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Chapter 1: Problem 16

Simplify the given algebraic expressions. $$-2 a-\frac{1}{2}(b-a)$$

Short Answer

Expert verified
The simplified expression is \(-\frac{3}{2}a - \frac{1}{2}b\).

Step by step solution

01

Distribute the Fraction

The expression to simplify is \(-2a - \frac{1}{2}(b-a)\). Start by distributing the \(\frac{1}{2}\) across the terms inside the parentheses. This gives us: \(-2a - \frac{1}{2}b + \frac{1}{2}a\).
02

Combine Like Terms

Now, look at the expression from Step 1: \(-2a + \frac{1}{2}a - \frac{1}{2}b\). Combine the like terms involving \(a\). Combine \(-2a\) and \(\frac{1}{2}a\) to get \(-2a + \frac{1}{2}a = -\frac{4}{2}a + \frac{1}{2}a = -\frac{3}{2}a\).
03

Write the Simplified Expression

After combining all like terms, the simplified expression is \(-\frac{3}{2}a - \frac{1}{2}b\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributing Fractions
When you're dealing with algebraic expressions that include fractions, it's important to understand how to properly distribute the fraction to each term inside the parentheses. Distributing involves multiplying the fraction by each term within the parentheses.
Consider the exercise: \(-2a - \frac{1}{2}(b-a)\). To distribute \(\frac{1}{2}\) here:
  • Multiply \(\frac{1}{2}\) by \(b\) to get \(\frac{1}{2}b\)
  • Multiply \(\frac{1}{2}\) by \(-a\) to get \(-\frac{1}{2}a\)
This will transform the expression to \(-2a - \frac{1}{2}b + \frac{1}{2}a\). Breaking down each step when distributing fractions ensures that you apply the fraction correctly across each term, helping you to simplify the expression accurately.
Combining Like Terms
Combining like terms is a key part of simplifying algebraic expressions. Like terms are terms that contain the same variable raised to the same power, which can be added or subtracted together.
After distributing the fraction, you end up with the expression:\(-2a - \frac{1}{2}b + \frac{1}{2}a\).
  • Identify like terms, in this case, \(-2a\) and \(\frac{1}{2}a\).
  • Combine them by adding: \(-2a + \frac{1}{2}a\).
  • First, rewrite \(-2a\) as \(-\frac{4}{2}a\) to match the denominator with \(\frac{1}{2}a\).
  • This combining step yields \(-\frac{3}{2}a\).
Using this method ensures that you compute correctly, resulting in \(-\frac{3}{2}a - \frac{1}{2}b\). By clearly identifying and combining like terms one by one, you simplify expressions while maintaining accuracy.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations (such as addition and multiplication). Knowing how to simplify these expressions eases solving equations and inequalities.
In the expression \(-2a - \frac{1}{2}(b-a)\), we saw elements like fractions, variables like \(a\) and \(b\), and operations that needed to be performed.
Simplification involves:
  • Distributing: Apply operations within parentheses.
  • Combining: Merge like terms to reduce the expression.
The final simplified expression, \(-\frac{3}{2}a - \frac{1}{2}b\), is the result of proper application of these processes. Understanding how to manipulate algebraic expressions not only aids in simplifying but also sets up a foundation for more complex algebraic tasks.

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Most popular questions from this chapter

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