1. **State the problem:** We need to add the numbers $5.1$ and $-6.1$. 2. **Recall the rule for adding positive and negative numbers:** When adding a positive number and a negative number, subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value. 3. **Apply the rule:** $$5.1 + (-6.1) = 5.1 - 6.1$$ 4. **Calculate the difference:** $$5.1 - 6.1 = \cancel{5.1} - \cancel{5.1} - 1.0 = -1.0$$ 5. **Interpret the result:** Since $6.1$ is larger and negative, the result is negative. **Final answer:** $$5.1 + (-6.1) = -1.0$$