Chapter 3 - Pair Of Linear Equations In Two Variables, RD Sharma Solutions - (Part-15) | RD Sharma Solutions for Class 10 Mathematics PDF Download

Page No 3.58

Q.27. Solve each of the following systems of equations by the method of cross-multiplication :

Ans. GIVEN:

To find: The solution of the systems of equation by the method of cross-multiplication:
Here we have the pair of simultaneous equation

By cross multiplication method we get

For y

Hence we get the value of x = b and y = -a

Q.28. Solve each of the following systems of equations by the method of cross-multiplication :

Ans. GIVEN:
 
To find: The solution of the systems of equation by the method of cross-multiplication:
Here we have the pair of simultaneous equation

By cross multiplication method we get

Hence we get the value of x = y = ab

Page No 3.73

Q.1. 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:
x − 3y = 3
3x − 9y = 2
Ans. GIVEN:
x - 3y = 3
3x - 9y = 2
To find: To determine whether the system has a unique solution, no solution or infinitely many solutions
We know that the system of equations
a₁x + b₁ y = c₁
a_{2 x +} b₂ y = c₂
For unique solution

For no solution

For infinitely many solution

Here,

Since  which meanshence the system of equation has no solution.
Hence the system of equation has no solution

Q.2. 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
2x + y = 5
4x + 2y = 10
Ans. GIVEN:
 2x + y = 5
4x + 2y = 10
To find: To determine whether the system has a unique solution, no solution or infinitely many solutions
We know that the system of equations
a₁ x + b₁ y = c₁
a₂ x ₊ b₂y = c<sub>2
For unique solution

For no solution

For infinitely many solution

Here,

Since  which means</sub> hence the system of equation has infinitely many solution.
Hence the system of equation has infinitely many solutions

Q.3. 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
3x − 5y = 20
6x − 10y = 40
Ans. GIVEN:
3x − 5y = 20
6x − 10y = 40
To find: To determine whether the system has a unique solution, no solution or infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c<sub>2
For unique solution

For no solution</sub>

For infinitely many solution

Here,

Since which means  hence the system of equation has infinitely many solution.
Hence the system of equation has infinitely many solutions

Q.4. 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
x − 2y = 8
5x − 10y = 10
Ans. GIVEN:
 x − 2y = 8
5x − 10y = 10
To find: To determine whether the system has a unique solution, no solution or infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c<sub>2
For unique solution

For no solution

For infinitely many solution 

Here,

Since which means</sub>hence the system of equation has no solution.
Hence the system of equation has no solution

Q.5. Find the value of k for which the following system of equations has a unique solution:
kx + 2y = 5
3x + y = 1
Ans.5. GIVEN:
kx + 2y = 5
3x + y = 1
To find: To determine to value of k for which the system has a unique solution.
We know that the system of equations
a₁x + b₁y = c₁
a₂ + b₂y = c₂
For unique solution

Here,

Hence for the system of equation has unique solution.

Q.6. Find the value of k for which the following system of equations has a unique solution:
4x +ky + 8 = 0
2x + 2y + 2 = 0
Ans. GIVEN:
4x +ky + 8 = 0
2x + 2y + 2 = 0
To find: To determine to value of k for which the system has a unique solution.
We know that the system of equations
a1x + b₁y = c₁
a₂x + b₂y = c₂
For unique solution

Here,

Hence for the system of equation has unique solution.

Q.7. Find the value of k for which the following system of equations has a unique solution:
4x − 5y = k
2x − 3y = 12
Ans.7. GIVEN:
4x − 5y = k
2x − 3y = 12
To find: To determine to value of k for which the system has a unique solution.
We know that the system of equations
a1x + b₁y = c₁
a₂x + b₂y = c₂
For unique solution

Here,

Hence already   for the system of equation to have unique solution but the value of k should be a real number
Hence for k = real number the system of equation has unique solution.

Q.8. Find the value of k for which the following system of equations has a unique solution:
x + 2y = 3
5x + ky + 7 = 0
Ans. GIVEN:
x + 2y = 3
5x + ky + 7 = 0
To find: To determine to value of k for which the system has a unique solution.
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂ y = c<sub>2
For unique solution

Here,

Hence for  the system of equation has unique solution</sub>

Q.9. Find the value of k for which each of the following system of equations have infinitely many solutions :
2x + 3y − 5 = 0
6x + ky − 15 = 0
Ans.9. GIVEN:
2x + 3y - 5 = 0
6x + ky - 15 = 0
To find: To determine for what value of k the system of equation has infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂ y = c_2
Here,

Hence for k = 9 the system of equation have infinitely many solutions

Q.10. Find the value of k for which each of the following system of equations have infinitely many solutions :
4x + 5y = 3
kx + 15y = 9
Ans. GIVEN:
4x + 5y = 3
kx + 15y = 9
To find: To determine for what value of k the system of equation has infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂
For infinitely many solution

Here,

Hence for k = 12 the system of equation have infinitely many solutions.

