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When simplifying algebraic expressions, one fundamental step is to combine like terms. Like terms are terms that contain the same variables, raised to the same power. For example, in the expression \(2a/ y + 3/y - a/y\), the terms \(2a/y\) and \(a/y\) are like terms because they both contain the variable \(a\) to the first power and share a common denominator, \(y\).

Combining like terms involves performing arithmetic operations on the coefficients (the numerical part of the terms) while keeping the variable part unchanged. To combine the like terms in our example, we subtract the coefficient of \(a/y\) (which is 1) from the coefficient of \(2a/y\) (which is 2), resulting in \(a/y\). We can leave the constant term \(3/y\) as it is because it does not have a like term to combine with. This process leads us to a more simplified expression.

Simplifying expressions means reducing them to their simplest form, making them easier to work with or interpret. Simplification includes combining like terms, as mentioned previously, and also can involve other operations such as distributing, canceling, and factoring.

After combining like terms, you might need to simplify the expression further. For instance, in our example \(a + 3/y\), there's nothing more to simplify since the terms do not have the same variables and thus cannot be combined. Simplifying expressions makes them neater and often reveals a more direct way to derive solutions for equations or inequalities.

A common denominator is necessary when adding or subtracting fractions. It is the shared multiple of the denominators of two or more fractions. In our exercise, where we have \(2a/y + 3/y - a/y\), the common denominator is \(y\).

Having a common denominator allows us to add or subtract fractions directly by combining their numerators. This is because fractions with the same denominator represent parts of the same whole. In contrast, fractions with different denominators represent parts of wholes that are of different sizes, and thus additional steps are needed to find a common denominator before combining them. In summary, finding a common denominator simplifies the process of adding or subtracting fractions by standardizing the parts to a single whole.