well, let's DO that. note that this corresponds to "removing x from 2 of the equations (eliminating x in 2 equations)". subtracting 1/2 times row 1 from row 2 we get:

next, we subtract 3/2 row 1 from row 3 to get:

then we multiply row 1 by 1/2 to get:

rather then figure out some weird fractions, let's multiply row 2 by -1/4 now:

this is apparently the point at which you get "stuck". but we just keep doing the same thing, except now our goal is to get rid of everything in column 2 except for the 1 in ROW 2. so our next step is to subtract row 2 from row 1:

and next we subtract row 2 from row 3:

to get the 1 we want in row 3, column 3, multiply row 3 by 8/7:

now, it's going to get a bit messy, but we need to add 3/8 times row 3 to row 2:

and to finish off, add 1/8 times row 3 to row 1:

so, if i have done my arithmetic correctly, x = 3/7, y = -5/7, z = -18/7 is a solution:

2(3/7) + 2(-5/7) - (-18/7) = 6/7 - 10/7 + 18/7 = 14/7 = 2 (check one)

3/7 - 3(-5/7) + (-18/7) = 3/7 + 15/7 - 18/7 = 0 (check two)

3(3/7) + 4(-5/7) - (-18/7) = 9/7 - 20/7 + 18/7 = 1 (check three, and verified as solution).

NOTE: i did the above row-reduction "idiot fashion" without looking for shortcuts. sometimes you can see easier ways to get the same thing done.