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

Students can download RD Sharma Solutions for Class 12 Maths Chapter 8 Solution of Simultaneous Linear Equations from the below provided links. By practising these solutions they are able to get their doubts cleared instantly. RD Sharma Solutions are essential reference books to score high in Mathematics board exams as well as in the competitive exams. We mainly focus on providing answers, which are in easy to remember format, which further helps students to clearly understand and remember the formulae. This chapter of RD Sharma Solutions for Class 12 mainly focuses on the homogeneous and non-homogeneous system of equations. Let us have a look at some of the important concepts that are discussed in this chapter.

  • 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 
  • Applications of simultaneous linear equations
  • Solution of homogeneous system of linear equations
    • The determinant of the coefficient matrix is non-singular
    • The determinant of the coefficient matrix is singular

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

RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 1
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 2
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 3
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 4
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 5
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 6
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 7
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 8
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 9
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 10
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 11
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 12
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 13
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 14
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 15
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 16
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 17
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 18
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 19
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 20
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 21
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 22
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 23
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 24
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 25
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 26
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 27
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 28
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 29
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 30
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 31
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 32
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 33
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 34
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 35
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 36
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 37
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 38
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 39
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 40
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 41
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 42
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 43
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 44
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 45
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 46
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 47
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 48
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 49
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 50
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 51
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 52
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 53
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 54
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 55
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 56
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 57
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 58
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 59
RD Sharma Class 12 Maths Solutions Chapter 8 Solutions Of Simultaneous Linear Equations 60

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

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

Exercise 8.2 Page No: 8.20

Solve the following systems of homogeneous linear equations by matrix method:

1. 2x – y + z = 0

3x + 2y – z = 0

x + 4y + 3z = 0

Solution:

Given

2x – y + z = 0

3x + 2y – z = 0

X + 4y + 3z = 0

The system can be written as

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 115

A X = 0

Now, |A| = 2(6 + 4) + 1(9 + 1) + 1(12 – 2)

|A| = 2(10) + 10 + 10

|A| = 40 ≠ 0

Since, |A|≠ 0, hence x = y = z = 0 is the only solution of this homogeneous equation.

2. 2x – y + 2z = 0
5x + 3y – z = 0
X + 5y – 5z = 0

Solution:

Given 2x – y + 2z = 0

5x + 3y – z = 0

X + 5y – 5z = 0

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 116

A X = 0

Now, |A| = 2(– 15 + 5) + 1(– 25 + 1) + 2(25 – 3)

|A| = – 20 – 24 + 44

|A| = 0

Hence, the system has infinite solutions

Let z = k

2x – y = – 2k

5x + 3y = k

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 117

3. 3x – y + 2z = 0
4x + 3y + 3z = 0
5x + 7y + 4z = 0

Solution:

Given 3x – y + 2z = 0

4x + 3y + 3z = 0

5x + 7y + 4z = 0

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 118

A X = 0

Now, |A| = 3(12 – 21) + 1(16 – 15) + 2(28 – 15)

|A| = – 27 + 1 + 26

|A| = 0

Hence, the system has infinite solutions

Let z = k

3x – y = – 2k

4x + 3y = – 3k

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 119

4. x + y – 6z = 0
x – y + 2z = 0
– 3x + y + 2z = 0

Solution:

Given x + y – 6z = 0

x – y + 2z = 0

– 3x + y + 2z = 0

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 120

A X = 0

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

|A| = – 4 – 8 + 12

|A| = 0

Hence, the system has infinite solutions

Let z = k

x + y = 6k

x – y = – 2k

RD Sharma Solutions for Class 12 Maths Chapter 8 Solutions of Simultaneous Linear Equations Image 121

Also, access RD Sharma Solutions for Class 12 Maths Chapter 8 Solution of Simultaneous Linear Equations

Exercise 8.1 Solutions

Exercise 8.2 Solutions

Leave a Comment

Your email address will not be published. Required fields are marked *