Always write systems of equations so that like terms 
line up vertically, like this:

     5y - 7z = 14
2x +  y + 4z = 10
2x + 6y - 3z = 30

The first equation contains only 2 letters. That is, 
x is already eliminated in the first equation.

Therefore we eliminate x from the other two equations:

We multiply the 2nd equation by -1 and add it term
by term to the 3rd equation:

-2x -  y - 4z = -10
 2x + 6y - 3z =  30
-------------------
      5y - 7z =  20

Now we put the first equation together with this one
and we now have only two equations in only two 
variables:

     5y - 7z = 14
     5y - 7z = 20

We eliminate y by multiplying the second equation by
-1 and adding:

     5y - 7z =  14
    -5y + 7z = -20    
  -----------------
          0z = -6

There can be no value of z such that when we multiply
by 0 we well get -6, so there is no solution.  The
system of equations is said to be INCONSISTENT.

Edwin