RD Sharma Solutions Class 12 Solution Of Simultaneous Linear Equations Exercise 8.1

For students who wish to excel in their exams and build their confidence level, practising RD Sharma Class 12 textbook is essential. On regular practice, students can speed up the method of solving problems by using short-cut tips to secure high marks in their examination. Students can download the pdf of RD Sharma Solutions for Class 12 Exercise 8.1 of Chapter 8 Solution of Simultaneous Linear Equations from the links provided below. This exercise mainly focused on the homogeneous and non-homogeneous system of linear equations. Let us have a look at some of the topics discussed in this exercise.

  • Definition and meaning of consistent system
  • Homogeneous and non-homogeneous systems
  • Matrix method for the solution of a non-homogeneous system
  • Solving the given system of linear equations when the coefficient matrix is non-singular
  • Solving the given system of equations when the coefficient matrix is singular
  • Solving a system of linear equations when the inverse of the coefficient matrix is obtained

RD Sharma Solutions For Class 12 Chapter 8 – Solution of Simultaneous Linear Equations Exercise 8.1:-Download PDF Here

RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations Ex8.1
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Access another exercise of RD Sharma Solutions For Class 12 Chapter 8 – Solution of Simultaneous Linear Equations

Exercise 8.2 Solutions

Access answers to Maths RD Sharma Solutions For Class 12 Chapter 8 – Solution of Simultaneous Linear Equations Exercise 8.1

Exercise 8.1 Page No: 8.14

1. Solve the following system of equations by matrix method:

(i) 5x + 7y + 2 = 0

4x + 6y + 3 = 0

(ii) 5x + 2y = 3

3x + 2y = 5

(iii) 3x + 4y – 5 = 0

x – y + 3 = 0

(iv) 3x + y = 19

3x – y = 23

(v) 3x + 7y = 4

x + 2y = -1

(vi) 3x + y = 7

5x + 3y = 12

Solution:

(i) Given 5x + 7y + 2 = 0 and 4x + 6y + 3 = 0

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 1
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 2

Hence, x = 9/2 and y = -7/2

(ii) Given 5x + 2y = 3

3x + 2y = 5

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 3
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 4

Hence, x = -1 and y = 4

(iii) Given 3x + 4y – 5 = 0

x – y + 3 = 0

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 5
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 6
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 7

Hence, X = 1 Y = – 2

(iv) Given 3x + y = 19

3x – y = 23

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 8
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 9
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 10

(v) Given 3x + 7y = 4

x + 2y = -1

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 11
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 12

(vi) Given 3x + y = 7

5x + 3y = 12

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 13
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 14

2. Solve the following system of equations by matrix method:

(i) x + y –z = 3
2x + 3y + z = 10
3x – y – 7z = 1

(ii) x + y + z = 3

2x – y + z = -1

2x + y – 3z = -9

(iii) 6x – 12y + 25z = 4

4x + 15y – 20z = 3

2x + 18y + 15z = 10

(iv) 3x + 4y + 7z = 14

2x – y + 3z = 4

x + 2y – 3z = 0

(v) (2/x) – (3/y) + (3/z) = 10

(1/x) + (1/y) + (1/z) = 10

(3/x) – (1/y) + (2/z) = 13

(vi) 5x + 3y + z = 16

2x + y + 3z = 19

x + 2y + 4z = 25

(vii) 3x + 4y + 2z = 8

2y – 3z = 3

x – 2y + 6z = -2

(viii) 2x + y + z = 2

x + 3y – z = 5

3x + y – 2z = 6

(ix) 2x + 6y = 2

3x – z = -8

2x – y + z = -3

(x) 2y – z = 1

x – y + z = 2

2x – y = 0

(xi) 8x + 4y + 3z = 18

2x + y + z = 5

x + 2y + z = 5

(xii) x + y + z = 6

x + 2z = 7

3x + y + z = 12

(xiii) (2/x) + (3/y) + (10/z) = 4,

(4/x) – (6/y) + (5/z) = 1,

(6/x) + (9/y) – (20/z) = 2, x, y, z ≠ 0

(xiv) x – y + 2z = 7

3x + 4y – 5z = -5

2x – y + 3z = 12

Solution:

(i) Given x + y –z = 3

2x + 3y + z = 10

3x – y – 7z = 1

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 15

= (– 20) – 1(– 17) – 1(11)

