2x + 4y - 3z = -8
3y + 2z + 6x =  3
4z - 8x - 7y =  5

Rearrange the terms on the left so that the terms are
all in alphabetical order with the letters:

(A)      2x + 4y - 3z = -8
(B)      6x + 3y + 2z =  3
(C)     -8x - 7y + 4z =  5

Pick a letter to eliminate and two of the equations to
eliminate it from:

I will pick the letter x to eliminate and I will pick
(A) and (B) to eliminate it from:

I will multiply (A) by -3 and add (B) to it

        -6x - 12y +  9z = 24
(B)      6x +  3y +  2z =  3
------------------------------
(D)           -9y + 11z = 27

Eliminate the SAME letter from a different pair of equations,
[that is, you will choose the equation you did not choose before
along with one of the equations that you did choose before.]

I will pick (A) and (C) to eliminate the SAME letter x from:

I will multiply (A) by 4 and add (C) to it


         8x + 16y - 12z = -32
(C)     -8x -  7y +  4z =   5
-------------------------------
(E)            9y -  8z = -27

Now we have reduced the 3 equations and 3 unknowns down
to just 2 equations (D) and (E) and 2 unknowns:

(D)          -9y + 11z =  27
(E)           9y -  8z = -27

The y's will eliminate by just adding them as they are:

(D)          -9y + 11z =  27
(E)           9y -  8z = -27
------------------------------
                    3z = 0
                     z = 
                     z = 0

Substitute z = 0 in (E)

            9y -  8(0) = -27
                9y - 0 = -27
                    9y = -27
                     y = -3

Substitute y = -3 and z = 0 into one of the originals.
I'll pick (A)

(A)      2x + 4y - 3z = -8
    2x + 4(-3) - 3(0) = -8
          2x - 12 - 0 = -8
              2x - 12 = -8
                   2x =  4
                    x =  2

Solution: (x,y,z) = (2,-3,0)

Edwin