Problem 347 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ \frac{1}{4}(3 q+12) $$

Step-by-Step Solution

Answer

\( \frac{3}{4}q + 3 \)

Step 1: Identify the Expression Inside the Parentheses

The given expression is \( \frac{1}{4}(3q + 12) \). Identify the terms inside the parentheses, which are \(3q\) and \(+12\).

Step 2: Apply the Distributive Property

The Distributive Property states that \(a(b + c) = ab + ac\). Here, \(a = \frac{1}{4}\), \(b = 3q\), and \(c = 12\). Distribute \(\frac{1}{4}\) to each term inside the parentheses: \[ \frac{1}{4} \times 3q + \frac{1}{4} \times 12 \]

Step 3: Perform the Multiplications

Now, multiply each term: \[ \frac{1}{4} \times 3q = \frac{3}{4}q \] and \[ \frac{1}{4} \times 12 = 3 \]

Step 4: Combine the Results

Combine the simplified terms to get the final simplified expression: \[ \frac{3}{4}q + 3 \]

Key Concepts

Algebraic Simplification

Simplifying an algebraic expression means making it as straightforward as possible without changing its value. This often involves using properties like the Distributive Property. It helps in making complex expressions more manageable.
For example, when you simplify \(\frac{1}{4}(3q + 12)\), you distribute the \(\frac{1}{4}\) across each term in the parentheses. This reduces the given expression to a simpler form that is easy to understand.
Understanding how to simplify algebraic expressions is crucial, as it forms the basis for solving more complex algebra problems. Practice this regularly to master the skill.

Fraction Multiplication

Multiplying fractions is another essential skill in algebra. When you see a fraction multiplied by a term or an expression, you'll need to distribute that fraction across every term.
For the exercise \(\frac{1}{4}(3q + 12)\), you multiply \(\frac{1}{4}\) by each term in the parentheses:

• \(\frac{1}{4} \times 3q = \frac{3}{4}q\)
• \(\frac{1}{4} \times 12 = 3\)

So, \(\frac{1}{4}(3q + 12)\) simplifies to \(\frac{3}{4}q + 3\).
It's important to understand how fraction multiplication works to avoid mistakes and ensure your overall expression is simplified correctly.

Combining Like Terms

Combining like terms is the process of simplifying expressions by adding or subtracting terms that have the same variables raised to the same power.
In the exercise we considered, there are no like terms directly to combine in the final result \(\frac{3}{4}q + 3\). However, knowing how to combine like terms is fundamental as you work on more complex expressions.
To combine like terms, look for terms that have the same variable and exponent:

• For example, \2x + 3x\ can be combined to \5x\.
• Terms like \2x\ and \3y\ cannot be combined because they have different variables.

This helps in further simplifying algebraic expressions, making them much easier to solve.