= – 20 + 17 + 11 = 8

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 – 21 + 1 = – 20

C21 = (– 1)2 + 1 – 7 – 1 = 8

C31 = (– 1)3 + 1 1 + 3 = 4

C12 = (– 1)1 + 2 – 14 – 3 = 17

C22 = (– 1)2 + 1 – 7 + 3 = – 4

C32 = (– 1)3 + 1 1 + 2 = – 3

C13 = (– 1)1 + 2 – 2 – 9 = – 11

C23 = (– 1)2 + 1 – 1 – 3 = 4

C33 = (– 1)3 + 1 3 – 2 = 1

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 16
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 17

(ii) Given x + y + z = 3

2x – y + z = -1

2x + y – 3z = -9

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 18

= (3 – 1) – 1(– 6 – 2) + 1(2 + 2)

= 2 + 8 + 4

= 14

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 3 – 1 = 2

C21 = (– 1)2 + 1 – 3 – 1 = 4

C31 = (– 1)3 + 1 1 + 1 = 2

C12 = (– 1)1 + 2 – 6 – 2 = 8

C22 = (– 1)2 + 1 – 3 – 2 = – 5

C32 = (– 1)3 + 1 1 – 2 = 1

C13 = (– 1)1 + 2 2 + 2 = 4

C23 = (– 1)2 + 1 1 – 2 = 1

C33 = (– 1)3 + 1 – 1 – 2 = – 3

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 19

(iii) Given 6x – 12y + 25z = 4

4x + 15y – 20z = 3

2x + 18y + 15z = 10

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 20

= 6(225 + 360) + 12(60 + 40) + 25(72 – 30)

= 3510 + 1200 + 1050

= 5760

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 (225 + 360) = 585

C21 = (– 1)2 + 1 (– 180 – 450) = 630

C31 = (– 1)3 + 1 (240 – 375) = – 135

C12 = (– 1)1 + 2 (60 + 40) = – 100

C22 = (– 1)2 + 1 (90 – 50) = 40

C32 = (– 1)3 + 1 (– 120 – 100) = 220

C13 = (– 1)1 + 2 (72 – 30) = 42

C23 = (– 1)2 + 1(108 + 24) = – 132

C33 = (– 1)3 + 1 (90 + 48) = 138

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 21

(iv) Given 3x + 4y + 7z = 14

2x – y + 3z = 4

x + 2y – 3z = 0

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 22

= 3(3 – 6) – 4(– 6 – 3) + 7(4 + 1)

= – 9 + 36 + 35

= 62

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 3 – 6 = – 3

C21 = (– 1)2 + 1 – 12 – 14 = 26

C31 = (– 1)3 + 112 + 7 = 19

C12 = (– 1)1 + 2 – 6 – 3 = 9

C22 = (– 1)2 + 1 – 3 – 7 = – 10

C32 = (– 1)3 + 1 9 – 14 = 5

C13 = (– 1)1 + 2 4 + 1 = 5

C23 = (– 1)2 + 1 6 – 4 = – 2

C33 = (– 1)3 + 1 – 3 – 8 = – 11

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 23
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 24

(v) Given (2/x) – (3/y) + (3/z) = 10

(1/x) + (1/y) + (1/z) = 10

(3/x) – (1/y) + (2/z) = 13

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 25

= 5(4 – 6) – 3(8 – 3) + 1(4 – 2)

= – 10 – 15 + 3

= – 22

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 (4 – 6) = – 2

C21 = (– 1)2 + 1(12 – 2) = – 10

C31 = (– 1)3 + 1(9 – 1) = 8

C12 = (– 1)1 + 2 (8 – 3) = – 5

C22 = (– 1)2 + 1 20 – 1 = 19

C32 = (– 1)3 + 1 15 – 2 = – 13

C13 = (– 1)1 + 2 (4 – 2) = 2

C23 = (– 1)2 + 1 10 – 3 = – 7

C33 = (– 1)3 + 1 5 – 6 = – 1

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 26

(vi) Given 5x + 3y + z = 16

2x + y + 3z = 19

x + 2y + 4z = 25

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 27

= 3(12 – 6) – 4(0 + 3) + 2(0 – 2)

