Question

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

Answer 1

Using Gaussian elimination:
x + y + z = 9  / * ( - 2 )    / ( - 1 ) ( we will multiply by - 2 and add to the 2nd         2 x - 3 y + 4 z = 7      equation and multiply by - 1 and add to the 3rd equation ) 
x - 4 y + 3 z = - 2
-----------------------------
x + y + z = 9
    - 5 y + 2 z = - 11  / * ( - 1 )
    - 5 y + 2 z = - 11
---------------------------
x + y + z = 9
   - 5 y + 2 z = - 11
              0 z = 0
There are infinitely many solutions to the system of equations.
z = t,
- 5 y + 2 t = - 11
5 y = 11 + 2 t
y = ( 11 + 2 t ) / 2
x + ( 11 + 2 t ) / 2  + t = 9    / * 2
2 x + 11 + 2 t + 2 t = 18
2 x = 18 - 11 - 4 t
x = ( 7 - 4 t ) / 2
( x, y , z ) = (   (7 - 4 t ) / 2,   ( 11 + 2 t ) / 2,    t  )