Ranking Method Examples 1





In this page ranking method examples 1 we are going to see solution of question 1 in rank method.

Question 1:

Solve the following linear equations by rank-method

2x + y + z = 5

x + y + z = 4

x - y + 2z = 1

Solution:

 
2 1 1 5
1 1 1 4
1 -1 2 1
 


ranking method examples 1

˜
 
2 1 1 5
1 1 1 4
1 -1 2 1
 

R₂ <-> R₂

˜
 
1 1 1 4
2 1 1 5
1 -1 2 1
 

R₂ => R₂ - 2R₁

       2          1         1         5

         2          2         2        8

        (-)       (-)       (-)       (-)

      _________________________

       0        -1       -1       -3

      __________________________

R => R - R₁

       1         -1         2         1

         1          1         1         4

        (-)       (-)       (-)       (-)

      ___________________________

       0        -2        1        -3

      _____________________________

ranking method examples 1

˜
 
1 1 1 4
0 -1 -1 -3
0 -2 1 -3
 

R₃ => R₃ - 2R

         0         -2         1         -3

         0         -2        -2         -6

        (-)       (+)       (+)       (+)

      ___________________________

       0         0          3         3

      ____________________________


˜
 
1 1 1 4
0 -1 -1 -3
0 0 3 -3
 


R₃ => R₃ - 2R₂

Rank (A) = 3

Rank [A,B] = 3

If rank (A) = rank of [A,B] = number of unknowns then we can say that the system is consistent and it has unique solution.

x + y + z = 4   --------(1)

-y - z = -3  --------(2)

       3z = 3 --------(3)

        z = 3/3

        z = 1

substitute z = 1 the second equation to get the value of y

          - y - 1 = -3

              - y = - 3 + 1

               - y = - 2

                  y = 2

substitute z = 1 and y = 2 in the first equation to get the value of x

       x + 2 + 1 = 4

       x + 3 = 4 

            x = 4 - 3

            x  = 1

Answer :

 x = 1

 y = 2

 z = 1


Questions



Solution


2) Find the following linear equations by using rank method of matrix

x + 2y + z = 7

2x - y + 2z = 4

x + y - 2z = -1

Solution

3) Find the following linear equations by using rank method of matrix

2x + 5y + 7z = 52

x + y + z = 9

2x + y - z = 0

Solution

4) Find the following linear equations by using rank method of matrix

3x + y - z = 2

2x - y + 2z = 6

2x + y - 2z = -2

Solution

5) Find the following linear equations by using rank method of matrix

2x - y + 3z = 9

x + y + z = 6

x - y + z = 2

ranking method examples 1 ranking method examples 1

Solution







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