CBSE Class 10 Mathematics Pair Of Linear Equations In 2 Variables Worksheet Set D

Question. A pair of linear equations which has a unique solution x = 2 and y = – 3 is
(a) x + y = 1 and 2x – 3y = – 5
(b) 2x + 5y = – 11 and 2x – 3y = – 22
(c) 2x + 5y = – 11 and 4x + 10y = – 22
(d) x – 4y – 14 = 0 and 5x – y – 13 = 0
Answer. (c) 2x + 5y = – 11 and 4x + 10y = – 22

Question. 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
Answer. (d) intersecting or coincident

Question. One of the common solution of ax + by = c and y axis is
(a) (0, c/b)
(b) (0, b/c)
(c) (c/b, 0)
(d) 0, - c/b)
Answer. (a) (0, c/b)

Question. For what value of p, system of equations 2x + py = 8 and x + y = 6 have no solution.
Answer. p = 2

Question. A motor cyclist is moving along the line x – y = 2 and another motor cyclist is moving along the line x – y = 4 find out their moving direction.
Answer. move parallel

Question. If 2x + 5y = 4, write another linear equation, so that lines represented by the pair are coincident.
Answer. 4x + 10y = 8

Question. The area of the triangle formed by the lines x = 3, y = 4 and x = y is _____ .
Answer. 1/2 sq. unit

Question. If the lines given by 3x + 2ky = 2 and 2x + 5y = 1 are parallel, then the value of k is _______ .
Answer. k = 15/4

Question. Write the solution of y = x and y = –x.
Answer. (0, 0)

Question. What is the value of p, for which the pair of linear equations x + y = 3 and 3x + py = 9 is inconsistent.
Answer. p = 3

SHORT ANSWER TYPE (I) QUESTIONS

Question. Solve the pair of linear equations by substitution method x – 7y + 42 = 0 and x – 3y – 6 = 0
Answer. 42, 12

Question. Solve for x and y
3x + 2y = 11 and 2x + 3y = 4
Also find p if p = 8x + 5y
Answer. x = 5, y = – 2, p = 30

Question. Form a pair of linear equations for: The sum of the numerator and denominator of the fraction is 3 less than twice the denominator. If the numerator and denominator both are decreased by 1, the numerator becomes half the denominator.
Answer. x – y = – 3, 2x – y = 1

Question. The difference of two numbers is 66. If one number is four times the other, find the numbers.
Answer. 88, 22

SHORT ANSWERS TYPE (II) QUESTIONS

Question. Solve for x and y
5/(x+y) + 1/(x-y) = 2
15/(x+y) - 5/(x-y) = -2
Answer. (3, 2)

Question. Sunita has some `₹ 50 and `₹ 100 notes amounting to a total of `₹ 15,500. If the total number of notes is 200, then find how many notes of `₹ 50 and `₹ 100 each, she has.
Answer. ₹ 50 notes = 90, ₹ 100 notes = 110

Question. Find the value of k for no. solutions
(3k + 1)x + 3y – 2 = 0
(k² + 1)x + (k – 2)y – 5 = 0
Answer. k = –1

Question. For what values of a and b the following pair of linear equations have infinite number of solutions?
2x + 3y = 7
a(x + y) – b(x – y) = 3a + b – 2
Answer. a = 5, b = 1

Question. Pinky 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 were deducted for each wrong answer, then pinky again would have scored 40 marks. How many questions were there in the test?
Answer. 40 questions

LONG ANSWER TYPE QUESTIONS

Question. A boat can travel 30 km upstream and 28 km downs stream 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.
Answer. 10 km/hr, 4 km/hr

Question. A man travels 600 km to his home partly, by train and partly by bus. He takes 8 hours, if he travels 120 km by train and rest by bus. Further, it takes 20 minute longer, if he travels 200 km by train and rest by bus. Find the speeds of the train and the bus.
Answer. 60 km/hr, 80 km/hr

Question. Solve the pair of equations by reducing them to a pair of linear equations 2^x = 8^y-1 and 9^y = 3^x–6
Answer. x = 24, y = 9

Question. Determine graphically, the vertices of the triangle formed by the times y = x, 3y = x and x + y = 8.
Answer. Vertices of the triangle are (0, 0) (4, 4) (6, 2).

