## Chapter 3 Pair of Linear Equations in Two Variables R.D. Sharma Solutions for Class 10th Math Exercise 3.5

**Exercise 3.5**

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. x - 3y - 3 = 0

3x - 9y - 2 = 0

**Solution**

The given system of equations may be written as

2. 2x + y - 5 = 0

4x + 2y - 10 = 0

**Solution**

The given system of equation may be written as

3. 3x - 5y = 20

6x - 10y = 40

**Solution**

4. x - 2y - 8 = 0

5x - 10y - 10 = 0

**Solution**

The given system of equation may be written as

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

5. kx + 2y - 5 = 0

3x + y - 1 = 0

**Solution**

The given system of equation is

6. 4x + ky + 8 = 0

2x + 2y + 2 = 0

**Solution**

7. 4x - 5y = k

2x - 3y = 12

**Solution**

The given system of equation is

So, the given system of equations will have a unique solution for all real values of k.

8. x + 2y = 3

5x + ky + 7 = 0

**Solution**

The given system of equation is

So, the given system of equations will have a unique solution for all real values of k other than 10.

Find the value of k for which each of the following systems of equations have infinitely many solution: (9-19)

9. 2x + 3y - 5 = 0

6x - ky - 15 = 0

**Solution**

The given system of equation is

Hence, the given system of equations will have infinitely many solutions, if 9.

10. 4x + 5y = 3

kx + 15y = 9

**Solution**

The given system of equation is

Hence, the given system of equations will have infinitely many solutions, if k = 12.

11. kx - 2y + 6 = 0

4x + 3y + 9 = 0

**Solution**

The given system of equation is

Hence, the given system of equations will have infinitely many solutions, if k = 8/3.

12. 8x + 5y = 9

kx + 10y = 18

**Solution**

The given system of equation is

Hence, the given system of equations will have infinitely many solutions, if k = 16

13. 2x - 3y = 7

(k+2) x - (2k + 1) y - 3 (2k-1)

**Solution**

The given system of equation may be written as

Hence, the given system of equations will have infinitely many solutions, if k = 4.

14. 2x + 3y = 2

(k+2) x + (2k+1) y - (k-1)

**Solution**

The given system of equation may be written as

Hence, the given system of equations will have infinitely many solutions, if k = 4 .

15. x = (k+1) y = 4

(k+1) x + 9y - (5k+2)

**Solution**

The given system of equation may be written as

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

16. kx + 3y - 2k + 1

2(k+1) x + 9y - (7k+1)

**Solution**

The given system of equation may be written as

17. 2x + (k-2) y = k

6x + (2k-1) y- (2k+5)

**Solution**

The given system of equation may be written as

Hence, the given system of equations will have infinitely many solutions, if k = 5.

18 . 2x + 3y = 7

(k+1) x + (2k-1) y - (4k+1)

**Solution**

The given system of equation may be written as

Hence, the given system of equations will have infinitely many solutions, if k = 5.

19. 2x + 3y = k

(k-1) x + (k+1) y - 3k

**Solution**

The given system of equation may be written as

Hence, the given system of equations will have infinitely many solutions, if k = 7.

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

20. kx - 5y = 2

6x + 2y = 7

**Solution**

Given

21. x + 2y = 0

2x + ky - 5 = 0

**Solution**

The given system of equation may be written as

Hence, the given system of equations has no solutions, when k = 4.

22. 3x - 4y + 7 = 0

kx + 3y - 5 = 0

**Solution**

The given system of equation may be written as

23. 2x - ky + 3 = 0

3x + 2y - 1 = 0

**Solution**

The given system of equation may be written as

24. 2x + ky = 11

5x - 7y = 5

**Solution**

The given system of equation is

25. kx + 3y = 3

12x + ky = 6

**Solution**

26. For what value of a, 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

Hence, the given system of equation is inconsistent, when k = 3

27. For what value of a, the system of equations

ax + 3y = a - 3

12x + ay = a

will have no solution ?

**Solution**

The given system of equation may be written as

ax + 3y - (a - 3) = 0

12x + ay - a = 0

Hence, the given system of equation will have no solution, if a = -6 .

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

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

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

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

Hence, the given system of equations will represent coincident lines, if k = 4

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

Hence, am ≠ bl is the required conidition.

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

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

Hence, the given system of equations will have infinitely many solutions, If a = -5 and b = -1 .

35. Find the values of p and q for which the following system of linear equations 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

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

(i) 2x = 3y = 7

(a-b)x + (a+b)y = 3a+b-2

**Solution**

Hence, the given system of equation will have infinitely many solutions,if a = 3 and b = 1/5.

(ii) 2x-(2a+5)y = 5

(2b + 1) x - 9y = 15

**Solution**

Hence, the given system of equations will have infinitely many solutions, If a = -1 and b = 5/2.

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

6x + (1+2b) y = 6

**Solution**

The given system of equations is

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

Putting b = 1 in a = 5b, we get

a = 5 × 1 = 5

Hence, the given system of equations will have infinitely many solutions, If a = 5 and b = 1.

(v) 2x + 3y = 7

(a-1) x + (a+1)y = (3a-1)

**Solution**

The given system of equations is

Hence, the given system of equations will have infinitely many solutions, If a = 5.

(vi) 2x + 3y = 7

(a-1) x + (a+2) y = 3a

**Solution**

The given system of equations is