Question 94554
Alright here goes the solution for your question on solving equations by the method of gaussian elimination


2x + y - 3z = 1


3x - y + 4z = 6


x + 2y - z = 9 


The given system can be written in the augmented form as: 



[2  1   -3 :  1]
[3  -1   4 :  6]
[1  2   -1 :  9]


The operations performed are:  


We shall first interchange the first row with the third row. 


[1  2   -1 :  9]
[3  -1   4 :  6]
[2  1   -3 :  1]


Next:  R2 - 3R1 and R3 - 2R1 are the operations which follow


[1  2   -1 :  9]
[0  -7   7 :  -21]
[0  -3   -1 :  -17] 


R3 - R2 is the next opeartion.


[1  2   -1 :  9]
[0  -7   7 :  -21]
[0  4   -8 :  4] 


Lets divide R2/7 and R3/4 


[1  2   -1 :  9]
[0  -1   1 :  -3]
[0  1   -2 :  1] 



Now R3 + R2 gives us the final step


[1  2   -1 :  9]
[0  -1   1 :  -3]
[0  0   -1 :  -2] 


From this the given system of equations reduces to: 



x + 2y - z = 9 
-y + z = -3 
-z = - 2


Hence, the value of z can be directly found out. 


==> z = 2 


now apply back substituion.


-y + z = -3 


-y + 2 = -3


-y = -3 - 2


-y = -5 


==> y = 5 


Using y and z in the first equation we find the value of x.


x + 2y - z = 9 


x + 10 - 2 = 9


x = 9 - 8 


x = 1


Hence, the solution.


===============================================================================
===============================================================================


Now the second question. 



The given set of equations are:  



x - y + z = 17

x + y - z = -11

x - y - z = 9 


This in the augmented form can be written as: 


[1   -1     1  : 17]
[1    1    -1  : -11]
[1   -1    -1  : 9]


Lets perform the firts row operation that is R2 - R1 and R3- R1



[1   -1     1  : 17]
[0   2    -2  : -28]
[0   0    -2  : -8]


Now we divide R2/2 and R3/2 



[1   -1     1  : 17]
[0   1    -1  : -14]
[0   0    -1  : -4]


Thus the above augmented matrix can be written as: 



x - y + z = 17 

y - z = -14 

-z = -4


This implies z = 4 


By back substitution, we find the values of y and x. 


y - 4 = -14 


y = -14 + 4 


==> y = -10 


Now we find the value of x. 


x - y + z = 17 


x + 10 + 4 = 17 


x + 14 = 17 


x = 17 - 14 


x = 3 


Hence, the solution.


I hope all the stpes are quite clear..


You can get back to me for an online help in math at any time of yours. 


Regards

Chitra