Question. A boat covers 32 km upstream and 36 km downstream, in 7 hours. Also it Covers 40 km upstream and 48 km downstream in 9 hours. Find the speed of boat in still water and that of the stream.
Answer. 10 km/hr, 2 km/hr

Question. A nirudh takes 3 hours more than Nishi to walk 30 km. But if Anirudh doubles his speed, he is ahead of Nishi by 1½ hours. Find their speed fo walking.
Answer. 10/3 km/hr, 5 km/hr

Question. Find the values of a and b for infinite solutions
(i) 2x – (a – 4)y = 2b + 1
4x – (a – 1)y = 5b – 1
(ii) 2x + 3y = 7
2ax + ay = 28 – by
Answer. (i) 7, 3
(ii) 4, 8

Question. Vijay had some bananas and he divided them into two lots A and B. He sold the first lot at the rate of `₹ 2 for 3 bananas and the second lot at the rate of `₹ 1 per banana and got a total of `₹ 400. If he had sold the first lot at the rate of `₹ 1 per banana and the second lot at the rate of `₹ 4 for 5 bananas, his total collection would have been `₹ 460. Find the total number of bananas he had.
Answer. Let the no. of bananas in lots A be x and in lots B be y
Case I : (2/3)x + y = 400 ⇒ 2x + 3y = 1200
Case 2 : x + (4/5)y = 460 ⇒ 5x + 4y = 2300
x = 300, y = 200, Total bananas = 500.

Question. In the equations 3x + 2y = 13xy and 4x -5y = 2xy , the value of xand y satisfy that the equations are:   
(A) (2,3)
(B) (3,2)
(C) (1/2,1/3)
(D) (1/3,1/2)

Answer: C

Question. A father is 7 times as old as his son. Two year ago, the father was 13 times as old as his son. Father’s present age is: 
(A) 24 years
(B) 28 years
(C) 30 years
(D) 32 years

Answer: B

Question. If x + y + 7and3x -2y=11. Then the value of x will be: 
(A) 5
(B) 6
(C) 7
(D) 8

Answer: A

Question. If 3y -2x = 4and4y - px = 2perpendicular to each other the value of ‘p’ will be:   
(A) 3/2
(B) 8/3
(C) 6
(D) -6

Answer: D

Question. A boat whose speed is 18km/hr in still water takes 1 hr more to go 24km upstream than to return downstream to the same spot. Find the speed of the stream. 
(A) 8 km/hr
(B) 6 km/hr
(C) 10 km/hr
(D) 5.5 km/hr 

Answer: B

Question. It is given that there is no solution to the system x + 2y = 3,ax +by = 4 . Which one of the following is true?   
(A) a has a unique value
(B) b has a unique value
(C) a can have more than one value
(D) a has exactly two different values

Answer: C

Question. In a two digit number, the number of ten’s place is double of the number of unit’s place. If we exchange the numbers mutually then the number decrease b 18, then the number is:-   
(A) 24
(B) 36
(C) 39
(D) 42

Answer: D

Question. The system of equation- x + 2y = 6,3x +6y=18   
(A) Is inconsistent
(B) Has an infinite number of solution
(C) Has a unique solution
(D) None of these

Answer: B

Question. A man can row three quarters of a km against the stream in 11(1/4) minutes and return in 7(1/2) minutes. The speed of man in still water is:   
(A) 2 km/h
(B) 3 km/h
(C) 4 km/h
(D) 5 km/h

Answer: D

Solved Problems on Class 10 Linear Equations

Question. What are the coordinates of points where two lines representing the given equations meet y-axis?

Answer: (0, –2), (0, 4)

Question. The sum of the numerator and denominator of a fraction is 3 less than twice the denominator. If the numerator and denominator are decreased by 1, the numerator becomes half the denominator. Determine the fraction.

