Let x be the length of the shortest board.  The second piece is 3 inches longer than the first, or (x+3).  The third piece is 4 inches longer than the second, or (x+3+4) which is (x+7).  The original board was 37 inches, so the three pieces have to add up to 37:

(x) + (x+3) + (x+7) = 37

Combine terms: 3x + 10 = 37

Subtract 10 from each side: 3x + 10 - 10 = 37 - 10

So: 3x = 27

Divide both sides by 3: 3x/3 = 27/3

So x = 9.

The shortest board was x, so it's 9 inches.  The second board was x+3, or 12 inches.  The third board was x+7, or 16 inches.  Add them together and you get the original 37 inches.