1. **State the problem:** We need to find the sum of $-\frac{5}{3}$ and $-\frac{3}{7}$. 2. **Formula and rules:** To add fractions, we need a common denominator. The sum of two fractions $\frac{a}{b} + \frac{c}{d}$ is given by $$\frac{ad + bc}{bd}$$. Since both fractions are negative, the sum will also be negative. 3. **Find the common denominator:** The denominators are 3 and 7. The least common denominator (LCD) is $3 \times 7 = 21$. 4. **Rewrite each fraction with the LCD:** $$-\frac{5}{3} = -\frac{5 \times 7}{3 \times 7} = -\frac{35}{21}$$ $$-\frac{3}{7} = -\frac{3 \times 3}{7 \times 3} = -\frac{9}{21}$$ 5. **Add the fractions:** $$-\frac{35}{21} + (-\frac{9}{21}) = \frac{-35 - 9}{21} = \frac{-44}{21}$$ 6. **Simplify if possible:** 44 and 21 have no common factors other than 1, so the fraction is already in simplest form. 7. **Final answer:** $$-\frac{44}{21}$$