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Adding and subtracting fractions might seem tricky at first, but it's all about understanding how the numerators and denominators work together. When fractions have the same denominators, they are called "like fractions." This makes your job much easier. You can simply add or subtract the numerators and keep the common denominator.

For example, looking at \(-\frac{7}{9}\) and \(+\frac{3}{9}\), since both have 9 as the denominator, you only need to focus on the numerators. Subtract 7 from 3 to get \(-4\), keeping the denominator of 9. The result is \(-\frac{4}{9}\).

This simple rule is true for any addition and subtraction of fractions: combine the tops and keep the bottom the same! But remember, this only applies when the denominators are the same. That's when we need to talk about "common denominators."

For fractions to be added or subtracted, they need to share the same denominator. This shared denominator is crucial because it essentially defines the "size" of one whole for each fraction. If fractions do not share a common denominator, we need to convert them so they do.

To find a common denominator, you often look to the least common multiple (LCM) of the two denominators. In the exercise, we have \(\frac{1}{3}\), which needs to be converted to have a common denominator with \(\frac{7}{9}\).

The LCM of 3 and 9 is 9. Therefore, we convert \(\frac{1}{3}\) into \(\frac{3}{9}\) by multiplying both the numerator and the denominator by 3 (since 3 * 3 = 9). Now both fractions, \(\frac{7}{9}\) and \(\frac{3}{9}\), have the common denominator of 9, making it possible to perform straightforward addition or subtraction.

Converting fractions is a powerful tool in both simplifying your calculations and ensuring all components are compatible for mathematical operations. This process involves making fractions equivalent to one another so operations like addition and subtraction can proceed smoothly.

Take \(\frac{1}{3}\) from the original exercise: it needs to be re-expressed with a denominator of 9. By multiplying the top and bottom by the same number, we don't change the fraction’s value, just how it looks. Here, multiplying \(\frac{1}{3}\) by \(\frac{3}{3}\) changes it into \(\frac{3}{9}\), now compatible with the other fraction, \(\frac{7}{9}\).

The idea is that multiplying both the numerator and the denominator by the same number creates an equivalent fraction. This technique is essential for solving fraction problems involving different denominators, simplifying fractions, or even comparing them.