Chapter 3: Pair of Linear Equations in Two Variables Exercise – 3.5

Question: 1

In each of the following systems of equation determine whether the system has a unique solution, no solution or infinite solutions. In case there is a unique solution,

x − 3y − 3 = 0,  x − 3y − 3 = 0

3x − 9y − 2 = 0, 3x − 9y − 2 = 0

Solution:

The given system may be written as

x − 3y − 3 = 0 

3x − 9y − 2 = 0

The given system of equation is of the form

a₂x + b₂y − c₂ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 1, b₁ = −3, c₁ = −3

a₂ = 3, b₂ = −9, c₂ = −2

We have,

Therefore, the given equation has no solution.

Question: 2

In each of the following systems of equation determine whether the system has a unique solution, no solution or infinite solutions. In case there is a unique solution,

2x + y − 5 = 0

4x + 2y − 10 = 0

Solution:

The given system may be written as

2x + y − 5 = 0 4x + 2y − 10 = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 2, b₁ = 1, c₁ = −5

a₂ = 4, b₂ = 2, c₂ = −10

We have,

Therefore, the given equation has infinitely many solution.

Question: 3

In each of the following systems of equation determine whether the system has a unique solution, no solution or infinite solutions. In case there is a unique solution,

3x − 5y = 20

6x − 10y = 40

Solution:

The given system may be written as

3x − 5y = 20 6x − 10y = 40

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 3, b₁ = −5, c₁ = − 20

a₂ = 6, b₂ = −10, c₂ = − 40

We have,

Therefore, the given equation has infinitely many solution.

Question: 4

In each of the following systems of equation determine whether the system has a unique solution, no solution or infinite solutions. In case there is a unique solution,

x − 2y − 8 = 0

5x − 10y − 10 = 0

Solution:

The given system may be written as

x − 2y − 8 = 0 5x − 10y − 10 = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 1, b₁ = −2, c₁ = −8

a₂ = 5, b₂ = −10, c₂ = −10

We have,

Therefore, the given equation has no solution.

Question: 5

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

kx + 2y − 5 = 0

3x + y − 1 = 0

Solution:

The given system may be written as

kx + 2y − 5 = 0 

3x + y − 1 = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 

a₂x + b₂y − c₂ = 0

Where, a₁ = k, b₁ = 2, c₁ = −5

a₂ = 3, b₂ = 1, c₂ = −1

For unique solution, we have

Therefore, the given system will have unique solution for all real values of k other than 6.

Question: 6

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

4x + ky + 8 = 0

2x + 2y + 2 = 0

Solution:

The given system may be written as

4x + ky + 8 = 0 2x + 2y + 2 = 0

The given system of equation is of the form

a₁x +b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 4, b₁ = k, c₁ = 8

a₂ = 2, b₂ = 2, c₂ = 2

For unique solution, we have

Therefore, the given system will have unique solution for all real values of k other than 4.

Question: 7

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

4x − 5y = k

2x − 3y = 12

Solution:

The given system may be written as

4x − 5y − k = 0 

2x − 3y − 12 = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 4, b₁ = −5, c₁ = −k

a₂ = 2, b₂ = -3, c₂ = -12

For unique solution, we have

⇒ k can have any real values.

Therefore, the given system will have unique solution for all real values of k.

Question: 8

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

x + 2y = 3

5x + ky + 7 = 0

Solution:

The given system may be written as

x + 2y = 3 

5x + ky + 7 = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where a₁ = 1, b₁ = 2, c₁ = −3

a₂ = 5, b₂ = k, c₂ = 7

For unique solution, we have

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

Question: 9

Find the value of k for which each of the following system of equations having infinitely many solution: 

2x + 3y − 5 = 0

6x − ky − 15 = 0

Solution:

The given system may be written as

2x + 3y − 5 = 0 6x − ky − 15 = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 2, b₁ = 3, c₁ = −5

a₂ = 6, b₂ = k, c₂ = −15

For unique solution, we have

Therefore, the given system of equation will have infinitely many solutions, if k = 9.

Question: 10

Find the value of k for which each of the following system of equations having infinitely many solution:

4x + 5y = 3

x + 15y = 9

Solution:

The given system may be written as

4x + 5y = 3 

kx +15y = 9

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 4, b₁ = 5, c₁ = 3

a₂ = k, b₂ = 15, c₂ = 9

For unique solution, we have

Therefore, the given system will have infinitely many solutions if k = 12.

