1. **State the problem:** We need to find three consecutive odd integers whose sum is 51. 2. **Define variables:** Let the first odd integer be $x$. Then the next two consecutive odd integers are $x+2$ and $x+4$ because odd numbers differ by 2. 3. **Set up the equation:** The sum of these three integers is given as: $$x + (x+2) + (x+4) = 51$$ 4. **Simplify the equation:** $$x + x + 2 + x + 4 = 51$$ $$3x + 6 = 51$$ 5. **Isolate $x$:** $$3x + 6 = 51$$ $$3x = 51 - 6$$ $$3x = 45$$ 6. **Divide both sides by 3:** $$\cancel{3}x = \cancel{3}15$$ $$x = 15$$ 7. **Find the three integers:** First integer: $x = 15$ Second integer: $x + 2 = 17$ Third integer: $x + 4 = 19$ 8. **Check the sum:** $$15 + 17 + 19 = 51$$ which matches the problem statement. **Final answer:** The three consecutive odd integers are $15$, $17$, and $19$.