Use GaussJordan elimination method to find all solutions of

Use Gauss-Jordan elimination method to find all solutions of the system of linear equations {2x + 3y = 12 2x - 3y = 0 5x - y = 13

Solution

Given that

2x + 3y = 12

2x - 3y = 0

5x - y = 13

The augmented matrix for above system of linear equations is ,

[    2   3   12

     2   -3   0

    5    -1   13 ]

Applying gauss-jordan elimination ,

      Swap matrix rows , R₁ <-> R₃

      [    5    -1    13

          2       3    0

          2      -3   12 ]

   R₃ <- R₂ - ( (2/5).R₁ )

   [      5        -1            13

         0       (-13/5)   (-26/5)

         2          3             12     ]

   R₃ <- R₃ - ( (2/5).R₁ )

    [     5         -1           13

          0       (-13/5)     (-26/5)

         0        (17/5)       (34/5)      ]

Swap matrix rows , R₂ <-> R₃

     [     5         -1           13

           0        (17/5)      (34/5)

           0       (-13/5)      (-26/5)    ]

     R₃ <- R₃ + ( (13/17).R₂ )

   [     5          -1            13

         0         (17/5)     (34/5)

         0            0            0           ]

Hence,

             The followung is the system of equations equivalent to the given system of equations ,

       i.e  

                5x - y = 13 ...................................1

               (17/5) y = ( 34/5 )..............................2

                 17y = 34

                     y = 34/17

                     y = 2

Substitute y = 2 in equation 1

         5x - y = 13

         5x = 13 + y

          5x = 13 + 2

          5x = 15

           x = 15/5

          x = 3

Therefore,

      The solution is :    x = 3 , y = 2