Question: 11

Find the value of k for which each of the following system of equations having infinitely many solution: 

kx − 2y + 6 = 0

4x + 3y + 9 = 0

Solution:

The given system may be written as

kx − 2y + 6 = 0 4x + 3y + 9 = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = k, b₁ = −2, c₁ = 6

a₂ = 4, b₂ = −3, c₂ = 9

For unique solution, we have

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

Question: 12

Find the value of k for which each of the following system of equations having infinitely many solution:

8x + 5y = 9

kx + 10y = 19

Solution:

The given system may be written as

8x + 5y = 9 kx + 10y = 19

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ =0

Where, a₁ = 8, b₁ = 5, c₁ = −9

a₂ = k, b₂ = 10, c₂ = −18 a₂ = k, b₂ = 10,c₂ = −18

For unique solution, we have

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

Question: 13

Find the value of k for which each of the following system of equations having infinitely many solution:

2x − 3y = 7

 (k + 2)x − (2k + 1)y = 3(2k − 1)

Solution:

The given system may be written as

2x − 3y = 7 (k + 2)x − (2k + 1)y = 3(2k − 1)

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 2, b₁ = −3, c₁ = −7

a₂ = k, b₂ = − (2k + 1), c₂ = −3(2k − 1)

For unique solution, we have

⇒ 2(2k + 1) = 3(k + 2) and 3 × 3(2k − 1) = 7(2k + 1)

⇒ 4k + 2 = 3k + 6 and 18k − 9 = 14k + 7 

⇒ k = 4 and 4k = 16

⇒ k = 4

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

Question: 14

Find the value of k for which each of the following system of equations having infinitely many solution: 

2x + 3y = 2

 (k + 2)x + (2k + 1)y = 2(k − 1)

Solution:

The given system may be written as

2x + 3y = 2 (k + 2)x + (2k + 1)y = 2(k − 1)

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 2, b₁ = 3,c₁ = −2

a₂ = (k + 2), b₂ = (2k + 1),c₂ = −2(k − 1)

For unique solution, we have

⇒ 2(2k + 1) = 3(k + 2) and 3(k − 1) = (2k + 1)

⇒ 4k + 2 = 3k + 6 and 3k − 3 = 2k + 1

⇒ k = 4 and k = 4

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

Question: 15

Find the value of k for which each of the following system of equations having infinitely many solution: 

x + (k + 1)y = 4

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

Solution:

The given system may be written as

x + (k + 1)y = 4 (k + 1)x + 9y = (5k + 2)

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 1, b₁ = (k + 1), c₁ = −4

a₂ = (k + 1), b₂ = 9, c₂ = − (5k + 2)

For unique solution, we have

⇒ 9 = (k + 1)² and (k + 1)(5k + 2) = 36

⇒ 9 = k² + 2k + 1 and 5k² + 2k + 5k + 2 = 36

⇒ k² + 2k − 8 = 0 and 5k² + 7k − 34 = 0

⇒ k² + 4k − 2k − 8 = 0 and 5k² + 17k − 10k − 34 = 0

⇒ k(k + 4) −2 (k + 4) = 0 and (5k + 17) − 2 (5k + 17) = 0

⇒ (k + 4)(k − 2) = 0 and (5k + 17)(k − 2) = 0

⇒ k = - 4 or k = 2 and k = -17/5 or k = 2

Thus, k = 2 satisfies both the condition.

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

Question: 16

Find the value of k for which each of the following system of equations having infinitely many solution:

kx + 3y = 2k + 1

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

Solution:

The given system may be written as

kx + 3y = 2k + 1 2(k + 1)x + 9y = (7k + 1)

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = k, b₁ = 3, c₁ = −(2k + 1)

a₂ = 2(k + 1), b₂ = 9, c₂ = −(7k + 1)

For unique solution, we have

⇒ 9k = 3 × 2(k + 1) and 3(7k + 1) = 9(2k + 1)

⇒ 9k − 6k = 6 and 21k − 18k = 9 − 3

⇒ 3k = 6 ⇒ k = 2 and k = 2

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

Question: 17

Find the value of k for which each of the following system of equations having infinitely many solution:

2x + (k − 2)y = k

6x + (2k − 1)y = (2k + 5)

Solution:

The given system may be written as

2x +( k − 2)y = k 6x + (2k − 1)y = (2k + 5)

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where,  a₁ = 2,b₁ = (k − 2),c₁ = −k

a₂ = 6,b₂ = (2k − 1),c₂ = −(2k + 5)

For unique solution, we have

⇒ 2k − 3k = −6 + 1 and k + k = 10

⇒ −k = −5 and 2k = 10 ⇒ k = 5 and k = 5

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

Question: 18

Find the value of k for which each of the following system of equations having infinitely many solution:

2x + 3y = 72x + 3y = 7

(k + 1)x + (2k − 1)y = (4k + 1)

Solution:

