# Pair Of Linear Equations In Two Variables Class 10

Pair of Linear Equations in Two Variables is a concept in algebra that deals with solving two linear equations simultaneously in two variables. The general form of a linear equation in two variables is given by:

ax + by = c; where a, b, and c are constants and x and y are variables.

In the case of two linear equations in two variables, we need to solve for the values of x and y that satisfy both equations at the same time. There are several methods to solve a pair of linear equations, given below that are very important for the CBSE Board exams as well as State Board exams

1. Graphical Method: In this method of Linear Equations In Two Variables, we plot the graphs of both equations on the same coordinate system and find the point of intersection of the two lines, which gives us the solution of the equations.
2. Substitution Method: In this method of Linear Equations In Two Variables, we solve one equation for one variable and substitute the expression obtained in the other equation. This helps us to reduce the equations to one variable and solve it.
3. Elimination Method: In this method, we eliminate one of the variables by multiplying one or both equations by suitable constants and then adding or subtracting them. This helps us to reduce the equations to one variable and solve it.
4. Cross Multiplication Method: in this method of Linear Equations In Two Variables, we use the concepts of cross multiplication to find the value of x and y.

## Pair Of Linear Equations In Two Variables Class 10 Extra Questions

1. Represent the following pair of equations graphically and write the coordinates of points where the lines intersect the y-axis. X+3Y = 6 ; 2X-3Y=12

2. Draw the graph of
2y = 4x – 6; 2x = y + 3 and determine whether this system of linear equations has a unique solution or not.

3. Check graphically whether the pair of equations 3x – 2y + 2 = 0 and 3/2 x – y + 3 = 0, is consistent. Also, find the coordinates of the points where the graphs of the equations meet the Y-axis.

4. Find whether the following pair of linear equations is consistent or inconsistent: 3x + 2y = 8; 6x – 4y = 9

5. Calculate the area bounded by the line x + y = 10 and both the coordinate axes.

6. For what value of k, does the pair of equations 4x – 3y = 9, 2x + ky = 11 have no solution?

7. If ax + by = a2 – b2 and bx + ay = 0, find the value of (x + y).

8. How many solutions do the pair of equations y = 0 and y = -5 have?

9. Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis.

10. Amit bought two pencils and three chocolates for ₹11 and Sumeet bought one pencil and two chocolates for ₹7. Represent this situation in the form of a pair of linear equations. Find the price of one pencil and that of one chocolate graphically.

11. 7x – 5y – 4 = 0 is given. Write another linear equation, so that the lines represented by the pair are:
(i) intersecting
(ii) coincident
(iii) parallel

12. Draw the graphs of the following equations:
2x – y = 1; x + 2y = 13
Find the solution of the equations from the graph and shade the triangular region formed by the lines and the y-axis.

13. Solve the following pair of linear equations graphically:
x + 3y = 6 ; 2x – 3y = 12
Also, find the area of the triangle formed by the lines representing the given equations with the y-axis.

14. Solve the following pair of equations for x and y: a2/x − b2/y = 0; a2b/x + b2a/y = a + b, x ≠ 0; y ≠ 0

15. Solve for x and y: 10/(x+y) + 2/(x-y) = 4; 15/(x+y) – 5/(x-y) = -2 ; x + y ≠ 0
x – y ≠ 0

16. Solve the following pair of linear equations for x and y:
141x + 93y = 189;
93x + 141y = 45

17. Solve the following pair of linear equations for x and y:

(b/a)x + (a/b)y = a2 + b2; x + y = 2ab

18. Solve by elimination:
3x = y + 5
5x – y = 11

19. Solve by elimination:
3x – y =7
2x + 5y + 1 = 0

20. Solve for x and y:
27x + 31y = 85;
31x + 2 7y = 89

21. Solve for x and y: x/a=y/b;
ax + by = a2+ b2

22. Solve the following pair of equations:
49x + 51y = 499
51x + 49 y = 501

23. Find the two numbers whose sum is 75 and the difference is 15.

24. Find the value of α and β for which the following pair of linear equations has an infinite number of solutions:
2x + 3y = 7;
αx + (α + β)y = 28

25. Solve the following pair of linear equations by the Substitution as well as elimination method: x + 2y = 2; x – 3y = 7.

26. A man earns ₹600 per month more than his wife. One-tenth of the man’s salary and l/6th of the wife’s salary amount to ₹1,500, which is saved every month. Find their incomes.

27. The sum of the digits of a two-digit number is 8 and the difference between the number and that formed by reversing the digits is 18. Find the number.

28. The age of the father is twice the sum of the ages of his 2 children. After 20 years, his age will be equal to the sum of the ages of his children. Find the age of the father.

29. A two-digit number is seven times the sum of its digits. The number formed by reversing the digits is 18 less than the given number. Find the given number.

