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When working with rational expressions, it's important to know how to add and subtract them. This process is similar to adding and subtracting fractions. The first step is to have a common denominator for the expressions, just like with regular fractions.

In our example, we start with \(\frac{8}{3-7y}-\frac{2}{7y-3}\). Notice that the denominators are different. To add or subtract these rational expressions, we first have to make the denominators the same. We observe that the denominators 3-7y and 7y-3 are actually negatives of each other. This allows us to adjust the second fraction so it has the same denominator as the first.

With a common denominator, adding or subtracting rational expressions becomes straightforward by combining the numerators over this shared base.

Simplifying denominators is a crucial step. It's the task of transforming different denominators into a common one. Initially, we had 3-7y and 7y-3. These look different but if we factor them, we notice:\br> - 3 - 7y can be written as -(7y - 3). \br> We can leverage this relationship to simplify our work in this problem. \br> By rewriting the second fraction \(\frac{2}{7y-3}\) as \(-\frac{2}{3-7y}\), we now have two fractions with the same denominator. This simplification allows us to move on to combining the terms.

Once the rational expressions have the same denominator, we can combine them by adding their numerators together. This is similar to adding like terms in algebra.

In this case:
\(\frac{8}{3 - 7y} + -\frac{2}{3 - 7y}\)

We combine the numerators as follows:
\(\frac{8 + 2}{3 - 7y} = \frac{10}{3 - 7y}\)

Thus, we get a single fraction with the numerator resulting from the sum of the initial numerators and the common denominator.

Finally, always check if the resulting fraction is in its lowest terms. In our case \(\frac{10}{3 - 7y}\) is already simplified as 10 and 3 - 7y have no common factors.