= 18 – 12 – 4

= 2

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 (12 – 6) = 6

C21 = (– 1)2 + 1(24 + 4) = – 28

C31 = (– 1)3 + 1(– 12 – 4) = – 16

C12 = (– 1)1 + 2 (0 + 3) = – 3

C22 = (– 1)2 + 1 18 – 2 = 16

C32 = (– 1)3 + 1 – 9 – 0 = 9

C13 = (– 1)1 + 2 (0 – 2) = – 2

C23 = (– 1)2 + 1 (– 6 – 4) = 10

C33 = (– 1)3 + 1 6 – 0 = 6

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 28
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 29

(vii) Given 3x + 4y + 2z = 8

2y – 3z = 3

x – 2y + 6z = -2

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 30

= 2(– 6 + 1) – 1(– 2 + 3) + 1(1 – 9)

= – 10 – 1 – 8

= – 19

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 – 6 + 1 = – 5

C21 = (– 1)2 + 1(24 + 4) = – 28

C31 = (– 1)3 + 1 – 1 – 3 = – 4

C12 = (– 1)1 + 2 – 2 + 3 = – 1

C22 = (– 1)2 + 1 – 4 – 3 = – 7

C32 = (– 1)3 + 1 – 2 – 1 = 3

C13 = (– 1)1 + 21 – 9 = – 8

C23 = (– 1)2 + 12 – 3 = – 1

C33 = (– 1)3 + 1 6 – 1 = 5

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 31

(viii) Given 2x + y + z = 2

x + 3y – z = 5

3x + y – 2z = 6

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 32

= 2(– 6 + 1) – 1(– 2 + 3) + 1(1 – 9)

= – 10 – 1 – 8

= – 19

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 – 6 + 1 = – 5

C21 = (– 1)2 + 1(24 + 4) = – 28

C31 = (– 1)3 + 1 – 1 – 3 = – 4

C12 = (– 1)1 + 2 – 2 + 3 = – 1

C22 = (– 1)2 + 1 – 4 – 3 = – 7

C32 = (– 1)3 + 1 – 2 – 1 = 3

C13 = (– 1)1 + 21 – 9 = – 8

C23 = (– 1)2 + 12 – 3 = – 1

C33 = (– 1)3 + 1 6 – 1 = 5

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 33
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 34

(ix) Given 2x + 6y = 2

3x – z = -8

2x – y + z = -3

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 35

= 2(0 – 1) – 6(3 + 2)

= – 2 – 30

= – 32

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 0 – 1 = – 1

C21 = (– 1)2 + 16 + 0 = – 6

C31 = (– 1)3 + 1 – 6 = – 6

C12 = (– 1)1 + 2 3 + 2 = 5

C22 = (– 1)2 + 1 2 – 0 = 2

C32 = (– 1)3 + 1 – 2 – 0 = 2

C13 = (– 1)1 + 2 – 3 – 0 = – 3

C23 = (– 1)2 + 1 – 2 – 12 = 14

C33 = (– 1)3 + 1 0 – 18 = – 18

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 36

(x) Given 2y – z = 1

x – y + z = 2

2x – y = 0

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 37

= 0 + 4 – 1

= 3

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 1 – 0 = 1

C21 = (– 1)2 + 11 – 2 = 1

C31 = (– 1)3 + 10 + 1 = 1

C12 = (– 1)1 + 2 – 2 – 0 = 2

C22 = (– 1)2 + 1 – 1 – 0 = – 1

C32 = (– 1)3 + 1 0 – 2 = 2

C13 = (– 1)1 + 2 4 – 0 = 4

C23 = (– 1)2 + 1 2 – 0 = – 2

C33 = (– 1)3 + 1 – 1 + 2 = 1

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 38
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 39

(xi) Given 8x + 4y + 3z = 18

2x + y + z = 5

x + 2y + z = 5

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 40

= 8(– 1) – 4(1) + 3(3)

= – 8 – 4 + 9

= – 3

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 1 – 2 = – 1

C21 = (– 1)2 + 1 4 – 6 = 2

C31 = (– 1)3 + 1 4 – 3 = 1

C12 = (– 1)1 + 2 2 – 1 = – 1

C22 = (– 1)2 + 1 8 – 3 = 5

C32 = (– 1)3 + 1 8 – 6 = – 2

C13 = (– 1)1 + 2 4 – 1 = 3

C23 = (– 1)2 + 1 16 – 4 = – 12

C33 = (– 1)3 + 1 8 – 8 = 0

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 41

(xii) Given x + y + z = 6

x + 2z = 7

3x + y + z = 12

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 42
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 43

= 1(– 2) – 1(1 – 6) + 1(1)