30. Sita Devi wants to make a rectangular pond on the roadside for the purpose of providing drinking water for street animals. The area of the pond will be decreased by 3 square feet if its length is decreased by 2 ft. and its breadth is increased by 1 ft. Its area will be increased by 4 square feet if the length is increased by 1 ft. and the breadth remains the same. Find the dimensions of the pond.

31. On reversing the digits of a two-digit number, the number obtained is 9 less than three times the original number. If the difference between these two numbers is 45, find the original number.

32. Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours and 30 minutes. Find the speed of the stream.

33. The owner of a taxi company decides to run all the taxis on CNG fuel instead of petrol/diesel. The taxi charges in the city comprise fixed charges together with the charge for the distance covered. For a journey of 12 km, the charge paid is 789, and for a journey of 20 km, the charge paid is ₹145.
What will a person have to pay for traveling a distance of 30 km?

34. A boat takes 4 hours to go 44 km downstream and it can go 20 km upstream at the same time. Find the speed of the stream and that of the boat in still water.

35. A man travels 300 km partly by train and partly by car. He takes 4 hours if the travels 60 km by train and the rest by car. If he travels 100 km by train and the remaining by car, he takes 10 minutes longer. Find the speeds of the train and the car separately.

36. The owner of a taxi company decides to run all the taxis on CNG fuel instead of petrol/diesel. The taxi charges in the city comprise fixed charges together with the charge for the distance covered. For a journey of 13 km, the charge paid is ₹129 and for a journey of 22 km, the charge paid is ₹210.
What will a person have to pay for traveling a distance of 32 km?

37. A fraction becomes 9/11 if 2 is added to both the numerator and the denominator. If 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction.

38. 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 a book she kept for five days. Find the fixed charge and the charge for each extra day.

39. 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.

40. The ratio of incomes of two persons is 9: 7 and the ratio of their expenditures is 4 : 3. If each of them manages to save Rs. 2000 per month, find their monthly incomes.

1. X+3Y = 6 intersects y-axis at (0,2) and 2x-3y=12 intersects y-axis at (0,-4)
2.  infinitely many solutions.
3. 3x – 2y + 2 = 0 meets the Y-axis at (0,1).
and  3/2x – y + 3 = 0 meets the Y-axis at (0, 3).
4. Consistent
5. 50 sq. cm
6. K= -3/2
7. X+y = a-b
8. No solution.
9. Vertices are (2,3); (-1,0); (4,0)
10. Pencil = rs1; Chocolate = rs 3
11. DIY
12. X=3; y=5
13. 18 sq units.
14. X=a2 y= b2
15. X=3; y=2
16. X=2 ; y=-1
17. X = ab; y = ab
18. X=3; y=4
19. X=2; y=-1
20. X=2; y=1
21. X=a; y=b
22. X= 11/2; y = 9/2
23. 45 and 30 or 30 and 45
24. α =8; β= 4
25. x=4; y=-1
26. wife =Rs 5400; Husband= RS 6000
27. 53
28. 40 years
29. 42
31. 27
32. 5km/hr
33. Rs 215
34. Still water = 8km/hr; Stream = 3km/hr
35. Train = 60km/hr; Bus= 40km/hr
36. Rs 300
37. 7/9
38. Fixed charge = 15; Per day = 3
39. Still water=  6km/hr; Stream = 4km/hr

## Pair Of Linear Equations In Two Variables Class 10 MCQ

1. Graphically, the pair of equations 7x – y = 5; 21x – 3y = 10 represents two lines which are

(a) intersecting at one point
(b) parallel
(c) intersecting at two points
(d) coincident

2. The value of k for which the equations (3k + l)x + 3y = 2; (k2 + l)x + (k – 2)y = 5 has no solution, then k is equal to

(a) 2
(b) 3
(c) 1
(d) -1

3. The pair of equations x = a and y = b graphically represents lines which are
(a) parallel
(b) intersecting at (b, a)
(c) coincident
(d) intersecting at (a, b)

4. Asha has only ₹1 and ₹2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is ₹75, then the number of ₹1 and ₹2 coins are, respectively
(a) 35 and 15
(b) 15 and 35
(c) 35 and 20
(d) 25 and 25

5. The father’s age is six times his son’s age. Four years hence, the age of the father will be four times his son’s age. The present ages of the son and the father are, respectively
(a) 4 and 24
(b) 5 and 30
(c) 6 and 36
(d) 3 and 24

6. The value of k, for which the system of equations x + (k + l)y = 5 and (k + l)x + 9y = 8k – 1 has infinitely many solutions is
(a) 2
(b) 3
(c) 4
(d) 5

7. The value of k, for which equations 3x + 5y = 0 and kx + lOy = 0 has a non-zero solution is
(a) 6
(b) 0
(c) 2
(d) 5

8. If in the equation x + 2y = 10, the value of y is 6, then the value of x will be
(a) -2
(b) 2
(c) 4
(d) 5

9. Two equations in two variables taken together are called
(a) linear equations