RD Chapter 6 Determinants Ex 6.4 Solutions
Question - 1 : - Solve the following system of linear equations by Cramer’s rule:
x – 2y = 4
-3x + 5y = -7
Answer - 1 : -
Given x – 2y = 4
-3x + 5y = -7
Let there be a systemof n simultaneous linear equations and with n unknown given by
Solving determinant,expanding along 1st row
⇒ D = 5(1) – (– 3)(– 2)
⇒ D = 5 – 6
⇒ D = – 1
Again,
Solving determinant,expanding along 1st row
⇒ D1 =5(4) – (– 7) (– 2)
⇒ D1 =20 – 14
⇒ D1 =6
And
Solving determinant,expanding along 1st row
⇒ D2 =1(– 7) – (– 3) (4)
⇒ D2 =– 7 + 12
⇒ D2 =5
Thus by Cramer’s Rule,we have
Question - 2 : - Solve the following system of linear equations by Cramer’s rule:
Answer - 2 : -
Given 2x – y = 1 and
7x – 2y = -7
Let there be a systemof n simultaneous linear equations and with n unknown given by
Solving determinant,expanding along 1st row
⇒ D1 =1(– 2) – (– 7) (– 1)
⇒ D1 =– 2 – 7
⇒ D1 =– 9
And
Solving determinant,expanding along 1st row
⇒ D2 =2(– 7) – (7) (1)
⇒ D2 =– 14 – 7
⇒ D2 =– 21
Thus by Cramer’s Rule,we have
Question - 3 : - Solve the following system of linear equations by Cramer’s rule:
Answer - 3 : -
Given 2x – y = 17 and
3x + 5y = 6
Let there be a systemof n simultaneous linear equations and with n unknown given by
Solving determinant,expanding along 1st row
⇒ D1 =17(5) – (6) (– 1)
⇒ D1 =85 + 6
⇒ D1 =91
Solving determinant,expanding along 1st row
⇒ D2 =2(6) – (17) (3)
⇒ D2 =12 – 51
⇒ D2 =– 39
Thus by Cramer’s Rule,we have
Question - 4 : - Solve the following system of linear equations by Cramer’s rule:
Answer - 4 : -
Let there be a systemof n simultaneous linear equations and with n unknown given by
Solving determinant,expanding along 1st row
⇒ D = 3(– 1) – (3)(1)
⇒ D = – 3 – 3
⇒ D = – 6
Again,
Solving determinant,expanding along 1st row
⇒ D1 =19(– 1) – (23) (1)
⇒ D1 =– 19 – 23
⇒ D1 =– 42
Solving determinant,expanding along 1st row
⇒ D2 =3(23) – (19) (3)
⇒ D2 =69 – 57
⇒ D2 =12
Thus by Cramer’s Rule,we have
Question - 5 : - Solve the following system of linear equations by Cramer’s rule:
Answer - 5 : -
Given 2x – y = -2 and
3x + 4y = 3
Let there be a systemof n simultaneous linear equations and with n unknown given by
Solving determinant,expanding along 1st row
⇒ D2 =3(2) – (– 2) (3)
⇒ D2 =6 + 6
⇒ D2 =12
Thus by Cramer’s Rule,we have
Question - 6 : - Solve the following system of linear equations by Cramer’s rule:3x + ay = 4
2x + ay = 2, a ≠ 0
Answer - 6 : -
Given 3x + ay = 4 and
2x + ay = 2, a ≠ 0
Let there be a systemof n simultaneous linear equations and with n unknown given by
3x + ay = 4
2x + ay = 2, a≠0
So by comparing withthe theorem, let’s find D, D1 and D2
Solving determinant,expanding along 1st row
⇒ D = 3(a) – (2)(a)
⇒ D = 3a – 2a
⇒ D = a
Again,
Solving determinant,expanding along 1st row
⇒ D1 =4(a) – (2) (a)
⇒ D = 4a – 2a
⇒ D = 2a
Solving determinant,expanding along 1st row
⇒ D2 =3(2) – (2) (4)
⇒ D = 6 – 8
⇒ D = – 2
Thus by Cramer’s Rule,we have
Question - 7 : - Solve the following system of linear equations by Cramer’s rule:
Answer - 7 : -
Let there be a systemof n simultaneous linear equations and with n unknown given by
Solving determinant,expanding along 1st row
⇒ D = 2 (6) – (3)(1)
⇒ D = 12 – 3
⇒ D = 9
Again,
Solving determinant,expanding along 1st row
⇒ D1 =10 (6) – (3) (4)
⇒ D = 60 – 12
⇒ D = 48
Solving determinant,expanding along 1st row
⇒ D2 =2 (4) – (10) (1)
⇒ D2 =8 – 10
⇒ D2 =– 2
Thus by Cramer’s Rule,we have
Question - 8 : - Solve the following system of linear equations by Cramer’s rule:
Answer - 8 : -
Let there be a systemof n simultaneous linear equations and with n unknown given by
Now, here we have
5x + 7y = – 2
4x + 6y = – 3
So by comparing withthe theorem, let’s find D, D1 and D2
Solving determinant,expanding along 1st row
⇒ D = 5(6) – (7)(4)
⇒ D = 30 – 28
⇒ D = 2
Again,
Solving determinant,expanding along 1st row
⇒ D1 =– 2(6) – (7) (– 3)
⇒ D1 =– 12 + 21
⇒ D1 =9
Solving determinant,expanding along 1st row
⇒ D2 =– 3(5) – (– 2) (4)
⇒ D2 =– 15 + 8
⇒ D2 =– 7
Thus by Cramer’s Rule,we have
Question - 9 : - Solve the following system of linear equations by Cramer’s rule:
Answer - 9 : -
Let there be a systemof n simultaneous linear equations and with n unknown given by
Solving determinant,expanding along 1st row
⇒ D = 3(9) – (5)(– 2)
⇒ D = 27 + 10
⇒ D = 37
Again,
Solving determinant,expanding along 1st row
⇒ D1 =10(3) – (8) (5)
⇒ D1 =30 – 40
⇒ D1 =– 10
Solving determinant,expanding along 1st row
⇒ D2 =9(8) – (10) (– 2)
⇒ D2 =72 + 20
⇒ D2 =92
Thus by Cramer’s Rule,we have
Question - 10 : - Solve the following system of linear equations by Cramer’s rule:
Answer - 10 : -
Let there be a systemof n simultaneous linear equations and with n unknown given by
Solving determinant,expanding along 1st row
⇒ D = 1(1) – (3)(2)
⇒ D = 1 – 6
⇒ D = – 5
Again,
Solving determinant,expanding along 1st row
⇒ D1 =1(1) – (2) (4)
⇒ D1 =1 – 8
⇒ D1 =– 7
Solving determinant,expanding along 1st row
⇒ D2 =1(4) – (1) (3)
⇒ D2 =4 – 3
⇒ D2 =1
Thus by Cramer’s Rule,we have
