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The Distributive Property is a crucial concept in algebra. It allows us to simplify expressions by distributing one term across terms inside a set of parentheses. This property is used to eliminate parentheses and enable straightforward arithmetic. The general form is: \[a(b + c) = ab + ac.\] In the original exercise, we first see this property applied to \[-2(6x - 3y).\] This means multiplying each term inside the parentheses by \(-2\). Here's how it looks step-by-step:

• Multiply \(-2\) by \(6x\), giving: \(-12x.\)
• Multiply \(-2\) by \(-3y\), giving: \(+6y.\)

The result after applying the distributive property is: \[-12x + 6y.\] A common hurdle students face is correctly distributing negative signs. Remember, if the term outside the parentheses is negative, it will change the sign of each term inside.

Once you have used the Distributive Property, it is time to simplify the algebraic expression. This involves combining like terms and making the expression as straightforward as possible. The simplification process makes the expression easier to read and solves problems more efficiently. In our provided exercise, after distributing the terms, we end up with: \[-12x + 6y - 5y + 4x.\] Now, the goal is to group similar terms together. "Like terms" means terms that have the same variable raised to the same power, even though their coefficients might differ. This process involves:

• Adding coefficients of the same variable types:
• \(-12x + 4x\) simplifies to \(-8x\).
• \(6y - 5y\) simplifies to \(+1y\) or \(y\).

By pooling together like terms, the expression becomes cleaner, making it easier to understand and work with.

Understanding and identifying like terms is vital when simplifying expressions in algebra. Like terms are those that contain the same variable component and are raised to the same power, though they can have different coefficients. In the expression \[-12x + 6y - 5y + 4x,\] "like terms" are grouped based on the variable and exponent, not the coefficient. For example:

• \(-12x\) and \(+4x\) are like terms because they both contain the variable \(x\).
• \(6y\) and \(-5y\) are like terms since they both have the variable \(y\).

This allows us to combine them by adding or subtracting their coefficients, leading to a simplified expression: \[-8x + y.\] Recognizing like terms quickly is a key skill in algebra that greatly aids in simplifying complex expressions and solving equations efficiently.