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The distributive property is a fundamental principle in algebra that allows you to simplify expressions by removing parentheses. It states that for any numbers or variables \(a, b,\) and \(c\), the expression \(a(b + c)\) is equivalent to \(ab + ac\).

For instance, in the given exercise, we need to distribute \(3\) across the terms inside the parentheses: \(3(x + y)\). This gives us \(3x + 3y\).

Similarly, we distribute \(4x\) across \(-y\) in the expression \(4x(-y)\) to get \(-4xy\). By using the distributive property properly, we break down complex expressions into simpler components, making our calculations more straightforward and organized.

Remember, the distributive property works both ways: it can be used to expand expressions (as shown here) or to factor them down.

Combining like terms is crucial for simplifying algebraic expressions further after distribution. Like terms are terms that have the same variable raised to the same power.

For example, \(2x\) and \(5x\) are like terms because both are simply multiples of \(x\).

In the exercise, after distributing the terms, we have: \(3x + 3y - 4xy\). Now, we look for terms that can be combined.

\(3x\) and \(3y\) are not like terms because they contain different variables. Similarly, \(3x\) and \(-4xy\) are not like terms because \(-4xy\) has an additional variable \(y\). Therefore, the expression has no like terms to combine further. It is crucial to acknowledge when you cannot simplify further and understand that the expression is in its simplest form.

Algebraic simplification involves transforming an expression into its simplest form. This process requires recognizing when to apply the distributive property and combine like terms.

Let's revisit the expression from the exercise: \(3(x + y) + 4x(-y)\).

• Step 1: Distribute the terms: \(3(x + y) = 3x + 3y\) and \(4x(-y) = -4xy\).
• Step 2: Combine the simplified terms: \(3x + 3y - 4xy\).

In this final expression, there are no like terms to combine. Therefore, it is considered simplified. Recognizing this ensures efficiency and correctness in algebra.

Simplification not only makes expressions easier to understand but also sets the stage for solving equations. Always verify that all possible terms are combined and the expression is fully simplified.