= – 2 + 5 + 1

= 4

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 0 – 2 = – 2

C21 = (– 1)2 + 1 1 – 1 = 0

C31 = (– 1)3 + 1 2 – 0 = 2

C12 = (– 1)1 + 2 1 – 6 = 5

C22 = (– 1)2 + 1 1 – 3 = – 2

C32 = (– 1)3 + 1 2 – 1 = – 1

C13 = (– 1)1 + 2 1 – 0 = 1

C23 = (– 1)2 + 1 1 – 3 = 2

C33 = (– 1)3 + 1 0 – 1 = – 1

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 44
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 45

(xiii) Given (2/x) + (3/y) + (10/z) = 4,

(4/x) – (6/y) + (5/z) = 1,

(6/x) + (9/y) – (20/z) = 2, x, y, z ≠ 0

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 46

AX = B

Now,

|A| = 2(75) – 3(– 110) + 10(72)

= 150 + 330 + 720

= 1200

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 120 – 45 = 75

C21 = (– 1)2 + 1 – 60 – 90 = 150

C31 = (– 1)3 + 1 15 + 60 = 75

C12 = (– 1)1 + 2 – 80 – 30 = 110

C22 = (– 1)2 + 1 – 40 – 60 = – 100

C32 = (– 1)3 + 1 10 – 40 = 30

C13 = (– 1)1 + 2 36 + 36 = 72

C23 = (– 1)2 + 1 18 – 18 = 0

C33 = (– 1)3 + 1 – 12 – 12 = – 24

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 47
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 48

(xiv) Given x – y + 2z = 7

3x + 4y – 5z = -5

2x – y + 3z = 12

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 49

A X = B

Now,

|A| = 1(12 – 5) + 1(9 + 10) + 2(– 3 – 8)

= 7 + 19 – 22

= 4

So, the above system has a unique solution, given by

X = A – 1B

Cofactors of A are

C11 = (– 1)1 + 1 12 – 5 = 7

C21 = (– 1)2 + 1 – 3 + 2 = 1

C31 = (– 1)3 + 1 5 – 8 = – 3

C12 = (– 1)1 + 2 9 + 10 = – 19

C22 = (– 1)2 + 1 3 – 4 = – 1

C32 = (– 1)3 + 1 – 5 – 6 = 11

C13 = (– 1)1 + 2 – 3 – 8 = – 11

C23 = (– 1)2 + 1 – 1 + 2 = – 1

C33 = (– 1)3 + 1 4 + 3 = 7

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 50

3. Show that each one of the following systems of linear equations is consistent and also find their solutions:

(i) 6x + 4y = 2

9x + 6y = 3

(ii) 2x + 3y = 5

6x + 9y = 15

(iii) 5x + 3y + 7z = 4

3x + 26y + 2z = 9

7x + 2y + 10z = 5

(v) x + y + z = 6

x + 2y + 3z = 14

x + 4y + 7z = 30

(vi) 2x + 2y – 2z = 1

4x + 4y – z = 2

6x + 6y + 2z = 3

Solution:

(i) Given 6x + 4y = 2

9x + 6y = 3

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 51

|A| = 36 – 36 = 0

So, A is singular, Now X will be consistence if (Adj A) x B = 0

C11 = (– 1)1 + 1 6 = 6

C12 = (– 1)1 + 2 9 = – 9

C21 = (– 1)2 + 1 4 = – 4

C22 = (– 1)2 + 2 6 = 6

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 52
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 53

(ii) Given 2x + 3y = 5

6x + 9y = 15

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 54

|A| = 18 – 18 = 0

So, A is singular,

Now X will be consistence if (Adj A) x B = 0

C11 = (– 1)1 + 1 9 = 9

C12 = (– 1)1 + 2 6 = – 6

C21 = (– 1)2 + 1 3 = – 3

C22 = (– 1)2 + 2 2 = 2

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 55
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 56

(iii) Given 5x + 3y + 7z = 4

3x + 26y + 2z = 9

7x + 2y + 10z = 5

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 57

|A| = 5(260 – 4) – 3(30 – 14) + 7(6 – 182)

= 5(256) – 3(16) + 7(176)

|A| = 0

So, A is singular. Thus, the given system is either inconsistent or it is consistent with infinitely many solution according to as:

(Adj A) x B≠0 or (Adj A) x B = 0

Cofactors of A are

C11 = (– 1)1 + 1 260 – 4 = 256

C21 = (– 1)2 + 1 30 – 14 = – 16

C31 = (– 1)3 + 1 6 – 182 = – 176

C12 = (– 1)1 + 2 30 – 14 = – 16

C22 = (– 1)2 + 1 50 – 49 = 1

C32 = (– 1)3 + 1 10 – 21 = 11

C13 = (– 1)1 + 2 6 – 182 = – 176

C23 = (– 1)2 + 1 10 – 21 = 11

C33 = (– 1)3 + 1 130 – 9 = 121

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 58

Now, AX = B has infinite many solution

Let z = k

Then, 5x + 3y = 4 – 7k

3x + 26y = 9 – 2k

This can be written as

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 59

(v) Given x + y + z = 6

x + 2y + 3z = 14

x + 4y + 7z = 30

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 60

|A| = 1(2) – 1(4) + 1(2)

= 2 – 4 + 2

|A| = 0

So, A is singular. Thus, the given system is either inconsistent or it is consistent with infinitely many solution according to as:

(Adj A) x B≠0 or (Adj A) x B = 0

Cofactors of A are

C11 = (– 1)1 + 1 14 – 12 = 2

C21 = (– 1)2 + 1 7 – 4 = – 3

C31 = (– 1)3 + 1 3 – 2 = 1

C12 = (– 1)1 + 2 7 – 3 = – 4

C22 = (– 1)2 + 1 7 – 1 = 6

C32 = (– 1)3 + 1 3 – 1 = 2

C13 = (– 1)1 + 2 4 – 2 = 2

C23 = (– 1)2 + 1 4 – 1 = – 3

C33 = (– 1)3 + 1 2 – 1 = 1

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 61

Now, AX = B has infinite many solution

Let z = k

Then, x + y = 6 – k

x + 2y = 14 – 3k

This can be written as:

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 62

(vi) Given x + y + z = 6

x + 2y + 3z = 14

x + 4y + 7z = 30

This can be written as

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 63

|A| = 2(14) – 2(14) – 2(0)

|A| = 0

So, A is singular. Thus, the given system is either inconsistent or it is consistent

with infinitely many solution according to as:

(Adj A) x B≠0 or (Adj A) x B = 0

Cofactors of A are:

C11 = (– 1)1 + 18 + 6 = 14

C21 = (– 1)2 + 1 4 + 12 = – 16

C31 = (– 1)3 + 1 – 2 + 8 = 6

C12 = (– 1)1 + 2 8 + 6 = – 14

C22 = (– 1)2 + 1 4 + 12 = 16

C32 = (– 1)3 + 1 – 2 + 8 = – 6

C13 = (– 1)1 + 2 24 – 24 = 0

C23 = (– 1)2 + 1 12 – 12 = 0

C33 = (– 1)3 + 1 8 – 8 = 0

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 64

4. Show that each one of the following systems of linear equations is consistent:

(i) 2x + 5y = 7

6x + 15y = 13

(ii) 2x + 3y = 5

6x + 9y = 10

(iii) 4x – 2y = 3

6x – 3y = 5

(iv) 4x – 5y – 2z = 2

5x – 4y + 2z = -2

2x + 2y + 8z = -1

(v) 3x – y – 2z = 2

2y – z = -1

3x – 5y = 3

(vi) x + y – 2z = 5

x – 2y + z = -2

-2x + y + z = 4

Solution:

(i) Given 2x + 5y = 7

6x + 15y = 13

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 65

|A| = 30 – 30 = 0

So, A is singular,

Now X will be consistence if (Adj A) x B = 0

C11 = (– 1)1 + 1 15 = 15

C12 = (– 1)1 + 2 6 = – 6

C21 = (– 1)2 + 1 5 = – 5

C22 = (– 1)2 + 2 2 = 2

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 66
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 67

(ii) Given 2x + 3y = 5

6x + 9y = 10

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 68

|A| = 18 – 18 = 0

So, A is singular,

Now X will be consistence if (Adj A) x B = 0

C11 = (– 1)1 + 1 9 = 9

C12 = (– 1)1 + 2 6 = – 6

C21 = (– 1)2 + 1 3 = – 3

C22 = (– 1)2 + 2 2 = 2

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 69

(iii) Given 4x – 2y = 3

6x – 3y = 5

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 70

|A| = – 12 + 12 = 0

So, A is singular,

Now X will be consistence if (Adj A) x B = 0

C11 = (– 1)1 + 1 – 3 = – 3

C12 = (– 1)1 + 2 6 = – 6

C21 = (– 1)2 + 1 – 2 = 2

C22 = (– 1)2 + 2 4 = 4

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 71

(iv) Given 4x – 5y – 2z = 2

5x – 4y + 2z = -2

2x + 2y + 8z = -1

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 72

|A| = 4(– 36) + 5(36) – 2(18)

|A| = 0

Cofactors of A are:

C11 = (– 1)1 + 1 – 32 – 4 = – 36

C21 = (– 1)2 + 1 – 40 + 4 = – 36

C31 = (– 1)3 + 1 – 10 – 8 = – 18

C12 = (– 1)1 + 2 40 – 4 = – 36

C22 = (– 1)2 + 1 32 + 4 = 36

C32 = (– 1)3 + 1 8 + 10 = – 18

C13 = (– 1)1 + 2 10 + 8 = 18

C23 = (– 1)2 + 1 8 + 10 = – 18

C33 = (– 1)3 + 1 – 16 + 25 = 9

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 73

(v) Given 3x – y – 2z = 2

2y – z = -1

3x – 5y = 3

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 74

|A| = 3(– 5) + 1(3) – 2(– 6)

|A| = 0

Cofactors of A are

C11 = (– 1)1 + 1 0 – 5 = – 5

C21 = (– 1)2 + 1 0 – 10 = 10

C31 = (– 1)3 + 1 1 + 4 = 5

C12 = (– 1)1 + 2 0 + 3 = – 3

C22 = (– 1)2 + 1 0 + 6 = 6

C32 = (– 1)3 + 1 – 3 – 0 = 3

C13 = (– 1)1 + 2 0 – 6 = – 6

C23 = (– 1)2 + 1 – 15 + 3 = 12

C33 = (– 1)3 + 1 6 – 0 = 6

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 75

(vi) Given x + y – 2z = 5

x – 2y + z = -2

-2x + y + z = 4

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 76

|A| = 1(– 3) – 1(3) – 2(– 3) = – 3 – 3 + 6

|A| = 0

Cofactors of A are:

C11 = (– 1)1 + 1 – 2 – 1 = – 3

C21 = (– 1)2 + 1 1 + 2 = – 3

C31 = (– 1)3 + 1 1 – 4 = – 3

C12 = (– 1)1 + 2 1 + 2 = – 3

C22 = (– 1)2 + 1 1 – 4 = – 3

C32 = (– 1)3 + 1 1 + 2 = – 3

C13 = (– 1)1 + 2 1 – 4 = – 3

C23 = (– 1)2 + 1 1 + 2 = – 3

C33 = (– 1)3 + 1 – 2 – 1 = – 3

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 77
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 78

x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 79
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 80
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 81

2x – 3y + 5z = 11, 3x + 2y – 4z = -5, x + y – 2z = -3.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 82

|A| = 2(0) + 3(– 2) + 5(1)

= – 1

Now, the cofactors of A

C11 = (– 1)1 + 1 – 4 + 4 = 0

C21 = (– 1)2 + 1 6 – 5 = – 1

C31 = (– 1)3 + 1 12 – 10 = 2

C12 = (– 1)1 + 2 – 6 + 4 = 2

C22 = (– 1)2 + 1 – 4 – 5 = – 9

C32 = (– 1)3 + 1 – 8 – 15 = 23

C13 = (– 1)1 + 2 3 – 2 = 1

C23 = (– 1)2 + 1 2 + 3 = – 5

C33 = (– 1)3 + 1 4 + 9 = 13

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 83
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 84
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 85
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 86

x + 2y + 5z = 10, x – y – z = -2, 2x + 3y – z = -11.

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 87

|A| = 1(1 + 3) + 2(– 1 + 2) + 5(3 + 2)

= 4 + 2 + 25

= 27

Now, the cofactors of A

C11 = (– 1)1 + 1 1 + 3 = 4

C21 = (– 1)2 + 1 – 2 – 15 = 17

C31 = (– 1)3 + 1 – 2 + 5 = 3

C12 = (– 1)1 + 2 – 1 + 2 = – 1

C22 = (– 1)2 + 1 – 1 – 10 = – 11

C32 = (– 1)3 + 1 – 1 – 5 = 6

C13 = (– 1)1 + 2 3 + 2 = 5

C23 = (– 1)2 + 1 3 – 4 = 1

C33 = (– 1)3 + 1 – 1 – 2 = – 3

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 88
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 89
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 90

