1. Pick a letter to eliminate
2. Pick 2 equations to eliminate that letter from
3. Pick one of those equations with the third equation and eliminate
   that SAME letter from.
4. Solve the resulting system of two equations in two unknowns.
5. Substitute those values in one of the original equations to
   find the first letter eliminated.

(1)     -x +  y + 2z = 7 
(2)     2x + 3y +  z = 1 
(3)    -3x - 4y +  z = 4


1. Pick a letter to eliminate.

I'll pick z


2. Pick 2 equations to eliminate that letter from

I'll pick (2) and (3):

(2)     2x + 3y +  z = 1 
(3)    -3x - 4y +  z = 4

I'll multiply (2) by -1 and add (3) to it

       -2x - 3y -  z = -1 
(3)    -3x - 4y +  z =  4
-------------------------
(4)    -5x - 7y      =  3

3. Pick one of those equations with the third equation and eliminate
   that SAME letter from.

I'll pick (2) and use it with (1) to eliminate the SAME letter z

(1)     -x +  y + 2z = 7 
(2)     2x + 3y +  z = 1

I'll multiply (2) by -2 and add it to (1)

(1)     -x +  y + 2z =  7 
       -4x - 6y - 2z = -2
-------------------------
(5)    -5x - 5y      =  5

4. Solve the resulting system of two equations in two unknowns.

(4)    -5x - 7y = 3
(5)    -5x - 5y = 5

Multiply (4) by -1 and add (5) to it:

        5x + 7y = -3
(5)    -5x - 5y =  5
--------------------
             2y =  2
              y =  1

Substitute y = 1 in (4) to find x

(4)    -5x - 7y = 3
     -5x - 7(1) = 3
        -5x - 7 = 3
            -5x = 10
              x = -2

5. Substitute those values in one of the original equations to
   find the first letter eliminated.

I'll substitute x = -2 and y = 1 in (2)

(2)     2x + 3y +  z = 1
    2(-2) + 3(1) + z = 1
          -4 + 3 + z = 1
              -1 + z = 1
                   z = 2

(x,y,z) = (-2,1,2)

Notice that different people would pick different letters
and diferent equations but in the end the solutions would
all be the same.

Also sometimes the process can be shortened because one of
the letters is already eliminated from one of the equations.
But that was not the case here because all three equations
cntained all three letters.

Edwin