The given system may be written as

2x + 3y = 7 (k + 1)x + (2k − 1)y = (4k + 1)

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where,  a₁ = 2, b₁ = 3, c₁ = −7

a₂ = k + 1, b₂ = 2k − 1, c₂ = −(4k + 1)

For unique solution, we have

Extra close brace or missing open brace

⇒ 4k − 2 = 3k + 3 and 12k + 3 = 14k − 7

⇒ k = 5 and 2k = 10 ⇒ k = 5 and k = 5

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

Question: 19

Find the value of k for which each of the following system of equations having infinitely many solution:

2x + 3y = k

(k − 1)x + (k + 2)y = 3k

Solution:

The given system may be written as

2x + 3y = k (k − 1)x + (k + 2)y = 3k

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 2,b₁ = 3, c₁ = −k

a₂ = k − 1, b₂ = k + 2, c₂ = −3k

For unique solution, we have

Extra close brace or missing open brace

⇒ 2k + 4 = 3k − 3 and 9 = k + 2 ⇒ 2k + 4 = 3k − 3 and 9 = k + 2 ⇒ k = 7 and k = 7

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

Question: 20

Find the value of k for which the following system of equation has no solution:

kx − 5y = 2

6x + 2y = 7

Solution:

The given system may be written as

kx − 5y = 2 6x + 2y = 7

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = k, b₁ = −5, c₁ = −2

a₂ = 6 b₂ = 2, c₂ = −7

For no solution, we have

⇒ 2k = -30 ⇒ k = -15

Therefore, the given system of equations will have no solutions, if k = −15.

Question: 21

Find the value of k for which the following system of equation has no solution:

x + 2y = 0

2x + ky = 5

Solution:

The given system may be written as

x₂y = 0 2x + ky = 5

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 1, b₁ = 2, c₁ = 0

a₂ = 2, b₂ = k, c₂ = −5

For no solution, we have

⇒ k = 4

Therefore, the given system of equations will have no solutions, if k = 4.

Question: 32

Find the value of k, for which the following system of equation have unique solution.

ax + by = c

lx + my = n

Solution:

The given system may be written as

ax + by − c = 0 lx + my − n = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = a, b₁ = b, c₁ = − c

a₂ = l, b₂ = m, c₂ = − n

For unique solution, we have

Therefore, the given system of equations will have unique solution, if am ≠ bl.

Question: 33

Find the value of a and b such that the following system of linear equation have infinitely many solution:

 (2a − 1)x + 3y − 5 = 0

3x + (b − 1)y − 2 = 0

Solution:

The given system of equation may be written as,

(2a − 1)x + 3y − 5 = 0 3x + (b − 1)y − 2 = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = (2a − 1), b₁ = 3, c₁ = −5

a₂ = 3, b₂ = b − 1, c₂ = −2

The given system of equation will have infinitely many solution, if

⇒ 2(2a − 1) = 15 and 6 = 5(b − 1)

⇒ 4a − 2 = 15 and 6 = 5b − 5 ⇒ 4a = 17 and 5b = 11

Question: 34

Find the value of a and b such that the following system of linear equation have infinitely many solution:

2x − 3y = 7

(a + b)x − (a + b − 3)y = 4a + b

Solution:

The given system of equation may be written as,

2x − 3y − 7 = 0 (a + b)x − (a + b − 3)y − (4a + b) = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 2, b₁ = −3, c₁ = −7

a₂ = (a + b), b₂ = −(a + b − 3), c₂ = −(4a + b)

The given system of equation will have infinitely many solution, if

⇒ 2(a + b − 3) = 3(a + b) and 3(4a + b) = 7(a + b − 3)

⇒ 2a + 2b − 6 = 3a + 3b and 12a + 3b = 7a + 7b − 21

⇒ a + b = −6 and 5a − 4b = −21

⇒ a = − 6 − b

Substituting the value of a in 5a − 4b = −21 we have

5( –  b – 6) – 4b = – 21

⇒ − 5b − 30 − 4b = − 21

⇒ 9b = − 9 ⇒ b = −1

As a = – 6 – b

⇒ a = − 6 + 1 = − 5

Hence the given system of equation will have infinitely many solution if

a = – 5 and b = –1.

Question: 35

Find the value of p and q such that the following system of linear equation have infinitely many solution:

2x − 3y = 9

(p + q)x + (2p − q)y = 3(p + q + 1)

Solution:

The given system of equation may be written as,

2x − 3y − 9 = 0 (p + q)x + (2p − q)y − 3(p + q + 1) = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ =2, b₁ = 3, c₁ = −9

a₂ = (p + q), b₂ = (2p − q), c₂ = -3(p + q + 1)

The given system of equation will have infinitely many solution, if

 2(2p - q) = 3(p + q) and (p + q + 1) = 2p - q

⇒ 4p - 2q = 3p + 3q and -p + 2q = -1

⇒ p = 5q and p - 2q = 1

Substituting the value of p in p - 2q = 1, we have

3q = 1

⇒ q = 1/3

Substituting the value of p in p = 5q we have

p = 5/3

Hence the given system of equation will have infinitely many solution if

p = 5/3 and q = 1/3.

