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The distributive property is a crucial concept in algebra that helps us deal with expressions containing parentheses. It allows us to remove parentheses by distributing a factor across terms inside the parentheses. In this exercise, we began with the expression \(3x - (5x - 9)\). Here, the negative sign before the parentheses acts as a factor, and we need to distribute it.

When you distribute a negative sign, it changes the sign of each term inside the parentheses. So, \((5x - 9)\) turns into \(-5x + 9\). Thus, the expression becomes \(3x - 5x + 9\).

This step is crucial as it simplifies the expression and allows other simplification steps like combining like terms to follow. Remember, the distributive property can also be used in more complex expressions where you distribute a specific number or variable over multiple terms inside the parentheses, such as \(a(b + c) = ab + ac\).

Combining like terms is an essential technique used to simplify algebraic expressions. Like terms are terms that have the same variable raised to the same power. In this exercise, after using the distributive property, we have the expression \(3x - 5x + 9\).

Here, \(3x\) and \(-5x\) are like terms because they both contain the variable \(x\). To combine like terms, sum or subtract their coefficients. In our example, we subtract because of the negative sign:

• \(3x - 5x = -2x\)

Once combined, we arrive at a simplified expression. Always ensure that terms are truly "like" by checking their variables and powers before combining. When done correctly, this step condenses your expression, making it easier to work with.

Simplifying expressions is about making an expression as short and easy to manage as possible, without changing its value. You achieve this by performing operations such as distributing property and combining like terms.

In the exercise at hand, the initial expression was simplified by first applying the distributive property, resulting in \(3x - 5x + 9\), and then combining like terms \(3x\) and \(-5x\) to create \(-2x + 9\).

After these simplifications, we achieve the most simplified version of the expression:

• \(-2x + 9\)

This process can be applied to much more complicated algebraic expressions. The key steps are always recognizing similar operations needed, like distribution and combination, to ultimately simplify accurately. Simplification not only refines an expression but also makes subsequent problem-solving much easier.