Lesson Solving system of linear equation in 19 unknowns

Solving system of linear equation in 19 unknowns

Problem 1

     = 1,        
     = 2,        
     = 3,        
        . . . . . . . . .        
     = 16,    
     = 17,    
     = 18,      
     = 19,      

Let's write the system of equations in full

     = 1           (1)
     = 2           (2)
     = 3           (3)
        . . . . . . . . .        
     = 16      (16)
     = 17       (17)
     = 18        (18)
     = 19         (19)

                                                          
Subtract equation (1) from equation (2):
     -  = 2 - 1
     = 1
     =  + 1

Subtract equation (2) from equation (3):
     -  = 3 - 2
     -  = 1
     =  + 1

In general, we have  = .

This means:
     =  + 1
     =  = 
     =  = 
     =  = 

Similarly,
     = 
     =  = 
     = 
     =  = 

also,
     = 
     =  = 
     =  = 
     =  = 

and
     = 
     =  = 
     =  = 


Now let's use equations (1), (17), (18), and (19).

From equation  (1),     = 1           (20)
From equation (17),     = 17       (21)
From equation (18),     = 18        (22)
From equation (19),     = 19         (23)


Substituting the expressions we find:

from (20):   = 17
             = 17
             = 5                            (17')

from (21):   = 18
             = 18
             = 10                           (18')

from (22):   = 19
             = 19
             = 15                           (19') 


Add equations (17'), (18') and (19').  You will get

      = 5 + 10 + 15 

      = 30.


 Divide both sides by 4

      = 7.5.      (24)


Now we are on the finish line.


From equation (17') subtract equation (24).  You will get   =  5 - 7.5 = -2.5.


From equation (18') subtract equation (24).  You will get   = 10 - 7.5 = 2.5.


From equation (19') subtract equation (24).  You will get   = 15 - 7.5 = 7.5.


From equation (23),  = 19 -  -  -  = 19 - 0 - (-2.5) - 2.5 - 7.5 =  11.5


For completeness, let's determine a-1 from the very first equation in this post

     =  = 1 - ((-2.5) + 2.5 + 7.5) = 1 - 7.5 = -6.5.


From this point, all 19 terms  , , , , , . . . ,  can be determined.

See this table below


    a_1    a_2    a_3    a_4    a_5    a_6    a_7    a_8    a_9    a_10   a_11    a_12    a_13    a_14    a_15    a_16    a_17    a_18    a_19  

   -2.5    2.5    7.5   -6.5   -1.5    3.5    8.5   -5.5   -0.5    4.5    9.5     -4.5    0.5     5.5     10.5    5.5     1.5     6.5     11.5


You may check that all given 19 original equations are satisfied.


Final Answer:  = 11.5.