# RD Sharma Solutions Chapter 3 Pair of Linear Equation in Two Variables Exercise 3.5 Class 10 Maths

 Chapter Name RD Sharma Chapter 3 Pair of Linear Equation in Two Variables Book Name RD Sharma Mathematics for Class 10 Other Exercises Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.6 Related Study NCERT Solutions for Class 10 Maths

### Exercise 3.5 Solutions

In each of the following systems of equations determine whether the system has a unique solution , no solution or infinitely many solutions. In case there is a unique solution, find it : (1, -4)

1. x - 3y - 3 = 0
3x - 9y - 2 = 0

Solution

The given system of equations may be written as
x - 3y - 3 = 0
3x - 9y - 2 = 0
The given system of equations is of the form

2. 2x + y - 5 = 0
4x + 2y - 10 = 0

Solution

The given system of equation may be written as
2x + y - 5 =0
4x + 2y - 10 = 0
The given system of equations is of the form

3. 3x - 5y = 20
6x - 10y = 40

Solution

3x - 5y = 20
6x - 10y = 40
Compare it with

4.  x - 2y - 8 = 0
5x - 10y - 10 = 0

Solution

The given system of equation may be written as
x - 2y - 8 = 0
5x - 10y - 10 = 0
The given system if equation is of the form

5. kx + 2y - 5= 0
3x + y - 1 = 0

Solution

The given system of equation is
kx + 2y - 5= 0
3x + y - 1 = 0
The system of equation is of the form

6. 4x +5y + 8 = 0
2x + 2y+2 = 0
Solution

7.  4x - 5y = k
2x - 3y = 12
Solution

The given system of equation is
4x - 5y - k = 0
2x - 3y - 12 = 0
The system of equation is of the form

8. x + 2y = 3
5x + ky + 7 = 0
Solution
The given system of equation is
x + 2y - 3 = 0
5x + ky + 7 = 0
The system of equation is of the form
Find the value of k for which each of the following systems of equations have definitely many solution : (9 - 19)

9. 2x + 3y - 5 = 0
6x - ky - 15 = 0

Solution

The given system of equation is
2x + 3y - 5 = 0
6x - ky - 15 = 0
The system of equation is of the form

10. 4x + 5y = 3
kx + 15y = 9

Solution

The given system of equation is
4x + 5y - 3 = 0
kx + 15y - 9 = 0
The system of equation is of the form

11. kx - 2y + 6 = 0
4x  + 3y + 9 = 0

Solution

The given system of equation is
kx - 2y + 6 = 0
4x + 3y + 9 = 0
The system of equation is of the form

12. 8x + 5y = 9
kx + 10y = 18

Solution

The given system of equation is
8x + 5y - 9  = 0
kx + 10y - 18 = 0
The system of equation is of the form

13. 2x - 3y = 7
(k + 2)x - (2k + 1)y - 3(2k - 1)
Solution

The given system of equation may be written as
2x - 3y - 7 = 0
(k + 2)x - (2k + 1)y - 3(2k - 1) = 0
The system of equation is of the form

14. 2x + 3y = 2
(k + 2)x + (2k + 1)y - 2(k - 1)
Solution
The given system of equation may be written as
2x + 3y - 2 = 0
(k + 2)x + (2k + 1)y - 2(k - 1) = 0
The system of equation is of the form

15.  x + (k + 1)y = 4
(k + 1)x + 9y - (5k + 2)
Solution

The given system of equation may be written as
x + (k + 1)y - 4 = 0
(k + 1)x + 9y - (5k + 2) = 0
The system of equation is of the form

16. kx + 3y - 2k + 1
2(k + 1)x + 9y - (7k + 1)
Solution

The given system of equation may be written as
kx + 3y - (2k + 1) = 0
2(k + 1)x + 9y - (7k + 1) = 0
The system of equation is of the form

Hence,  the given system of equations will have infinitely many solutions, if k = 2.

17. 2x + (k - 2)y = k
6x + (2k - 1)y - (2k + 5)
Solution

The given system of equation may be written as
2x + (k - 2)y - k = 0
6x + (2k - 1)y - (2k + 5) = 0
The system of equation is of the form

18. 2x + 3y = 7
(k + 1)x + (2k - 1)y - (4k + 1)

Solution

The given system of equation may be written as
2x + 3y - 7  = 0
(k + 1)x + (2k - 1)y - (4k + 1) = 0
The system of equation is of the form

19. 2x +3y = k
(k - 1)x + (k + 2)y - 3k
Solution

The given system of equation may be written as
2x + 3y - k = 0
(k - 1)x + (k + 2)y - 3k = 0
The system of equation is of the form

Find the value of k for which the following system of equations has no solultion: (20 - 25)

20. kx - 5y = 2
6x + 2y = 7
Solution
Given,
kx - 5y = 2
6x + 2y = 7
condition for system of equations having no solution

21. x + 2y = 0
2x + ky - 5 = 0
Solution
The given system of equation may be written as

x + 2y = 0
2x + ky - 5 = 0
The system of equation is of the form

22. 3x - 4y + 7 = 0
kx  + 3y - 5 = 0
Solution
The given system of equation may be written as

