Problem 37 - College Algebra

Question

Solve for the variable. $$ \frac{1}{4}\left(8 w-4^{2}\right)=0 $$

Step-by-Step Solution

Answer

The solution is \( w = 2 \).

Step 1: Distribute the Fraction

First, distribute the fraction \( \frac{1}{4} \) across the terms within the parentheses. Apply it to both \( 8w \) and \( 4^2 \). This gives:\[\frac{1}{4} \times 8w - \frac{1}{4} \times 4^2 = 0\]Simplifying each term:\[2w - 4 = 0\]

Step 2: Rearrange the Equation

The goal is to isolate \( w \), the variable. Start by adding 4 to both sides of the equation:\[2w - 4 + 4 = 0 + 4\]This simplifies to:\[2w = 4\]

Step 3: Solve for w

Now, divide both sides by 2 to solve for \( w \):\[\frac{2w}{2} = \frac{4}{2}\]This results in:\[w = 2\]

Key Concepts

Distributive Property

When we encounter expressions with parentheses, particularly when a number or fraction is placed outside, the distributive property helps us simplify. In this exercise, we have \( \frac{1}{4}(8w - 4^2) \). We need to apply the fraction to both terms inside the parentheses. This means:

• Multiply \( \frac{1}{4} \times 8w \)
• And \( \frac{1}{4} \times 4^2 \)

Performing these operations gives us \( 2w - 4 \), since \( \frac{1}{4} \times 8w \) simplifies to \( 2w \), and \( \frac{1}{4} \times 16 \) simplifies to \( 4 \). This step makes our expression easier to handle, setting us up for the next steps in solving the equation.

Isolating Variables

After simplifying the expression using the distributive property, we need to focus on isolating the variable \( w \). This process involves performing operations to both sides of the equation until the variable is by itself. Initially, our expression is \( 2w - 4 = 0 \).To isolate \( w \), we start by eliminating the constant term \(-4\) on the left side. Adding 4 to both sides achieves this:

• \( 2w - 4 + 4 = 0 + 4 \)

This makes the equation: \( 2w = 4 \). Our goal is closer as \( w \) is now only paired with 2. In the next phase, we will focus on solving for \( w \) entirely.

Simplifying Algebraic Expressions

Simplifying is crucial throughout algebra to make equations easier to solve. After isolating terms, as seen in our equation \( 2w = 4 \), we simplify further to find the value of \( w \).To do this, divide each term by 2:

• \( \frac{2w}{2} = \frac{4}{2} \)

This operation leaves us with \( w = 2 \). Through a series of simplifications, starting from distributing fractions and isolating variables, we simplify the algebraic expression to its most basic form. It illustrates why understanding these core concepts in algebra can be powerful tools for solving equations effectively.