graph, the vertices of the triangle AOP formed by the given lines are A(4, 4), O(0, 0) and P(6, 2).
Question 54
Draw the graph of the equations x = 3, x = 5 and 2x - y - 4 = 0. Also, find the area of the quadrilateral formed by the lines and the x-axis.
Solution 54
The graph of x = 3 is a straight line parallel to Y-axis at a distance of 3 units to the right of Y-axis.
The graph of x = 5 is a straight line parallel to Y-axis at a distance of 5 units to the right of Y-axis.
Question 55
Draw the graphs of the lines x = -2, and y = 3. Write the vertices of the figure formed by these lines, the x-axis and the y-axis. Also, find the area of the figure.
Solution 55
The graph of x = -2 is a straight line parallel to Y-axis at a distance of 2 units to the left of Y-axis.
The graph of y = 3 is a straight line parallel to X-axis at a distance of 3 units above X-axis.
Question 56
Draw the graphs of the pair of linear equations x - y + 2 = 0 and 4x - y - 4 = 0. Calculate the area of the triangle formed by the lines so drawn and the x-axis.
Solution 56
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.3
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solve the following systems of equation:
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31
Question 32
Solution 32
Question 33
Solution 33
Question 34
Solution 34
Question 35
Solution 35
Question 36
Solution 36
Question 37
Solution 37
Question 38
Solution 38
Question 39
Solve the pair of equations:
Solution 39
Question 40
Solution 40
Question 41
Solution 41
Question 42
Solution 42
Question 43
Solution 43
Question 44
Solution 44
Question 45
Solution 45
Question 46
Solution 46
Question 47
Solution 47
Question 48
Solve the following systems of equation:
21x + 47y = 110
47x + 21y = 162
Solution 48
Question 49
If x + 1 is a factor of 2x3 + ax2 + 2bx + 1, the find the values of a and b given that 2a - 3b = 4.
Solution 49
Question 50
Find the solution of the pair of equations 
and
. Hence, find λ, if y = λx
+ 5.
Solution 50
Question 51
Find the values of x and y in the following rectangle.
Solution 51
Question 52
Write an equation of a line passing through the point representing solution of the pair of linear equations x + y = 2 and 2x - y = 1. How many such lines can we find?
Solution 52
Question 53
Write a pair of linear equations which has the unique solution x = -1, y = 3. How many such pairs can you write?
Solution 53
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.4
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solve each of the following systems of equations by the method of cross-multiplication:
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.5
Question 1
Determine whether the following system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
x - 3y = 3
3x - 9y = 2
Solution 1
Question 2
Determine whether the following system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
2x + y = 5
4x + 2y = 10
Solution 2
Question 3
Determine whether the following system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
3x - 5y = 20
6x - 10y = 40
Solution 3
Question 4
Determine whether the following system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
x - 2y = 8
5x - 10y = 10
Solution 4
Question 5
Find the value of k for which the following system of equations has a unique solution:
kx + 2y = 5
3x + y = 1
Solution 5
Question 6
Find the value of k for which the following system of equations has a unique solution:
4x + ky + 8 = 0
2x + 2y + 2 = 0
4x + ky + 8 = 0
2x + 2y + 2 = 0
Solution 6
Question 7
Find the value of k for which the following system of equations has a unique solution:
4x - 5y = k
2x - 3y = 12
Solution 7
Question 8
Find the value of k for which the following system of equations has a unique solution:
x + 2y = 3
5x + ky + 7 = 0
Solution 8
Question 9
Find the value of k for which the following systems of equations have infinitely many solutions:
2x + 3y - 5 = 0
6x + ky - 15 = 0
Solution 9
Question 10
Find the value of k for which the following systems of equations have infinitely many solutions:
4x + 5y = 3
kx + 15y = 9
Solution 10
Question 11
Find the value of k for which the following systems of equations have infinitely many solutions:
kx - 2y + 6 = 0
4x - 3y + 9 = 0
Solution 11
Question 12
Find the value of k for which the following systems of equations have infinitely many solutions:
8x + 5y = 9
kx + 10y = 18
Solution 12
Question 13
Find the value of k for which the following systems of equations have infinitely many solutions:
2x - 3y = 7
(k + 2)x - (2k + 1)y = 3(2k - 1)
Solution 13
Question 14
Find the value of k for which the following systems of equations have infinitely many solutions:
2x + 3y = 2
(k + 2)x + (2k + 1)y = 2(k - 1)
Solution 14
Question 15
Find the value of k for which the following systems of equations have infinitely many solutions:
x + (k + 1)y = 4
(k + 1)x + 9y = (5k + 2)
Solution 15
Question 16
Find the value of k for which the following systems of equations have infinitely many solutions:
kx + 3y = 2k + 1
2(k + 1)x + 9y = 7k + 1
Solution 16
Question 17
Find the value of k for which the following systems of equations have infinitely many solutions:
2x + (k - 2)y = k
6x + (2k - 1)y = 2k + 5
Solution 17
Question 18
Find the value of k for which the following systems of equations have infinitely many solutions:
2x + 3y = 7
(k + 1)x + (2k - 1)y = 4k + 1
Solution 18
Question 19
Find the value of k for which the following systems of equations have infinitely many solutions:
2x + 3y = k
(k - 1)x + (k + 2)y = 3k
Solution 19
Question 20
Find the value of k for which the following system of equations has no solution:
kx - 5y = 2
6x + 2y = 7
kx - 5y = 2
6x + 2y = 7
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Find he value of k for which of the following system of equation has no solution:
kx + 3y = k - 3
12x + ky = 6
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
For what value of k, the following system of equations will represent the coincident lines?
