Examples of 3unknowns





In this page examples of 3unknowns we are going to see example problems in the topic determinant method.

Question 1:

 2x + y + z = 5

 x + y + z = 4

 x - y + 2z = 1

Δ

=
 
2 1 1
1 1 1
1 -1 2
 
 

  

   Δ = 2 [2-(-1)] - 1 [2-1] + 1 [-1-1]

      = 2 [2+1] - 1 [1] + 1 [-2]

      = 2 [3] - 1 - 2

      = 6 - 1 - 2

      = 6 - 3

      = 3 ≠ 0


x=
 
5 1 1
4 1 1
1 -1 2
 
 


  Δx = 5 [2-(-1)] - 1 [8-1] + 1 [-4-1]

      = 5 [2+1] - 1 [7] + 1 [-5]

      = 5 [3] - 7 - 5

      = 15 - 7 - 5

      = 15 - 12

      = 3 ≠ 0


y=
 
2 5 1
1 4 1
1 1 2
 
 

 

 Δy = 2 [8-1] - 5 [2-1] + 1 [1-4]

      = 2 [7] - 5 [1] + 1 [-3]

      = 14 - 5 -3

      = 14 - 8

      = 6 ≠ 0


z=
 
2 1 5
1 1 4
1 -1 1
 
 

 

 Δz  = 2 [1-(-4)] - 1 [1-4] + 5 [-1-1]

      = 2 [(1+4)] - 1 [-3] + 5 [-2]

      = 2 [5] - 1 [-3] + 5 [-2]

      = 10 + 3 - 10

      =  3 ≠ 0


By Cramer's rule

x = Δx/Δ

  x = 3/3

  x = 1

y = Δy/Δ

  y = 6/3

  y = 2

z = Δz/Δ

  z = 3/3

  z = 1

Solution:

 x = 1

 y = 2

 z = 1


Some interesting facts about matrix:

       First let us see about the history of matrices. Even though the history goes to ancient times, the term 'Matrix' is started applied to the concept only from 1850.

      Matrix represents the Latin word womb and it retained the same meaning in English also.

     An important Chinese text between 300 BC and 200 AD, '  Nine Chapters Of Mathematical Art' first gives examples of uses of matrix methods in solving simultaneous equations.

The term 'Matrix' is first introduced by the mathematician James Joseph Sylvester.


 Students can try to solve the above problem on their own, and can verify their work with the above calculation. If you have any doubt you can contact us through mail, we will help you to clear your doubts. 

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Examples of 3unknowns to Inversion Method