1. The problem is to add the fractions $\frac{2}{3}$ and $\frac{5}{7}$. 2. To add fractions, we need a common denominator. The denominators here are 3 and 7. 3. The least common denominator (LCD) of 3 and 7 is their product since they are coprime: $$3 \times 7 = 21$$ 4. Convert each fraction to an equivalent fraction with denominator 21: $$\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}$$ $$\frac{5}{7} = \frac{5 \times 3}{7 \times 3} = \frac{15}{21}$$ 5. Now add the numerators: $$\frac{14}{21} + \frac{15}{21} = \frac{14 + 15}{21} = \frac{29}{21}$$ 6. The fraction $\frac{29}{21}$ is an improper fraction and can be expressed as a mixed number: $$29 \div 21 = 1 \text{ remainder } 8$$ So, $$\frac{29}{21} = 1 \frac{8}{21}$$ 7. Final answer: $$\boxed{\frac{29}{21} \text{ or } 1 \frac{8}{21}}$$