3x - 4y + 7 = 0
kx  + 3y - 5 = 0
The system of equation is of the form

23.  2x - ky +3 = 0
3x + 2y - 1 = 0
Solution
The given system of equation may be written as

2x - ky +3  = 0
3x + 2y - 1 = 0
The system of equation is of the form

24. 2x + ky - 11 = 0
5x - 7y - 5 = 0

Solution
The given system of equation is
2x + ky - 11 = 0
5x - 7y - 5 = 0
The system of equation is of the form

25. kx + 3y = 3
12x + ky = 6
Solution

kx + 3y = 3
12x + ky = 6

26. For what value of Î±, the following system of equations will be inconsistent?
4x + 6y - 11 = 0
2x + ky - 7 = 0
Solution
The given system of equation may be written as
4x + 6y - 11 = 0
2x + ky - 7 = 0

The system of equation is of the form

27. For what value of Î±, the system of equation
Î±x + 3y = Î± - 3
12x + Î±y = Î±
will have no solution ?
Solution
The given system of equation may be written as
Î±x + 3y - (Î± - 3) = 0
12x + Î±y - Î±  = 0
The system of equation is of the form

28. Find the value of k for which the system
kx + 2y = 5
3x + y = 1
has (i) a unique solution, and (ii) no solution.

Solution

The given system of equation may be written as
kx + 2y - 5 = 0
3x +y - 1 = 0
It is of the form

So, the given system of equations will have a unique solution , if k ≠ 6

(ii) The given system will have no solution, if

29. Prove that there is a value of c (≠0) for which the system
6x + 3y = c - 3
12x + cy = c
has infinitely many solutions. Find this value.

Solution

The given system of equation may be written as
6x + 3y - (c - 3)= 0
12x + cy - c = 0
This is of the form

30. Find the values of k for which the system
2x + ky = 1
3x - 5y = 7
will have (i) a unique solution, and (ii) no solution . Is there a value of k for which the system has infinitely many solutions ?
Solution

The given system of  equation may be written as
2x + ky - 1 = 0
3x - 5y - 7 = 0
It is of the form

31. For what value of k, the following system of equations will represent the coincident lines ?
x + 2y + 7 = 0
2x + ky + 14 = 0
Solution

The given system of equations may be written as
x + 2y + 7 = 0
2x + ky + 14 = 0

The given system of equations is of the form

32.  Obtain the condition for the following system  of linear equations to have a unique solution
ax + by = c
lx + my = n
Solution

The given system of equations may be written as

ax + by - c = 0
lx + my - n = 0
It is of the form

33. Determine the values of a and b so that the following system of linear equations have infinitely many solutions:
(2a - 1)x + 3y - 5 = 0
3x + (b-1)y - 2 = 0
Solution

The given system of equations may be written as
(2a - 1)x + 3y - 5= 0
3x +(b-1)y - 2 = 0
It is of the form

34. Find the values of a and b for which the following system of linear equations has infinite number of solutions :
2x - 3y = 7
(a+b)x - (a+ b - 3)y = 4a + b
Solution

The given system of equations may be written as
2x - 3y -7 = 0
(a+b)x - (a+ b - 3)y - (4a + b) = 0
It is of the form

35. Find the values of p and q for which the following system of linear equation has infinite number of solutions:
2x - 3y = 9
(p + q)x + (2p - q)y = 3(p + q + 1)

Solution

The given system of equations may be written as
2x - 3y - 9 = 0
(p + q)x + (2p - q)y - 3(p + q + 1) = 0
It is of the form

36. 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
2x + 3y - 7 = 0
(a-b)x + (a+ b)y - (3a + b - 2 ) = 0

(i) (2a - 1)x - 3y = 5
3x + (b - 2)y = 3
Solution

The given system of equations is

(2a - 1)x - 3y - 5 = 0
3x + (b - 2)y - 3 = 0
It is of the form

(ii) 2x - (2a + 5)y = 5
(2b + 1)x - 9y = 15
Solution

The given system of equations is

2x - (2a + 5)y - 5 = 0
(2b + 1)x - 9y - 15 = 0
It is of the form

(iii) (a -)x + 3y = 2
6x +(1 +2b)y = 6
Solution

The given system of equations is
(a -)x + 3y - 2 = 0
6x +(1 +2b)y  - 6  = 0
It is of the form
Hence, the given system of equations will have infinitely many solutions,
If a = 3 and b = -4.

(iv) 3x + 4y = 12
(a +b)x + 2(a -b)y = (5a -1)
Solution

The given system of equations is
3x + 4y - 12
(a +b)x + 2(a -b)y - (5a -1) = 0
It is of the form

(v) 2x + 3y = 7
(a - 1)x + (a + 1)y = (3a - 1)
Solution

The given system of equations is
2x + 3y - 7 = 0
(a - 1)x + (a + 1)y - (3a - 1) = 0
It is of the form

(vi) 2x + 3y = 7
(a-1)x + (a+ 2)y = 3a
Solution

The given system of equations is
2x + 3y - 7 = 0
(a-1)x + (a+ 2)y - 3a  = 0
It is of the form