To find the three consecutive integers that add up to 51, we can follow these steps:

1. Define the consecutive integers:

   Let's denote the three consecutive integers as $x$, $x + 1$, and $x + 2$.

2. Set up the equation:

   According to the problem, the sum of these integers is 51. This can be written as: $x + (x + 1) + (x + 2) = 51$

3. Simplify the equation:

   Combine like terms to simplify the equation: $x + x + 1 + x + 2 = 51$ $3x + 3 = 51$

4. Solve for $x$:

   Subtract 3 from both sides of the equation to isolate the term involving $x$: $3x = 48$

   Divide both sides by 3 to solve for $x$: $x = 16$

5. Determine the three consecutive integers:

   The first integer is $x = 16$. Therefore, the three consecutive integers are: $16, 17, \text{ and } 18$

6. Check the solution:

   To verify, calculate the sum of 16, 17, and 18: $16 + 17 + 18 = 51$ Since the sum is indeed 51, our solution is correct.

So, the three consecutive integers are 16, 17, and 18.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What are consecutive integers, and how can they be represented algebraically?
2. How can you find four consecutive integers that sum up to a given number?
3. What if the sum of three consecutive even integers is given instead?
4. How do you solve equations involving consecutive odd integers?
5. Can you generalize a formula for the sum of any number of consecutive integers?

Tip: For problems involving consecutive integers, always remember that they differ by a fixed amount (usually 1 for integers or 2 for even/odd integers), which simplifies the setup of equations.