1. Stating the problem: We need to add the fractions $\frac{10}{7}$ and $\frac{9}{7}$.\n\n2. Formula: When adding fractions with the same denominator, add the numerators and keep the denominator the same.\n\n3. Applying the formula: $$\frac{10}{7} + \frac{9}{7} = \frac{10 + 9}{7}$$\n\n4. Simplify the numerator: $$\frac{19}{7}$$\n\n5. The fraction $\frac{19}{7}$ is an improper fraction and can be expressed as a mixed number: $$19 \div 7 = 2 \text{ remainder } 5$$ so $$\frac{19}{7} = 2 \frac{5}{7}$$\n\nFinal answer: $$\frac{10}{7} + \frac{9}{7} = \frac{19}{7} = 2 \frac{5}{7}$$