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Understanding how to solve equations is a crucial part of algebra. An equation shows the relationship between two expressions, using an equal sign. To solve an equation, you need to find the value of the variable that makes the equation true. This process includes several steps:

• Identify the equation.
• Transform the equation using algebraic operations to isolate the variable.
• Perform the necessary calculations to find the solution.

In the given example, you start with the equation \[ \frac{C}{r} = 2\pi \]. The goal is to find the value of \( r \) that makes this equation true. Remember that each algebraic manipulation must maintain the balance of the equation.

Isolating a variable means getting the variable by itself on one side of the equation. This often involves reversing the operations that are currently applied to the variable. In the given exercise, we need to isolate the variable \( r \) from the equation. Here are the steps we take:
First, the original equation is \[ \frac{C}{r} = 2\pi \]. To isolate \( r \), multiply both sides by \( r \). This gives us: \[ r \cdot \frac{C}{r} = 2\pi \cdot r \]
This simplifies to: \[ C = 2\pi r \]
Now, \( r \) is no longer in the denominator and is multiplied by \( 2\pi \). To complete the isolation, divide both sides by \( 2\pi \): \[ \frac{C}{2\pi} = r \]
This shows \( r \) isolated and solved for: \[ r = \frac{C}{2\pi} \]

Algebraic manipulation is the process of rearranging and simplifying equations using algebraic rules. This includes addition, subtraction, multiplication, division, and using properties of equality. Let's breakdown our manipulation from the given example:
1. Start with the equation: \[ \frac{C}{r} = 2\pi \]
2. To remove \( r \) from the denominator, multiply both sides by \( r \): \[ r \cdot \frac{C}{r} = 2\pi \cdot r \]
This simplifies to: \[ C = 2\pi r \]
3. Finally, divide both sides by \( 2\pi \) to isolate \( r \): \[ \frac{C}{2\pi} = r \]

• Combining like terms and factoring when necessary.
• Keeping the equation balanced by performing the same operation on both sides.

These steps show the careful and logical approach needed to perform algebraic manipulations. Understanding these methods helps in solving more complicated algebraic problems.