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
