Problem 37 - College Algebra

Question

For the following exercises, 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: Simplify the Equation

First, we simplify the expression inside the parentheses: \[ \frac{1}{4}(8w - 4^2) = 0 \]Calculate \(4^2\), which is 16, and substitute it back into the equation: \[ \frac{1}{4}(8w - 16) = 0 \].

Step 2: Distribute the Fraction

Distribute \(\frac{1}{4}\) to both terms in the parentheses:\[ \frac{1}{4} \times 8w - \frac{1}{4} \times 16 = 0 \]This results in:\[ 2w - 4 = 0 \].

Step 3: Solve for \(w\)

Add 4 to both sides of the equation to isolate the \(2w\) term:\[ 2w = 4 \]Next, divide both sides by 2 to solve for \(w\):\[ w = 2 \].

Key Concepts

Distribution Property

The distribution property is a key concept in algebra, which allows you to multiply a term distributed across terms inside parentheses. This property is crucial for simplifying equations, especially when dealing with algebraic expressions.

• When you have an expression like \(a(b + c)\), the distribution property lets you rewrite it as \(ab + ac\).
• In our example, we applied this rule by distributing the fraction \(\frac{1}{4}\) across \(8w - 16\), changing the expression into \(2w - 4\).

Understanding this property helps simplify complex expressions and make solving equations more straightforward.

Isolating Variables

Isolating variables is a central concept in solving equations, allowing us to find the value for a variable. The main goal is to have the variable on one side of the equation, making it easier to determine its value.

• In an equation like \(2w - 4 = 0\), you need to manipulate the equation to isolate \(w\).
• This involves performing operations such as addition, subtraction, multiplication, or division on both sides of the equation to maintain balance.
• In our step-by-step solution, we added 4 to both sides to get \(2w = 4\), and then divided by 2 to solve for \(w\).

By practicing isolating variables, you become adept at solving for unknowns in any equation.

Simplification

Simplification is the process of making a mathematical expression easier to work with or solve. It involves reducing the expression to its simplest form without changing its value.

• Initially, complex expressions with multiple operations can be daunting to solve.
• By simplifying, you can reduce errors and ease the computation. In our example, we simplified \(\frac{1}{4}(8w - 4^2)\) into \(\frac{1}{4}(8w - 16)\) by calculating \(4^2\) as 16.

Simplifying is a first crucial step in solving equations that makes the other steps more manageable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Understanding how to manipulate these expressions is fundamental in algebra.

• Our example began with the expression \(\frac{1}{4}(8w - 4^2)\).
• An algebraic expression can become an equation if it includes an equality sign \(=\), meaning the expression must be balanced on either side.
• Knowing how to interpret and reorganize parts of these expressions is essential for solving equations.

Grasping the basics of algebraic expressions prepares you to tackle a wide variety of mathematical problems beyond simple equations.