x + 2y + 7 = 0
2x + ky + 14 = 0
x + 2y + 7 = 0
2x + ky + 14 = 0
Solution 31
Question 32
Solution 32
Question 33
Solution 33
Question 34
Solution 34
Question 35
Solution 35
Question 36
Solution 36
Question 37
Solution 37
Question 38
Solution 38
Question 39
Solution 39
Question 40
Find the values of a and b for which the following system of equations has infinitely many solutions:
2x + 3y = 7
(a - b) x + (a + b)y = 3a + b - 2
Solution 40
Question 41
Solution 41
Question 42
Solution 42
Question 43
Find the values of a and b for which the following system of equations has infinitely many solutions:
x + 2y = 1
(a - b)x + (a + b)y = a + b - 2
Solution 43
Question 44
Find the values of a and b for which the following system of equations has infinitely many solutions:
2x + 3y = 7
2ax + ay = 28 - by
Solution 44
Question 45
For which value(s) of λ, do the pair of linear equations λx + y = λ2 and x + λy = 1 have no solution?
Solution 45
Question 46
For which value(s) of λ, do the pair of linear equations λx + y = λ2 and x + λy = 1 have infinitely many solutions?
Solution 46
Question 47
For which value(s) of λ, do the pair of linear equations λx + y = λ2 and x + λy = 1 have a unique solution?
Solution 47
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.6
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Jamila sold a table and a chair for Rs.1050, thereby making a profit of 10% on a table and 25% on the chair. If she had taken profit of 25% on the table and 10% on the chair she would have got Rs.1065. Find the cost price of each.
Solution 7
Question 8
Susan invested certain amount of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. She received Rs.1860 as annual interest. However, had she interchanged the amount of investment in the two schemes, she would have received Rs.20 more as annual interest. How much money did she invest in each scheme?
Solution 8
Question 9
The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, he buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.
Solution 9
Question 10
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 10
Question 11
The cost of 4 pens and 4 pencils boxes is Rs.100. Three times the cost of a pen is Rs.15 more than the cost of a pencil box. Form the pair of linear equations for the above situation. Find the cost of a pen and a pencil box.
Solution 11
Question 12
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?
Solution 12
Question 13
A and B each have a certain number of mangoes. A says to B, "if you give 30 of your mangoes, I will have twice as many as left with you. "B replies, "if you give me 10, I will have thrice as many as left with you. "How many mangoes does each have?
Solution 13
Question 14
Vijay had some bananas, and he divided them into two lots A and B. He sold first lot at the rate of Rs.2 for 3 bananas and the second lot at the rate of Rs.1 per banana and got a total of Rs.400. If he had sold the first lot at the rate of Rs.1 per banana and the second lot at the rate of Rs.4 per five bananas, his total collection would have been Rs.460. Find the total number of bananas he had.
Solution 14
Question 15
Solution 15
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.7
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
The sum of two-digit number and the number formed by reversing the order of digits is 66. If the two digits differ by 2, find the number. How many such numbers are there?
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
The sum of 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 14
Question 15
Solution 15
Question 16
Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find the numbers.
Solution 16
Question 17
A two-digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number.
Solution 17
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.8
Question 1
Solution 1
Question 2
A fraction becomes 9/11 if 2 is added to both numerator and the denominator. If 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction.
Solution 2
Question 3
Solution 3
Question 4
If we add 1 to the numerator and subtract 1 from the denominator, a fraction becomes 1. It also becomes 1/2 if we only add 1 to the denominator. What is the fraction?
Solution 4
Question 5
The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. Find the fraction.
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.9
Question 1
Solution 1
Question 2
Solution 2
Question 3
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 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
The present age of a father is three more than three times the age of the son. Three years hence father's age will be 10 years more than twice the age of the son. Determine their present ages.
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
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 differs by 30 years. Find the ages of Ani and Biju.
Solution 11
The difference between the ages of Ani and Biju is given as 3 years. So, either Biju is 3 years older than Ani or Ani is 3 years older than Biju.
Let the age of Ani and Biju be x years and y years respectively.
Age of Dharam = 2 × x = 2x years 
Case I: Ani is older than Biju by 3 years
x - y = 3 ... (1)
x - y = 3 ... (1)
4x - y = 60 ....(2)
Subtracting (1) from (2), we obtain:
3x = 60 - 3 = 57
Age of Ani = 19 years
Age of Biju = 19 - 3 = 16 years
Case II: Biju is older than Ani by 3 years
y - x = 3 ... (3)
Age of Biju = 19 - 3 = 16 years
Case II: Biju is older than Ani by 3 years
y - x = 3 ... (3)
4x - y = 60 ... (4)
Adding (3) and (4), we obtain:
3x = 63
x = 21
Age of Ani = 21 years
Age of Biju = 21 + 3 = 24 years
Concept Insight: In this problem, ages of Ani and Biju are the unknown quantities. So, we represent them by variables x and y. Now, note that here it is given that the ages of Ani and Biju differ by 3 years. So, it is not mentioned that which one is older. So, the most important point in this question is to consider both cases Ani is older than Biju and Biju is older than Ani. For second condition the relation on the ages of Dharam and Cathy can be implemented . Pair of linear equations can be solved using a suitable algebraic method.
Question 12
Two years ago, Salim was thrice as old as his daughter and six years later, he will be four years older than twice her age. How old are they now?
Solution 12
Question 13
The age of the father is twice the sum of the ages of his two children. After 20 years, his age will be equal to the sum of the ages of his children. Find the age of the father.
Solution 13
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