Problem 12 - Algebra A Combined Function

Question

Solve each formula for the specified variable. \(x=4 \pi y\) for \(y\)

Step-by-Step Solution

Answer

\( y = \frac{x}{4\pi} \)

Step 1: Understand the Equation

We have the equation \( x = 4\pi y \). Our goal is to solve for \( y \), which means we need to express \( y \) in terms of \( x \) and \( \pi \).

Step 2: Isolate the Variable

To isolate \( y \), we need to divide both sides of the equation by \( 4\pi \). This will remove \( 4\pi \) from the right-hand side of the equation.

Step 3: Perform the Division

Divide both sides by \( 4\pi \) to get:\[ y = \frac{x}{4\pi} \]

Step 4: Simplify the Expression

The expression \( y = \frac{x}{4\pi} \) is as simple as it can get. Here, \( y \) is now expressed in terms of \( x \) and \( \pi \).

Key Concepts

Isolating Variables

Isolating variables is a fundamental skill in algebra that involves rearranging an equation so that a particular variable stands alone on one side of the equation. When solving for a variable, like in the problem where we solve for \( y \) in the equation \( x = 4\pi y \), our objective is to have \( y \) on one side of the equation by itself.

Steps to isolate variables include:

• Identify the variable you want to solve for.
• Use inverse operations to get rid of other terms that are around the variable.
• Ensure you apply the same operation on both sides to maintain the balance of the equation.

In the current exercise, we achieve isolation by dividing both sides of the equation by \( 4\pi \), letting \( y \) be equal to \( \frac{x}{4\pi} \) and thus completely isolated.

Algebraic Manipulation

Algebraic manipulation refers to the use of mathematical techniques to simplify or rearrange equations or expressions. It's all about using various operations to achieve a clearer, more straightforward equation that serves the purpose we need.

There are several basic steps you can follow:

• Simplifying both sides of the equation.
• Applying inverse operations such as addition/subtraction or multiplication/division.
• Using distributive property if needed.

In our step-by-step solution, we manipulated the equation \( x = 4\pi y \) by dividing both sides by \( 4\pi \), which is algebraically manipulating the equation to achieve our goal. These manipulations allowed us to conveniently express \( y \) in terms of \( x \) and \( \pi \).

Mathematical Expressions

Understanding mathematical expressions is key to effectively solving algebraic problems. A mathematical expression is a combination of numbers, variables, and mathematical operations (like addition or multiplication). They form the backbone of equations that we solve.

Here’s how you can approach them:

• Recognize different types of terms like constants, coefficients, and variables in the expression.
• Identify the mathematical operations and their order.
• See how these terms interrelate to each other to form an equation.

In the problem at hand, the expression \( x = 4\pi y \) initially shows how \( x \) is a product of \( 4\pi \) and \( y \). We then rearrange these terms to make sense of the relationship defined by the equation, ultimately expressing \( y \) as \( \frac{x}{4\pi} \). Understanding these components truly aids in the re-evaluation and resolution of mathematical problems in algebra.