Question: 36

Find the values of a and b for which the following system of equation has infinitely many solution:

(i)  (2a − 1)x + 3y = 5

3x + (b − 2)y = 3

(ii)  2x − (2a + 5)y = 5

(2b + 1)x − 9y = 15

(iii) (a − 1)x + 3y = 2

6x + (1 − 2b)y = 6

(iv) 3x + 4y = 12

(a + b)x + 2(a − b)y = 5a – 1

(v) 2x + 3y = 7

(a − 1)x + (a + 1)y = 3a − 1

(vi)  2x + 3y = 7

(a − 1)x + (a + 2)y = 3a

Solution:

(i) The given system of equation may be written as,

 (2a − 1)x + 3y − 5 = 0 3x + (b − 2)y − 3 = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 2a − 1, b₁ = 3, c₁ = −5

a₂ = 3, b₂ = b − 2, c₂ = -3(p + q + 1)

The given system of equation will have infinitely many solution, if

2a - 1 = 5 and - 9 = 5(b - 2)

⇒ a = 3 and -9 = 5b - 10 

a = 3 and b = 1/5

Hence the given system of equation will have infinitely many solution if

a = 3 and b = 1/5.

(ii)  The given system of equation may be written as,

2x − (2a + 5)y = 5 (2b + 1)x − 9y = 15

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 2, b₁ = - (2a + 5), c₁ = −5

a₂ = (2b + 1), b₂ = −9, c₂ = −15

The given system of equation will have infinitely many solution, if

Hence the given system of equation will have infinitely many solution if

a = - 1 and b = 5/2.

(iii) The given system of equation may be written as,

(a − 1)x + 3y = 2 6x + (1 − 2b)y = 6

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = a-1, b₁ = 3, c₁ = −2

a₂ = 6, b₂ = 1 − 2b, c₂ = −6

The given system of equation will have infinitely many solution, if

⇒ a – 1 = 2 and 1 – 2b = 9

⇒ a - 1 = 2 and 1 - 2b = 9 

⇒ a = 3 and b = -4

⇒ a = 3 and b = -4

Hence the given system of equation will have infinitely many solution if

a = 3 and b = −4.

(iv) The given system of equation may be written as,

3x + 4y − 12 = 0 (a + b)x + 2(a − b)y − (5a − 1) = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 3, b₁ = 4, c₁ = −12

a₂ = (a + b), b₂ = 2(a − b), c₂ = – (5a − 1)

The given system of equation will have infinitely many solution, if

⇒ 3(a - b) = 2a + 2b and 2(5a - 1) = 12(a - b)

⇒ a = 5b and -2a = -12b + 2

Substituting a = 5b in -2a = -12b + 2, we have

-2(5b) = -12b + 2

⇒ −10b = −12b + 2 ⇒ b = 1

Thus a = 5

Hence the given system of equation will have infinitely many solution if

a = 5 and b = 1.

(v) The given system of equation may be written as,

2x + 3y − 7 = 0 (a − 1)x + (a + 1)y − (3a − 1) = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 2, b₁ = 3, c₁ = −7

a₂ = (a − 1), b₂ = (a + 1), c₂ = - (3a − 1)

The given system of equation will have infinitely many solution, if

⇒ 2(a + 1) = 3(a - 1) and 3(3a - 1) = 7(a + 1)

⇒ 2a - 3a = -3 - 2 and 9a - 3 = 7a + 7

⇒ a = 5 and a = 5

Hence the given system of equation will have infinitely many solution if

a = 5 and b = 1.

(vi) The given system of equation may be written as,

2x + 3y − 7 = 0 (a − 1)x + (a + 2)y − 3a = 0

The given system of equation is of the form

a₁x + b₁y − c₁ = 0 a₂x + b₂y − c₂ = 0

Where, a₁ = 2, b₁ = 3, c₁ = −7

a₂ = (a − 1), b₂ = (a + 2), c₂ = −3a

The given system of equation will have infinitely many solution, if

⇒ 2(a + 2) = 3(a - 1) and 3(3a) = 7(a + 2)

⇒ 2a + 4 = 3a - 3 and 9a = 7a + 14

⇒ a = 7 and a = 7

Hence the given system of equation will have infinitely many solution if

a = 7and b = 1.