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Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes the equation true. In the given exercise, you are asked to solve the formula for the variable 'r'.
The steps involve isolating the variable and then finding its value.
Breaking down the steps can help in understanding the process more clearly.
For example:
• Identify the formula and the variable to solve for.
• Use algebraic operations to isolate the variable.
• Simplify the equation as much as possible.
• Solve for the variable and remember to consider the context (like ensuring the radius is non-negative).
These steps can be applied to solve various types of equations in algebra.

Algebraic manipulation involves using mathematical operations to rearrange and simplify equations. In the given formula, you perform algebraic manipulation to isolate the variable 'r'.
Here's the detailed breakdown:
1. Start with the equation: \( S = 4 \text{Ï€} r^{2} \)
2. Divide both sides by \( 4 \text{Ï€} \) to isolate \( r^{2} \): \( \frac{S}{4 \text{Ï€}} = r^{2} \)
3. Take the square root of both sides to solve for \( r \): \( r = \sqrt{ \frac{S}{4 \text{Ï€}} } \)
Each step uses basic algebraic operations like division and taking the square root. These operations help simplify the equation and solve for the variable.
Mastering algebraic manipulation is crucial for solving various mathematical problems, not just equations.

Formulas in algebra provide a way to represent relationships between different variables. The formula given in the exercise, \( S = 4 \text{Ï€} r^{2} \), represents the surface area of a sphere in terms of its radius. When solving formulas for a specified variable:
• Identify the variables involved.
• Understand the relationship between the variables as given by the formula.
• Apply algebraic operations to solve for the desired variable.
In this exercise, you learned how to solve for 'r' when given the surface area 'S'.
Formulas often arise in geometry, physics, and real-world applications. Understanding how to manipulate and solve them is essential for tackling complex problems in various fields.