SOLUTION: Please help me with this. The question was not given from a book. Solve for X, Y, and Z in the following sysyems of three equations: a. x+2y+z=6 x+y=4 3x+y+z=8 b. 10x+

Please help me with this. The question was not given from a book.
Solve for X, Y, and Z in the following sysyems of three equations:

I'll just do (b), and give the answers for the first and
third.  The first and third are done the same way. The first one is
shorter because z has already been eliminated from the second 
equation so you only need to eliminate z from the 1st and 3rd
equations.  If you get stuck on (a) or (c), post again.

a.
   x + 2y + z = 6
   x +  y     = 4
  3x +  y + z = 8 

Answer: (x, y, z) = (2, 2, 0)


b.
   10x +   y +  z = 12
    8x +  2y +  z = 11
   20x - 10y - 2z =  8

Pick any of the three letters to eliminate first. I'll pick
y.  Now pick any pair of equations that contain y, to use
to eliminate y.  I'll pick the first two equations:

   10x +   y +  z = 12
    8x +  2y +  z = 11

To eliminate y multiply the first equation thru by -2
and add the two equations term by term:

  -20x -  2y - 2z = -24
    8x +  2y +  z =  11
 -----------------------
  -12x       -  z = -13


Now pick a different pair of equations that contain y,
to use to eliminate the SAME letter y.  I'll pick the 
first and third equations:

   10x +   y +  z = 12
   20x - 10y - 2z =  8

To eliminate y multiply the first equation thru by 10
and add the two equations term by term: 

  100x + 10y + 10z = 120
   20x - 10y -  2z =   8
 ------------------------
  120x       +  8z = 128

Now you have two equations in two unknowns

  -12x -  z = -13
  120x + 8z = 128

Eliminate x by multiplying the first by 10
and adding the the second:

 -120x - 10z = -130
  120x +  8z =  128
--------------------
         -2z =   -2
           z = 1  

Substitute z = 1 in 

  -12x -  z  = -13
  -12x - (1) = -13
        -12x = -12
           x = 1

Now pick any original equation
and substitute z = 1 and x = 1   

I'll pick 

    8x +  2y +  z = 11
    8x + 2(1) + 1 = 11
       8x + 2 + 1 = 11
           8x + 3 = 11
               8x = 8
                x = 1

Answer: (x, y, z) = (1, 1, 1)

 
c. 22x + 5y +  7z = 12
   10x + 3y +  2z =  5
    9x + 2y + 12z = 14

This one is done the same way as b.  Use the
above for a model to solve this one by.

Answer: (x, y, z) = (0, 1, 1)

Edwin