If you were to ignore the last sentence and solve without using 34=21+2w, you would get the answer as follows.
The perimeter of a rectangle is 2w+2l=P, with w representing the width of the rectangle, l representing the length, and P representing the perimeter. Since you know the length is five times more than the width, you can replace l with 5w, so you only have one variable in the equation and you can solve it. If you plug in your 5w for l and 34 for P, the equation now reads 2w+2(5w)=34. Multiply out what's in parentheses, and you get 2w+10w=34, which simplifies to 12w=34. Divide both sides to get w by itself, and you get w=17/6, or w=2.83. Plug this back into the l=5w to get your length value, and you get length equal to 14.17.
Length: 14.17 units
Width: 2.83 units
Hope this helps!