## 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

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 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

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 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

x = 1

y = 2

z = 1

 Questions Solution 2) Find the following linear equations by using rank method of matrix x + 2y + z = 72x - y + 2z = 4x + y - 2z = -1 Solution 3) Find the following linear equations by using rank method of matrix 2x + 5y + 7z = 52x + y + z = 92x + y - z = 0 Solution 4) Find the following linear equations by using rank method of matrix 3x + y - z = 22x - y + 2z = 62x + y - 2z = -2 Solution 5) Find the following linear equations by using rank method of matrix 2x - y + 3z = 9x + y + z = 6x - y + z = 2 ranking method examples 1 ranking method examples 1 Solution  