Q.11. Find the value of k for which each of the following system of equations have infinitely many solutions :
kx − 2y + 6 = 0
4x − 3y + 9 = 0
Ans. GIVEN:
kx − 2y + 6 = 0
4x − 3y + 9 = 0
To find: To determine for what value of k the system of equation has infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂
For infinitely many solution

Here,

Hence for  the system of equation have infinitely many solutions.

Q.12. Find the value of k for which each of the following system of equations have infinitely many solutions :
8x + 5y = 9
kx + 10y = 18
Ans. GIVEN:
8x + 5y = 9
kx + 10y = 18
To find: To determine for what value of k the system of equation has infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂
For infinitely many solution

Here,

Hence for k = 16 the system of equation have infinitely many solutions

Q.13. Find the value of k for which each of the following system of equations have infinitely many solutions :
2x − 3y = 7
(k + 2) x − (2k + 1)y = 3(2k − 1)
Ans. GIVEN:
2x − 3y = 7
(k + 2) x − (2k + 1)y = 3(2k − 1)
To find: To determine for what value of k the system of equation has infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c<sub>2
For infinitely many solution

Here

Consider the following

Now consider the following

Hence for k = 4 the system of equation have infinitely many solutions.</sub>

Q.14. Find the value of k for which each of the following system of equations have infinitely many solutions :
2x + 3y = 2
(k + 2)x + (2k + 1)y = 2(k − 1)
Ans. GIVEN:
 2x + 3y = 2
(k + 2)x + (2k + 1)y = 2(k − 1)
To find: To determine for what value of k the system of equation has infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂
For infinitely many solution

Here,

Consider the following for k

Now consider the following

Hence for k = 4 the system of equation have infinitely many solutions.

Q.15. Find the value of k for which each of the following system of equations have infinitely many solutions :
x + (k + 1) y = 4
(k + 1) x + 9 y = 5k + 2
Ans.
GIVEN:
x + (k + 1) y = 4
(k + 1) x + 9 y = 5k + 2
To find: To determine for what value of k the system of equation has infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂
For infinitely many solution

Here,

Hence for k = 2 the system of equation have infinitely many solutions.

Q.16. Find the value of k for which each of the following system of equations have infinitely many solutions :
kx + 3y = 2k + 1
2(k + 1)x + 9y = 7k + 1
Ans. GIVEN:
kx + 3y = 2k + 1
2(k + 1)x + 9y = 7k + 1
To find: To determine for what value of k the system of equation has infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂
For infinitely many solution

Here,

Consider the following relation to find k

Now consider the following

Hence for k = 2 the system of equation have infinitely many solutions

Q.17. Find the value of k for which each of the following system of equations have infinitely many solutions :
2x + (k − 2)y = k
6x + (2k − 1)y = 2k + 5
Ans. GIVEN:
2x + (k − 2)y = k
6x + (2k − 1)y = 2k + 5
To find: To determine for what value of k the system of equation has infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂
For infinitely many solution

Here,

Consider the following relation to find k

Now again consider the following

Hence for k = 5  the system of equation have infinitely many solutions

Q.18. Find the value of k for which each of the following system of equations have infinitely many solutions :
2x + 3y = 7
(k + 1)x + (2k − 1)y = 4k + 1
Ans. GIVEN:
2x + 3y = 7
(k + 1)x + (2k − 1)y = 4k + 1
a₁x + b₁y = c₁
a₂x + b₂y = c₂
For infinitely many solution

Here


Now again consider the following to find k

Hence for k = 5 the system of equation have infinitely many solutions

Q.19. Find the value of k for which each of the following system of equations have infinitely many solutions :
2x + 3y = k
(k − 1)x + (k + 2)y = 3k
Ans. GIVEN:
2x + 3y = k
(k − 1)x + (k + 2)y = 3k
To find: To determine for what value of k the system of equation has infinitely many solutions
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂
For infinitely many solution

Here,

Consider the following to find out k

Now again consider the following relation

So the common solution is 7
Hence for k = 7 the system of equation have infinitely many solutions

Q.20. Find the value of k for which each of the following system of equations have no solution :
kx − 5y = 2
6x + 2y = 7
Ans.
kx − 5y = 2
6x + 2y = 7
To find: To determine for what value of k the system of equation has no solution
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂
For no solution

Here,

Hence for k = - 15 the system of equation have infinitely many solutions.

Q.21. Find the value of k for which each of the following system of equations have no solution :
x + 2y = 0
2x + ky = 5
Ans.
GIVEN:
x + 2y = 0
2x + ky = 5
To find: To determine for what value of k the system of equation has no solution
We know that the system of equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂
For no solution

Here,

Hence for k = 4 the system of equation has no solution.