Solving a 6x6 System of Equations

Date: 03/24/98 at 13:55:48
From: Jason Goodspeed
Subject: 6x6 equations

Could you please look at the following equation - is there a Cramer's 
rule for a 6x6 equation?

     a +  b - 2c +  d + 3e -  f  =    4
    2a -  b +  c + 2d +  e - 3f  =   20
     a + 3b - 3c -  d + 2e +  f  =  -15
    5a + 2b -  c -  d + 2e +  f  =  - 3
   -3a -  b + 2c + 3d +  e + 3f  =   16
    4a + 3b +  c - 6d - 3e - 2f  =  -27

  
Our 12th grade class and teachers can't come up with a true answer 
for each variable. Most of us came up with a dependent system. 
Could you help with a solution to this problem so I could show the 
class?

Date: 03/24/98 at 15:29:39
From: Doctor Rob
Subject: Re: 6x6 equations

Yes, there is a Cramer's Rule for any system of linear equations.

This system of equations is not dependent. The determinant of the 
matrix of coefficients (the denominator in Cramer's Rule) is -852.  
If the system were dependent, the determinant would be zero.

Here is a solution which does not use Cramer's Rule. I hate 
evaluating large determinants. This method is called Gaussian 
Elimination, or Row Reduction.

You can eliminate "a"s from all equations but the first by subtracting   

    2 times the first equation from the second, 
    1 times the first equation from the third, 
    5 times the first equation from the fourth, 
   -3 times the first equation from the fifth, and 
    4 times the first equation from the sixth.

This gives you the following five equations in five unknowns 
b, c, d, e, and f:

   a + 1*b - 2*c +  1*d +  3*e - 1*f  =   4
      -3*b + 5*c        -  5*e - 1*f  =  12
       2*b - 1*c -  2*d -  1*e + 2*f  = -19
      -3*b + 9*c -  6*d - 13*e + 6*f  = -23
       2*b - 4*c +  6*d - 10*e + 0*f  =  28
      -1*b + 9*b - 10*d - 15*e + 2*f  = -43


You can divide the fourth of these equations by 2, to get

         b - 2*c +  3*d -  5*e + 0*f  =  14

and rearrange the equations to make this the second one:

   a + 1*b - 2*c +  1*d +  3*e - 1*f  =   4
         b - 2*c +  3*d -  5*e + 0*f  =  14
      -3*b + 5*c        -  5*e - 1*f  =  12
       2*b - 1*c -  2*d -  1*e + 2*f  = -19
      -3*b + 9*c -  6*d - 13*e + 6*f  = -23
      -1*b + 9*c - 10*d - 15*e + 2*f  = -43

Now you can eliminate b from all equations but the second one by 
subtracting multiples of the second equation from all the rest.  
This gives:

   a       + 0*c -  2*d -  2*e - 1*f  = -10
       b   - 2*c +  3*d -  5*e + 0*f  =  14
             3*c -  8*d - 11*e + 2*f  = -47
             3*c +  3*d +  2*e + 6*f  =  19
            -1*c +  9*d + 10*e - 1*f  =  54
             7*c -  7*d - 10*e + 2*f  = -29

Now divide the fifth of these equations by -1, and move it to the 
third row, to get 

   a       + 0*c -  2*d -  2*e - 1*f  = -10
       b   - 2*c +  3*d -  5*e + 0*f  =  14
               c -  9*d - 10*e + 1*f  = -54
             3*c -  8*d - 11*e + 2*f  = -47
             3*c +  3*d +  2*e + 6*f  =  19
             7*c -  7*d - 10*e + 2*f  = -29

Now you can eliminate c from all equations but the third one by 
subtracting multiples of the third equation from all the rest. 
This gives:

   a               -2*d -  2*e - 1*f  = -10
       b          -15*d - 15*e + 2*f  = -94
           c      - 9*d - 10*e + 1*f  = -54
                   19*d + 19*e - 1*f  = 115
                   30*d + 32*e + 3*f  = 181
                   56*d + 60*e - 5*f  = 349

Now if we subtract three times the fourth row from the sixth, we get

                   -1*d +  3*e - 2*f  =   4

which we multiply by -1 and move to the fourth position:

   a               -2*d -  2*e - 1*f  = -10
       b          -15*d - 15*e + 2*f  = -94
           c      - 9*d - 10*e + 1*f  = -54
                      d -  3*e + 2*f  =  -4
                   19*d + 19*e - 1*f  = 115
                   30*d + 32*e + 3*f  = 181

Now we can eliminate d from all equations but the fourth by 
subtracting multiples of the fourth from all the other rows.  
This gives:

   a                    -  8*e +  3*f =  -18
       b                - 60*e + 32*f = -154
           c            - 37*e + 19*f =  -90
               d        -  3*e +  2*f =   -4
                          76*e - 39*f =  191
                         122*e - 57*f =  301

Now subtract twice the fifth equation from the sixth:

                         -30*e + 21*f =  -81

Now divide this by -3:

                          10*e -  7*f =   27

Subtract 7 times this equation from the fifth:

                           6*e + 10*f =    2

Now divide this by 2:

                           3*e +  5*f =    1

Now subtract 3 times this from the equation preceding it by two, to 
get:

                             e - 22*f =   24

Now the system of equations looks like:

   a                    -  8*e +  3*f =  -18
       b                - 60*e + 32*f = -154
           c            - 37*e + 19*f =  -90
               d        -  3*e +  2*f =   -4
                             e - 22*f =   24
                           3*e +  5*f =    1

Now we can eliminate e from all equations but the fifth by subtracting
multiples of the fifth equation from each of them:

   a                         -  173*f =  174
       b                     - 1288*f = 1286
           c                 -  795*f =  798
               d             -   64*f =   68
                   e         -   22*f =   24
                                 71*f =  -71

Now divide the last equation by 71. Eliminate f from all equations but 
the last one by subtracting multiplies of the last equation from each 
of the others:

   a                                  =  1
       b                              = -2
           c                          =  3
               d                      =  4
                   e                  =  2
                       f              = -1

There is your solution. You can check it. Notice that the absolute 
values of the numbers we divided by were 2, 3, 2, and 71, whose 
product is 852. Where have we seen that number before?

-Doctor Rob,  The Math Forum
Check out our Web site http://mathforum.org/dr.math/   

Date: 03/24/98 at 16:16:32
From: Blane and Meg Goodspeed
Subject: Re: 6x6 equations

Thanks. Your help was greatly appreciated.   Jason