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Isolating the target variable is a key step in any equation solving process. In our exercise, we are required to solve the equation for the variable \(h\). This means that we must manipulate the equation such that \(h\) is alone on one side of the equation. It requires moving all other terms away from \(h\) to the opposite side.
To start, identify the term with \(h\) in it. In the equation \(S = 2 \pi r^2 + 2 \pi r h\), \(2 \pi r h\) is the term involving \(h\). To isolate \(h\), we need to get rid of everything else on the same side of the equation. Start by eliminating constant terms that do not involve \(h\), by performing opposite operations like addition or subtraction.
Once you have collected all \(h\)-related terms on one side, divide or multiply to get \(h\) by itself. This technique of moving terms from one side to another is at the heart of isolating variables.

Algebraic manipulation involves applying various mathematical operations to simplify equations and solve for variables. In our example, after expressing the equation as \(S = 2 \pi r^2 + 2 \pi r h\), the next logical step is to

• Subtract \(2\pi r^2\) from both sides.

This subtraction is a fundamental algebraic manipulation step that helps remove constant terms from the side of the equation where the variable we are solving for, \(h\), is located.
Next, algebraic manipulation calls for dividing every term by \(2 \pi r\) to further isolate \(h\). By understanding operations in algebra, such as subtraction and division, we can transform equations into simpler forms. These operations can be reversed and applied to both sides of an equation without changing the balance, which is key in algebraic manipulation.
Mastering these simple steps will enable you to tackle more complex equations over time.

Solving equations involves a series of methodical steps. The objective is to find the value of the unknown variable by rearranging the equation appropriately. The original step-by-step solution illustrates how to solve the given equation for the variable \(h\). Here’s a quick review of the essential steps involved in solving an equation like this:

• **Identify the Variable:** Recognize the variable you need to solve for. In our problem, it's \(h\).
• **Remove Constant Terms:** Start by subtracting or adding constants to both sides, as needed. In our case, subtract \(2\pi r^2\).
• **Isolate the Variable:** Use multiplication or division to isolate the variable, \(h\), by dividing by \(2 \pi r\).

These steps are simplifications of more complex algebraic processes. By following this approach, you can methodically solve for unknowns in equations. Keep practicing these steps with different equations, and they will become second nature.