Exercise 1 solution

Start: N2 + H2 -> NH3

Step 1, count: left N = 2, H = 2; right N = 1, H = 3. Not balanced.
Step 2, fix nitrogen with 2 NH3: N2 + H2 -> 2NH3. Left N = 2, H = 2; right N = 2, H = 6.
Step 3, fix hydrogen with 3 H2: N2 + 3H2 -> 2NH3. Left N = 2, H = 6; right N = 2, H = 6.

Atom-count table (counts run in Python for the final equation):
                 before (N2 + H2 -> NH3)   after (N2 + 3H2 -> 2NH3)
  N  left/right        2 / 1                      2 / 2
  H  left/right        2 / 3                      6 / 6

Changing NH3 to NH2 would be changing a subscript. That makes a different substance
(a different molecular formula, Chapter 5), so it is no longer the ammonia you meant.
Only coefficients may be changed to balance.

Note: that the product is ammonia, NH3, is given in the exercise; the balancing is the
point. Nothing here comes from a page beyond the balancing rule of the Chemical equation
article: each side must have the same number of atoms of each element (check, general
knowledge for the substances).