A 2 meter long bookshelf is filled end-to-end with 46 books. Some of the books are 3 centimeters thick while all the others are 5 centimeters thick. Find the number of books on the shelf that are 3 centimeters thick.
1. Define the variables:
   - Let \( x \) be the number of 3 cm thick books.
   - Let \( y \) be the number of 5 cm thick books.

2. Set up the system of equations based on the given information:
   - The total number of books is 46:
     \[
     x + y = 46
     \]
   - The total thickness of the books is 200 cm (since 2 meters = 200 cm):
     \[
     3x + 5y = 200
     \]

3. Solve the system of equations:
   - First, solve the first equation for \( y \):
     \[
     y = 46 - x
     \]

   - Substitute \( y = 46 - x \) into the second equation:
     \[
     3x + 5(46 - x) = 200
     \]

4. Simplify and solve for \( x \):
   \[
   3x + 230 - 5x = 200
   \]
   \[
   3x - 5x = 200 - 230
   \]
   \[
   -2x = -30
   \]
   \[
   x = 15
   \]

5. Substitute \( x = 15 \) back into the first equation to find \( y \):
   \[
   y = 46 - 15
   \]
   \[
   y = 31
   \]

6. Verify the solution:
   - Check the total number of books:
     \[
     x + y = 15 + 31 = 46
     \]
   - Check the total thickness:
     \[
     3x + 5y = 3(15) + 5(31) = 45 + 155 = 200
     \]

Since both conditions are satisfied, the solution is correct.

The final answer is \( \boxed{15} \).