Answer: 4/7

Question. Two years ago, a father was five times as old as his son. Two years later, his age will be 8 more than three times the age of the son. Find the present ages of father and son.

Answer: Father’s age = 42 years, Son’s age = 10 years

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

Answer: (0, 0), (4, 4) (6, 2)

Question. If a1/a2≠b1/b2≠c1/c2  , then what does the system of linear equations, represent graphically?

Answer: Two coincident lines

Question. The cost of 4 pens and 4 pencil boxes is ` 100. Three times the cost of a pen is ` 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.

Answer: 10, ` 15

Question. 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.

Answer: 36

Question. Define consistent system of linear equations.

Answer: A pair of linear equations which has either unique or infinitely many solutions.

Question. What is the solution of given pair of equations? Read from graph.

Answer: Unique solution: (3, 1)

Question. The sum of a two digit number and the number obtained by reversing the order of its digits is 165. If the digits differ by 3, find the number.

Answer: 69 or 96

Question 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 the stream.

Answer: 2.5 km/h

Question. Points A and B are 70 km apart on a highway. A car starts from A and another car starts from B simultaneously. If they travel in the same direction, they meet in 7 hours, but if they travel towards each other, they meet in one hour. Find the speed of the two cars.

Answer: Speed from point A = 40 km/ h, from point B = 30 km/h

Question. Yash scored 35 marks in a test, getting 2 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?

Answer: 25

Question. Draw the graphs of the equations y = -1, y = 3 and 4x - y = 5. Also, find the area of the quadrilateral formed by the lines and the y-axis.

Answer: 6 square units

Question. What does a linear equation in two variables represent geometrically?

Answer: A straight line.

Question. What is the area of triangle formed by given lines and x-axis?

Answer: 1 sq. unit 

Question. A man travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes him 6 hours and 30 minutes. But, if he travels 200 km by train and the rest by car, he takes half an hour longer. Find the speed of the train and that of the car.

Answer: 100 km/h, 80 km/h

Question. Solve graphically the system of linear equations:
(ii) 4x-3y+4=0
4x-3y+20= 0
Find the area bounded by these lines and x-axis.

Answer: x = 2, y = 4, 12 sq. units

Question. Ankita travels 14 km to her home partly by rickshaw and partly by bus. She takes half an hour if she travels 2 km by rickshaw, and the remaining distance by bus. On the other hand, if she travels 4 km by rickshaw and the remaining distance by bus, she takes 9 minutes longer. Find the speed of the rickshaw and of the bus.

Answer: 10 km/h, 40 km/h

Question. When is a system of linear equations called inconsistent?

Answer: When it has no solution.

Question.A part of monthly hostel charges in a college are fixed and the remaining depend on the number of days one has taken food in the mess. When a student A takes food for 15 days, he has to pay 1200 as hostel charges whereas a student B, who takes food for 24 days, pays 1560 as hostel charges. Find the fixed charge and the cost of food per day.

Answer: ₹ 600, ₹ 40

Question. Solve the following system of linear equations graphically and shade the region between the two lines and x-axis.
(i) 3x+2y-4=0 (ii) 3x+2y-11=0
2x-3y+10= 0 2x-3y-7= 0

Answer: (i) x = 2, y = -1 (ii) x =1, y = 4

Question. Do the equations x +2y -7 = 0 and 2x + 4 y +5= 0 represent a pair of parallel lines?

Answer: Yes

Question. 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, and that taken by 1 man alone to finish the embroidery.

Answer: One man in 36 days, One woman in 18 days

Question. 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 1860 as annual interest. However, had she interchanged the amount of investment in the two schemes, she would have received 20 more as annual interest. How much money did she invest in each scheme?

Answer: Scheme A  ₹ 12000, Scheme B ₹ 10,000

Question. What is the area of triangle formed by given lines and y-axis?

Answer: 9 sq. units

Question. The car hire charges in a city comprise of a fixed charges together with the charge for the distance covered. For a journey of 12 km, the charge paid is 89 and for a journey of 20 km, the charge paid is person have to

Answer: ₹ 215

Question. Is it true to say that the pair of equations x +2y -3 = 0 and 3x +6 y -9 = 0 are dependent?

