RD SHARMA SOLUTION CHAPTER- 3 Pair of Linear Equations in Two Variables | CLASS 10TH MATHEMATICS-EDUGROWN

Chapter 3 Pairs of Linear Equations in Two Variables Exercise Ex. 3.1

Question 1Akhila 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 2Aftab 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 2Let 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:

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(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 7The 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 7Let 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:

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. 3.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

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 15Show graphically that each one of the following systems of equations is in-coinsistent (i.e. has no solution):

3x – 5y = 20

6x – 10y = – 40Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19(i)

Solution 19(i)

Question 19(ii)

Solution 19(ii)

Question 20

Solution 20

Question 21(i)

Solution 21(i)

Question 21(ii)

Solution 21(ii)

Question 22(i)

Solution 22(i)

Question 22(ii)

Solution 22(ii)

Question 22(iii)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 = 0Solution 22(iii)

Question 22(iv)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 = 0Solution 22(iv)

Question 22(v)

Solution 22(v)

Question 22(vi)

Solution 22(vi)

Question 23(i)

Solution 23(i)

Question 23(ii)

Solution 23(ii)

Question 23(iii)

Solution 23(iii)

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28(i)

Solution 28(i)

Question 28(ii)

Solution 28(ii)

Question 28(iii)

Solution 28(iii)

Question 28(iv)

Solution 28(iv)

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

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 32

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 33

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 33

(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 34(i)

Solution 34(i)

Question 34(ii)

Solution 34(ii)

Question 35

Solution 35

Question 36

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 linesSolution 36

(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 37(i)

Solution 37(i)

Question 37(ii)

Solution 37(ii)

Question 38

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 38

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 39

Determine, graphically, the vertices of the triangle formed by the lines y = x, 3y = x, x + y = 8.Solution 39

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 40

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 40

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 41

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 41

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 42

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 42

Question 5

Solve the following equations graphically:

x – y + 1 = 0

3x + 2y – 12 = 0Solution 5

Given equations are:

x – y + 1 = 0 … (i)

3x + 2y – 12 = 0 … (ii)

From (i) we get, x = y – 1

When x = 0, y = 1

When x = -1, y = 0

When x = 1, y = 2

We have the following table:

x	0	-1	1
y	1	0	2

From (ii) we get,  

When x = 0, y = 6

When x = 4, y = 0

When x = 2, y = 3

We have the following table:

x	0	4	2
y	6	0	3

Graph of the given equations is:

As the two lines intersect at (2, 3). 

Hence, x = 2, y = 3 is the solution of the given equations.

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 = 162Solution 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. 3.5

Question 29

Find c if the system of equations cx + 3y + 3 – c = 0, 12x + cy – c = 0 has infinitely many solutions.Solution 29

The given system of equations will have infinite number of solutions if

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 = 2Solution 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 = 10Solution 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 = 40Solution 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 = 10Solution 4

Question 5

Find the value of k for which the following system of equations has a unique solution:

kx + 2y = 5

3x + y = 1Solution 5

Question 6Find the value of k for which the following system of equations has a unique solution:

4x + ky + 8 = 0

2x + 2y + 2 = 0Solution 6

Question 7

Find the value of k for which the following system of equations has a unique solution:

4x – 5y = k

2x – 3y = 12Solution 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 = 0Solution 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 = 0Solution 9

Question 10

Find the value of k for which the following systems of equations have infinitely many solutions:

4x + 5y = 3

kx + 15y = 9Solution 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 = 0Solution 11

Question 12

Find the value of k for which the following systems of equations have infinitely many solutions:

8x + 5y = 9

kx + 10y = 18Solution 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 + 1Solution 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 + 5Solution 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 + 1Solution 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 = 3kSolution 19

Question 20Find the value of k for which the following system of equations has no solution:

kx – 5y = 2

6x + 2y = 7Solution 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 = 6Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 30

