Line them up so the letters, signs and equal signs are in columns:

equation 1:     4x + 2y - 6z = -38
equation 2:     5x - 4y +  z = -18
equation 3:      x + 3y + 7z =  38

We choose the equation 1 and 2 to eliminate y from.
If we multiply equation 1 by 2 the 2y will become 4y
and cancel with the -4y in equation 2:

We multiply equation 1 by 2 and write equation 2 under it
and add them term by term and call it equation 4:

                8x + 4y - 12z = -76
equation 2:     5x - 4y +   z = -18
-----------------------------------
equation 4:    13x      - 11z = -94

Now we have to use the unused equation 3 with either of
the equations we just used, 1 or 2, and eliminate that
SAME variable, y.  I will pick equations 1 and 3

equation 1:     4x + 2y - 6z = -38
equation 3:      x + 3y + 7z =  38

To eliminate y, we must make the y-terms cancel out.
Their coefficients are 2 and 3.  The smallest whole
number 2 and 3 will both go into is 6.  So we want
to make the 2y term into +6y and the 3y term into -6y.
So we multiply equation 1 by 3 and equation 2 by -2, and
add them term by term, and call it equation 5:

               12x + 6y - 18z = -114
               -2x - 6y - 14z =  -76
              ----------------------
equation 5     10x      - 32z = -190

Now we take equations 4 and 5:

equation 4:    13x - 11z =  -94
equation 5     10x - 32z = -190

Notice that equation 5 can be divided through by 2.
That will make it a little easier. We'll call it equation 6.

equation 4:    13x - 11z = -94
equation 6:     5x - 16z = -95

We'll pick one of the letters x or y to eliminate.
Let's pick x to eliminate. To eliminate x, we must 
make the x-terms cancel out. Their coefficients are 
13 and 5.  The smallest whole number 13 and 5 will 
both go into is 65.  So we want to make the 13x term 
into +65x and the 5x term into -65x.
So we multiply equation 4 by 5 and equation 6 by -13, and
add them term by term, and solve for y:

               65x -  55z = -470
              -65x + 208z = 1235
              ------------------
                     153z = 765
                        z = 
                        z = 5

Now we substitute z = 5 into equation 5 or 6.  We'll
choose equation 6:

equation 6:     5x - 16z = -95
              5x - 16(5) = -95
                 5x - 80 = -95
                      5x = -15
                       x = 
                       x = -3

Finally we substitute x = -3 and z = 5 into any one
of the original equations 1,2, or 3.  We'll choose
equation 1:

equation 1:     4x + 2y - 6z = -38
           4(-3) + 2y - 6(5) = -38
               -12 + 2y - 30 = -38
                    -42 + 2y = -38
                          2y = 4
                           y = 
                           y = 2

So the solution is (x,y,z) = (-3,2,5)

Edwin