Answer: Yes

Question. A two digit number is 4 times the sum of its digits and twice the product of the digits. Find the number.

Answer: 36

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

Answer: 60 km/h, 40 km/h

Question. If lines corresponding to two given linear equations are coincident, what can you say about the solution of the system of given equations?

Answer: It has infinitely many solutions.

Question. What are the coordinates of points where two lines representing the given equations meet x-axis?

Answer: (2, 0), (4, 0)

VERY SHORT ANSWER TYPE QUESTIONS

Question. For what value of k, the equations 2x – ky + 3 = 0 and 3x + 2y – 1 = 0 has no solution?
Solution. -4/3

Question. Is x = 2 and y = – 1 a solution of the pair of linear equations x + 2y = 0 and 3x + 4y = 20?
Solution. No

Question. What are the coordinates of the point of intersection of two lines 3x + 2y = 0 and 2x – y = 0?
Solution. x = 0, y = 0

Question. How many solutions do two linear equations in two variables have, if their graph are parallel?
Solution. No solution

Question. At what point does the line 4x + 5y = 20 intersects x-axis?
Solution. (5, 0)

Question. How many solutions will the following pair of linear equations have?
7x – 4y + 11 = 0
2x – 9y + 15 = 0
Solution. unique

Question. The graph of y = – 3 is a straight line parallel to which axis?
Solution. x-axis

Question. Draw the graph of the equations
4x – y= 4
4x + y = 12
Determine the vertices of the triangle formed by the lines representing these equations and the x-axis. Shade the triangular region so formed
Solution. (2, 4)(1,0 )(3,0 )

Question. Find the value of ‘a’ so that the point (2,9) lies on the line represented by ax-3y=5
Solution. a=16

Question. Determine the value of ‘a’ if the system of linear equations 3x+2y-4=0and ax–y–3=0 will represent intersecting lines.
Solution. a ≠ 3/2

Question. Write any one equation of the line which is parallel to √2x–√3y=5
Solution. 5√2x - 5√3 y = 5√5

Question. Check whether given pair of lines is consistent or not 5x–1=2y,y= -1/2 + 5/2 x .
Solution. Consistent

Question. Find the fraction which becomes to 2/3 when the numerator is increased by 2 and equal to 4/7 when the denominator is increased by 4.
Solution. 28/45

Question. Solve Graphically
x – y = –1
3x + 2y = 12.
Calculate the area bounded by these lines and the x- axis.
Solution. x = 2, y = 3 and area = 7.5 unit²

Question. Solve for u& v
4u–v=14uv
3u + 2v = 16uv where u≠0, v≠ 0
Solution. u = ½ , v = ¼

Question. Ritu can row downstream 20 km in 2 hours, and upstream 4km in 2hours. Find her speed of rowing in still water and the speed of the current.
Solution. Speed of the rowing in still water = 6km/hr
Speed of the current=4km/hr.

Question. Write the condition so that a₁x + b₁y = c₁ and a₂x + b₂y = c₂ have unique solution.
Solution. a₁/a₂ ≠ b₁/b₂

Question. 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. Cost of one pencil=Rs.3
Cost of one pen=Rs.5

Question. Form a pair of linear equations for the following problem :
Fourteen students of class X took part in a quiz and the number of boys is 2 more than the number of girls.
Solution. Let x and y denote the number of boys and girls respectively. Then, x = y + 2 and x + y = 14.

