SOLVING LINEAR EQUATIONS WITH 3 VARIABLES USING RANK METHOD

Solving Linear Equations in Three Variables Using Rank Method :

  • If there are n unknowns in the system of equations and ρ (A) = ρ ([A| B]) = n, then the system AX = B, is consistent and has a unique solution.
  • If there are n unknowns in the system AX = B, and ρ (A) = ρ ([A| B]) = n − k, k ≠  0 then the system is consistent and has infinitely many solutions and these solutions form a k − parameter family. In particular, if there are 3 unknowns in a system of equations and ρ (A) = ρ ([A| B]) = 2, then the system has infinitely many solutions and these solutions form a one parameter family. 
  • In the same manner, if there are 3 unknowns in a system of equations and ρ (A) = ρ ([A| B]) = 1, then the system has infinitely many solutions and these solutions form a two parameter family.
  • If ρ(A)  ρ ([A| B]), then the system AX = B is inconsistent and has no solution.

Problem 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

R₃ => R₃ - 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  =  1

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

- y - 1  =  -3 and y  =  2

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

x + 3  =  4 

x  =  1

Solution :

 x = 1,  y = 2 and z = 1

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. Problems on Finding Derivative of a Function

    Mar 29, 24 12:11 AM

    Problems on Finding Derivative of a Function

    Read More

  2. How to Solve Age Problems with Ratio

    Mar 28, 24 02:01 AM

    How to Solve Age Problems with Ratio

    Read More

  3. AP Calculus BC Integration of Rational Functions by Partical Fractions

    Mar 26, 24 11:25 PM

    AP Calculus BC Integration of Rational Functions by Partical Fractions (Part - 1)

    Read More