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 91

|A| = 1(1 + 6) + 2(2 – 0) + 0

= 7 + 4

= 11

Now, the cofactors of A

C11 = (– 1)1 + 1 1 + 6 = 7

C21 = (– 1)2 + 1 – 2 – 0 = 2

C31 = (– 1)3 + 1 – 6 – 0 = – 6

C12 = (– 1)1 + 2 2 – 0 = – 2

C22 = (– 1)2 + 1 1 – 0 = 1

C32 = (– 1)3 + 1 3 – 0 = – 3

C13 = (– 1)1 + 2 – 4 – 0 = – 4

C23 = (– 1)2 + 1 – 2 – 0 = 2

C33 = (– 1)3 + 1 1 + 4 = 5

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 92
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 93
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 94

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 95

|A| = 3(3 – 0) + 4(2 – 5) + 2(0 – 3)

= 9 – 12 – 6

= – 9

Now, the cofactors of A

C11 = (– 1)1 + 1 3 – 0 = 3

C21 = (– 1)2 + 1 – 4 – 0 = 4

C31 = (– 1)3 + 1 – 20 – 6 = – 26

C12 = (– 1)1 + 2 2 – 5 = 3

C22 = (– 1)2 + 1 3 – 2 = 1

C32 = (– 1)3 + 1 15 – 4 = – 11

C13 = (– 1)1 + 2 0 – 3 = – 3

C23 = (– 1)2 + 1 0 + 4 = – 4

C33 = (– 1)3 + 1 9 + 8 = 17

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 96
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 97
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 98

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 99
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 100
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 101
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 102

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 103

|A| = 1(– 1 – 1) – 2(– 2 – 0) + 0

= – 2 + 4

= 2

Now, the cofactors of A

C11 = (– 1)1 + 1 – 1 – 1 = – 2

C21 = (– 1)2 + 1 2 – 0 = 2

C31 = (– 1)3 + 1 – 2 – 0 = – 2

C12 = (– 1)1 + 2 2 – 0 = – 2

C22 = (– 1)2 + 1 1 – 0 = 1

C32 = (– 1)3 + 1 – 1 – 0 = 1

C13 = (– 1)1 + 2 – 2 – 0 = – 2

C23 = (– 1)2 + 1 – 1 – 0 = 1

C33 = (– 1)3 + 1 – 1 + 4 = 3

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 104
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 105
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 106

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 107
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 108
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 109

9. The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.

Solution:

Let the numbers are x, y, z

x + y + z = 2 
…… (i)

Also, 2y + (x + z) + 1

x + 2y + z = 1 …… (ii)

Again,

x + z + 5(x) = 6

5x + y + z = 6 …… (iii)

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 110

A X = B

|A| = 1(1) – 1(– 4) + 1(– 9)

= 1 + 4 – 9

= – 4

Hence, the unique solution given by x = A – 1B

C11 = (– 1)1 + 1 (2 – 1) = 1

C12 = (– 1)1 + 2 (1 – 5) = 4

C13 = (– 1)1 + 3 (1 – 10) = – 9

C21 = (– 1)2 + 1 (1 – 1) = 0

C22 = (– 1)2 + 2 (1 – 5) = – 4

C23 = (– 1)2 + 3 (1 – 5) = 4

C31 = (– 1)3 + 1 (1 – 2) = – 1

C32 = (– 1)3 + 2 (1 – 1) = 0

C33 = (– 1)3 + 3 (2 – 1) = 1

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 111
RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 112

10. An amount of ₹10,000 is put into three investments at the rate of 10, 12 and 15% per annum. The combined incomes are ₹1310 and the combined income of first and second investment is ₹ 190 short of the income from the third. Find the investment in each using matrix method.

Solution:

Let the numbers are x, y, and z

x + y + z = 10,000 …… (i)

Also,

0.1x + 0.12y + 0.15z = 1310 …… (ii)

Again,

0.1x + 0.12y – 0.15z = – 190 …… (iii)

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 113

A X = B

|A| = 1(– 0.036) – 1(– 0.03) + 1(0)

= – 0.006

Hence, the unique solution given by x = A – 1B

C11 = – 0.036

C12 = 0.27

C13 = 0

C21 = 0.27

C22 = – 0.25

C23 = – 0.02

C31 = 0.03

C32 = – 0.05

C33 = 0.02

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 114

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