Question. What is the minimum and maximum number of solutions that a system of simultaneous linear equations can have, if it is consistent system?
Solution. One and infinite

Question. Give an equation of a line which passes through the origin.
Solution. y = mx

Question. For what value of k, will the equations 4x + my = 8 and 3x – 5y + 7 = 0 represent parallel lines?
Solution. k = -20/3

Question. If a system of equation is inconsistent, then what type of graph the equations will have?
Solution. parallel lines

Question. For what value of k, the equations kx + 3y = k – 3 and 12 x + ky = k has infinitely many solutions?
Solution. 6

Question. Given a linear equation 3x – 5y = 15. Write another linear equation in two variables, such that the geometric representation of the pair so formed is coincident lines.
Solution. 6x – 10y = 30

Question. For what value of k, will the equations 2x – y + 8 = 0 and 4x – ky + 16 = 0 represent coincident lines?
Solution. k = 2

Question. What are the points of intersection of the line x/a + y/b + 3 = 0 with x-axis and with y-axis?
Solution. (–3a, 0), (0, –3b)

Question. For what value of k, will the equations 2x + ky = 7 and 3x + 9y = 13 may have a unique solution?
Solution. All real numbers except 6.

Question. Find a point on y-axis satisfying 3x – 4y = 12.
Solution. (0, –3)

Question. The sum of a two-digit number and the number obtained by reversing the order of digits is 165. If the digits differ by 3, find the number. 
Solution. Let the unit digit be x and tens digit be y.
Then, number = 10 y + x.
Number obtained by reversing the order of the digits = 10x + y.
According to the given question, we have
(10y + x) + (10 x + y) = 165 ...(1)
and, x – y = 3 ...(2)
OR
y – x = 3 ...(3)
From eqn. (1), we get
11x + 11y = 165 ⇒ x + y = 15 ...(4)
Solving eqn. (2) and (4) together, we get
x = 9, y = 6
Solving eqn. (3) and (4) together, we get
x = 6, y = 9
Substituting the values of x and y, for the number 10 y + x, we get number = 69 or 96. Ans.

Question. A fraction is such that if the numerator is multiplied by 3 and the denominator is reduced by 3, we get 18/11 , but if the numerator is increased by 8 and the denominator is doubled, we get 2/5 .Find the fraction.
Solution. Let the fraction be x/y
Then, according to given equestion, we have :
3x/y-3 = 18/11 and x + 8/2y = 2/5
⇒ 11x = 6y – 18 and 5x + 40 = 4y
⇒ 11x – 6y = – 18 ...(1)
and 5x – 4y = – 40 ...(2)
multiplying eqn. (1) by 2 and eqn. (2) by 3, we get :
22x – 12 y = – 36 ...(3)
15x – 12y = – 120 ...(4)
Subtracting eqn. (4) from eqn. (3), we get
7x = 84 ⇒ x = 12
Substituting x = 12 in eqn. (1), we get
11(12) – 6y = – 18
⇒ –6y = – 150 ⇒ y = 25.
Hence, the required fraction is 12/35 . Ans.

Question. A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay Rs. 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs. 1180 as hostel charges.
Find the fixed charges and the cost of food per day. 
Solution. Let the fixed charges be Rs. x and charges per day be Rs. y.
According to given question,
x + 20 y = 1000 ...(1)
x + 26 y = 1180 ...(2)
Subtracting eq. (1) from eqn. (2), we get
6y = 180 ⇒ y = 30
from eqn. (1), x + 20(30) = 1000
⇒ x = 1000 – 600 = 400
∴ Fixed charges = Rs. 400 and cost of food per day = Rs. 30.

Question. 2 tables and 3 chairs together cost Rs. 2000 whereas 3 tables and 2 chairs together cost Rs. 2500. Find the total cost of 1 table and 5 chairs.
Solution. Let the cost of a table be Rs. x and that of a chair be Rs. y. According to given question,
2x + 3y = 2000 ...(1)
and 3x + 2y = 2500 ...(2)
Adding eqn. (1) and (2), we get
5x + 5y = 4500 ⇒ x + y = 900 ...(3)
Subtracting eqn. (1) from eqn. (2), we get
x – y = 500 ...(4)
Adding eqn. (3) and eqn. (4), we get
2x = 1400 ⇒ x = 700
Using x = 700 in eqn. (3), we get
700 + y = 900 ⇒ y = 200
⇒ Cost of 1 table = Rs. 700 and cost of 1 chair = Rs. 200.
Hence, cost of 1 table and 5 chairs
= Rs. (x + 5y) = Rs. 700 + 5 (200) = Rs. 1700 Ans.