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When adding fractions, a common denominator is crucial because it allows us to combine the numerators directly. Without a common denominator, the fractions represent parts of different wholes and cannot be mathematically combined in a straightforward manner. In our example, \(\frac{-4}{7}\) and \(\frac{-2}{7}\) already have a common denominator of 7. This makes the process much simpler. Using a common denominator ensures that the parts we are adding together are of the same size.
To sum up, always check if the fractions have the same denominator before moving on to the other steps. If they don't, you'll need to find the least common multiple and adjust the fractions accordingly. In this case, step one is already complete for us.

With a common denominator established, the next step is numerator addition. Since the denominators are the same, we simply add the numerators together. This means we only focus on the numbers at the top of the fractions. In our example, adding the numerators from \(\frac{-4}{7}\) and \(\frac{-2}{7}\) looks like this: \(-4 + (-2) = -6\). The denominator remains unchanged during this step because we're combining parts of the same whole.
This method is efficient because it uses the shared denominator as a constant base. Addition becomes straightforward arithmetic with the numerators while maintaining the integrity of the fractions.

Finally, we need to check if the resulting fraction can be simplified. Fraction simplification means reducing the fraction to its simplest form where the numerator and the denominator have no common factors other than 1. For example, after we added the numerators in our problem, we got \(\frac{-6}{7}\). Here, -6 and 7 have no common factors, so the fraction is already in its simplest form.
Simplification is crucial for clear communication and ease of understanding in math. Always check your final answer to see if it can be simplified. This makes your work cleaner and ensures that anyone reading it will easily understand the result.