Solution 30

Question 31For what value of k, the following system of equations will represent the coincident lines?

x + 2y + 7 = 0

2x + ky + 14 = 0Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36 (i)

Solution 36 (i)

Question 36 (ii)

Solution 36 (ii)

Question 36 (iii)

Solution 36 (iii)

Question 36 (iv)

Solution 36 (iv)

Question 36 (v)

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 – 2Solution 36 (v)

Question 36 (vi)

Solution 36 (vi)

Question 36 (vii)

Solution 36 (vii)

Question 36(viii)

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 – 2Solution 36(viii)

Question 36(ix)

Find the values of a and b for which the following system of equations has infinitely many solutions:

2x + 3y = 7

2ax + ay = 28 – bySolution 36(ix)

Question 37(i)

For which value(s) of λ, do the pair of linear equations λx + y = λ² and x + λy = 1 have no solution?Solution 37(i)

Question 37(ii)

For which value(s) of λ, do the pair of linear equations λx + y = λ² and x + λy = 1 have infinitely many solutions?Solution 37(ii)

Question 37(iii)

For which value(s) of λ, do the pair of linear equations λx + y = λ² and x + λy = 1 have a unique solution?Solution 37(iii)

Chapter 3 Pairs of Linear Equations in Two Variables Exercise Ex. 3.8

Question 9

A fraction becomes 1/3 when 2 is subtracted from the numerator and it becomes 1/2 when 1 is subtracted from the denominator. Find the fraction.Solution 9

Let the fraction be 

According to the given conditions, we have

Subtracting (ii) from (i), we get x = 7

Substituting the value of x in (ii), we get

y = 15Question 1

Solution 1

Question 2A 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 4If 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 9Old

Solution 9Old

Question 10

Solution 10

Question 11

Solution 11

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. 3.6

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 5The 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 14The 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. 3.9

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 6The 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)

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)

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

Chapter 3 Pairs of Linear Equations in Two Variables Exercise Ex. 3.10

Question 1

Solution 1

Question 2

Solution 2

Question 3The 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 12Ritu 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 15A 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 15Let 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.

Question 16Places 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 = 120^o
B = 3y – 5 = 3(25) – 5 = 70^o
C = -4x = -4(-15) = 60^o
D = -7x + 5 = -7(-15) + 5 = 110^o

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 180^o. 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 12The 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 13A 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 15The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.Solution 15

Question 162 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 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 6

Solution 6

Question 7

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 solutionSolution 7

So, the correct option is (a).Question 8

Solution 8

So, the correct option is (b).Question 9

Solution 9

So, the correct option is (a).Question 10

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=0Solution 10

So, the correct option is (c).Question 11

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 coincidentSolution 11

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 12

Solution 12

So, the correct option is (c).Question 13

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. unitsSolution 13

So, the correct option is (b).Question 14

If the system of equations 2x + 3y=5,  4x + ky =10 has infinitely many solutions, then k=

(a) 1     

(b)     

(c) 3    

(d) 6Solution 14

So, the correct option is (d).Question 15

If the system of equations  kx – 5y = 2,  6x +2y=7 has no solution, then k=

(a) -10     

(b) -5      

(c) -6     

(d)-15Solution 15

So, the correct option is (d).Question 16

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 theseSolution 16

So, the correct option is (a).

Chapter 3 – Pairs of Linear Equations in Two Variables Exercise 3.116

Question 17

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 17

So, the correct option is (b).Question 18

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

1. 25
2. 72
3. 63
4. 36

Solution 18

Question 19

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

1. 3 and 1
2. 3 and 5
3. 5 and 3
4. -1 and -3

Solution 19

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 20

For what value k, do the equations 3x – y + 8 = 0 and 6x – ky + 16 = 0 represent coincident lines?

1. 
2. 
3. 2
4. -2

Solution 20

Question 21

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

1. 35 and 15
2. 35 and 20
3. 15 and 35
4. 25 and 25

Solution 21