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 
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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
The given system of equation is
kx + 2y  5= 0
3x + y  1 = 0
The system of equation is of the form
2x + 2y+2 = 0
2x  3y = 12
The given system of equation is
4x  5y  k = 0
2x  3y  12 = 0
The system of equation is of the form
5x + ky + 7 = 0
x + 2y  3 = 0
5x + ky + 7 = 0
The system of equation is of the form
6x  ky  15 = 0
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
The given system of equation is
4x + 5y  3 = 0
kx + 15y  9 = 0
The system of equation is of the form
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
The given system of equation is
8x + 5y  9 = 0
kx + 10y  18 = 0
The system of equation is of the form
(k + 2)x  (2k + 1)y  3(2k  1)
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)
2x + 3y  2 = 0
(k + 2)x + (2k + 1)y  2(k  1) = 0
The system of equation is of the form
(k + 1)x + 9y  (5k + 2)
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
2(k + 1)x + 9y  (7k + 1)
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.
6x + (2k  1)y  (2k + 5)
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)
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
(k  1)x + (k + 2)y  3k
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)
6x + 2y = 7
kx  5y = 2
6x + 2y = 7
condition for system of equations having no solution
2x + ky  5 = 0
x + 2y = 0
2x + ky  5 = 0
The system of equation is of the form
kx + 3y  5 = 0
3x  4y + 7 = 0
kx + 3y  5 = 0
The system of equation is of the form
3x + 2y  1 = 0
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
2x + ky  11 = 0
5x  7y  5 = 0
The system of equation is of the form
12x + ky = 6
kx + 3y = 3
12x + ky = 6
4x + 6y  11 = 0
2x + ky  7 = 0
4x + 6y  11 = 0
2x + ky  7 = 0
The system of equation is of the form
Î±x + 3y = Î±  3
12x + Î±y = Î±
will have no solution ?
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.
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
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.
The given system of equation may be written as
6x + 3y  (c  3)= 0
12x + cy  c = 0
This is of the form
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 ?
The given system of equation may be written as
2x + ky  1 = 0
3x  5y  7 = 0
It is of the form
x + 2y + 7 = 0
2x + ky + 14 = 0
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
ax + by = c
lx + my = n
The given system of equations may be written as
ax + by  c = 0
lx + my  n = 0
It is of the form
(2a  1)x + 3y  5 = 0
3x + (b1)y  2 = 0
The given system of equations may be written as
(2a  1)x + 3y  5= 0
3x +(b1)y  2 = 0
It is of the form
2x  3y = 7
(a+b)x  (a+ b  3)y = 4a + b
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)
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
2x + 3y = 7
(ab)x + (a+ b)y = 3a + b  2
(ab)x + (a+ b)y  (3a + b  2 ) = 0
3x + (b  2)y = 3
The given system of equations is
3x + (b  2)y  3 = 0
It is of the form
(2b + 1)x  9y = 15
The given system of equations is
(2b + 1)x  9y  15 = 0
It is of the form
6x +(1 +2b)y = 6
(a )x + 3y  2 = 0
6x +(1 +2b)y  6 = 0
It is of the form
If a = 3 and b = 4.
(a +b)x + 2(a b)y = (5a 1)
The given system of equations is
3x + 4y  12
(a +b)x + 2(a b)y  (5a 1) = 0
It is of the form
(a  1)x + (a + 1)y = (3a  1)
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
(a1)x + (a+ 2)y = 3a
The given system of equations is
2x + 3y  7 = 0
(a1)x + (a+ 2)y  3a = 0
It is of the form