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Simplifying expressions is an essential skill in algebra that allows you to rewrite expressions in a clearer and more concise manner. It helps in understanding the relationships between different variables and coefficients. Simplification involves removing or combining terms to make an expression easier to evaluate or compare.
You usually achieve this by:

• Applying mathematical operations like addition or subtraction.
• Using properties like the distributive property and commutative property.
• Combining like terms to get more manageable terms.

In our exercise, the goal was to transform a given expression into its simplest form: \(-2a - \frac{1}{2}(b-a)\). By identifying and combining like terms, and applying distributions effectively, you ensure calculations are streamlined and results are consistent.

The distributive property is a helpful algebraic principle that lets you multiply a single term by the terms inside a parenthesis. In mathematical terms, it is expressed as: \(a(b + c) = ab + ac\). This property ensures that every term inside the parenthesis is multiplied by the term outside, simplifying multiplication of terms within other operations like addition or subtraction.
In our exploration, we used the distributive property to address the term \(-\frac{1}{2}(b-a)\). By distributing \(-\frac{1}{2}\) across \(b\) and \(-a\), we reap the correlation:
- \(-\frac{1}{2} \cdot b = -\frac{1}{2}b\)
- \(-\frac{1}{2} \cdot -a = \frac{1}{2}a\)
This step eliminates the parenthesis in the expression, readying the terms for further simplification, and making it easier to combine terms in subsequent steps.

Combining like terms is the process of simplifying an expression by merging terms that have the same variable raised to the same power. This concept is essential in making expressions more readable and typically involves:

• Identifying terms with the same variable and exponents.
• Adding or subtracting their coefficients.

For example, in our expression \(-2a - \frac{1}{2}b + \frac{1}{2}a\), both \(-2a\) and \(\frac{1}{2}a\) are like terms because they contain the variable \(a\). We combine them by adding or subtracting their coefficients:
\[-2a + \frac{1}{2}a = -\frac{4}{2}a + \frac{1}{2}a = -\frac{3}{2}a\]
This method of combining ultimately simplifies the algebraic expression further, resulting in \[-\frac{3}{2}a - \frac{1}{2}b\]. Through combining like terms, the expression streamlines into a format that's easier to understand or solve for different values of the variables.