Let's break down the expression:

$-3 \frac{6}{7} + 2 \frac{7}{10}$

Step 1: Convert the mixed numbers to improper fractions.

1. $-3 \frac{6}{7}$
   Convert to an improper fraction:
   $-3 \frac{6}{7} = \frac{-3 \times 7 - 6}{7} = \frac{-21 - 6}{7} = \frac{-27}{7}$

2. $2 \frac{7}{10}$
   Convert to an improper fraction:
   $2 \frac{7}{10} = \frac{2 \times 10 + 7}{10} = \frac{20 + 7}{10} = \frac{27}{10}$

So, the expression becomes:

$\frac{-27}{7} + \frac{27}{10}$

Step 2: Find a common denominator.

The least common denominator (LCD) of 7 and 10 is 70.

Now, convert both fractions to have a denominator of 70:

$\frac{-27}{7} = \frac{-27 \times 10}{7 \times 10} = \frac{-270}{70}$ $\frac{27}{10} = \frac{27 \times 7}{10 \times 7} = \frac{189}{70}$

Step 3: Perform the addition.

Now that both fractions have the same denominator, add them:

$\frac{-270}{70} + \frac{189}{70} = \frac{-270 + 189}{70} = \frac{-81}{70}$

Step 4: Convert the result to a mixed number.

$\frac{-81}{70} = -1 \frac{11}{70}$

So, the final answer is:

$-1 \frac{11}{70}$

Would you like further details or have any questions?

Here are some related questions:

1. How do you convert improper fractions to mixed numbers?
2. What is the least common denominator, and why is it important in adding fractions?
3. How do you subtract mixed numbers without converting them to improper fractions?
4. Can you explain how to simplify a mixed number if needed?
5. What are other methods to add or subtract fractions?

Tip: Always check if you can simplify the numerator and denominator in a fraction to ensure the simplest form.