Problem 57 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ \frac{1}{3}(9 x-6)-(x-2) $$

Step-by-Step Solution

Answer

The simplified expression is \(2x\).

Step 1: Distribute the Fraction Inside the Parentheses

First, apply the distributive property to remove the parentheses that involve the fraction. This means multiplying each term inside the parentheses by \( \frac{1}{3} \). Thus, \( \frac{1}{3}(9x - 6) \) becomes \((\frac{1}{3} \times 9)x - (\frac{1}{3} \times 6)\). This simplifies to \(3x - 2\).

Step 2: Remove Parentheses and Simplify

Next, remove the parentheses from the second expression, \(-(x - 2)\). This is done by distributing the negative sign across the terms inside the parentheses. So, \(-(x - 2)\) becomes \(-x + 2\).

Step 3: Combine Like Terms

Now, combine like terms from the expression \(3x - 2 - x + 2\). Group the \(x\) terms and the constant terms separately. So, \(3x - x\) is \(2x\), and \(-2 + 2\) is \(0\). This simplifies the expression to \(2x\).

Key Concepts

Distributive Property

In algebra, the Distributive Property is a fundamental tool for simplifying expressions. It states that if you have an expression in the form of \( a(b + c) \), you can distribute \( a \) across the terms inside the parentheses. This means multiplying \( a \) with every term within the parentheses. To illustrate:

• \( a(b + c) = ab + ac \)

By applying this property, you effectively "distribute" the multiplier to each term. Let's break down its use in the context of the exercise:The exercise \( \frac{1}{3}(9x - 6) \) employs the distributive property by multiplying each term inside the parentheses by \( \frac{1}{3} \). Breaking it down:

• \( \frac{1}{3} \times 9x = 3x \)
• \( \frac{1}{3} \times -6 = -2 \)

Thus, the expression simplifies to \( 3x - 2 \), showing how the distributive property helps in removing parentheses and simplifying terms.

Simplifying Expressions

Simplifying Expressions is an essential step in algebra. It involves reducing a complex expression to its simplest form while maintaining its original value. This is often done using various algebraic properties, such as the distributive property, and involves getting rid of any unnecessary parentheses and combining terms.In our given expression \( \frac{1}{3}(9x - 6) - (x - 2) \), simplifying begins with removing parentheses. We first use the distributive property on \( \frac{1}{3}(9x - 6) \) to get \( 3x - 2 \). Next, simplify \( -(x - 2) \) by distributing the negative sign:\

• This changes \( -(x - 2) \) to \(-x + 2\).

Once parentheses are removed, the expression becomes \( 3x - 2 - x + 2 \). By simplifying, you prepare the expression for the final step of combining like terms. Avoid skipping these simplification steps as they align the terms, making the subsequent process straightforward. As a result, understanding and applying these techniques is foundational in algebra.

Combining Like Terms

Combining Like Terms is the final step in simplifying algebraic expressions and involves combining terms that have the same variable raised to the same power. This step is crucial for reducing expressions to their simplest form.Consider the expression from the exercise after using the distributive property and removing parentheses: \( 3x - 2 - x + 2 \). Here, you have terms with \( x \) and constant terms.

• First, focus on the \( x \) terms: \( 3x - x \) which simplifies to \( 2x \).
• Then, look at the constant terms: \( -2 + 2 \) which equals \( 0 \).

This leads us to the simplified expression: \( 2x \).By combining like terms, you effectively consolidate the expression, making it much easier and cleaner. It's important to carefully group similar terms and carefully perform arithmetic operations to ensure accuracy in your results.