Line up the equations like this:

2u - 3v     =  13
     3v + w =  -3
4u      - w =   2

Eliminate w by adding the second and 3rd equations:

     3v + w =  -3
4u      - w =   2
-----------------
4u + 3v     =  -1

Solve the system consisting of this equation
and the first original equation:

4u + 3v = -1
2u - 3v = 13

Eliminate v by adding these equations:

4u + 3v = -1
2u - 3v = 13
------------
6u      = 12
      u = 2

Substitute that in 

  2u - 3v = 13
2(2) - 3v = 13
   4 - 3v = 13
      -3v = 9
        v = -3
       
Substitute that in 

     3v + w = -3

  3(-3) + w = -3

     -9 + w = -3
  
          w = 6

(u, v, w) = (2, -3, 6)  

Edwin