Problem 74 - Elementary Algebra

Question

Solve for the indicated variable. Solve for \(S: h=S 2 \pi r-r\)

Step-by-Step Solution

Answer

S = \( \frac{h + r}{2\pi r} \)

Step 1: Arrange the equation

The given equation is \( h = S 2 \pi r - r \). Our goal is to solve for \( S \). We start by rearranging it into a standard equation format by isolating the terms with \( S \) on one side: \( h + r = S 2\pi r \).

Step 2: Isolate S

To isolate \( S \), we divide both sides of the equation by \( 2\pi r \): \( S = \frac{h + r}{2\pi r} \).

Step 3: Simplify the solution

Now the equation is solved for \( S \): \( S = \frac{h + r}{2\pi r} \). Ensure all terms are confirmed and correctly simplified in this final expression.

Key Concepts

Variable Isolation

When solving equations, one crucial step often involves isolating a specific variable. In this context, isolating a variable means getting it alone on one side of the equation without any other variables or numbers attached. By focusing on keeping only the target variable, you make the equation easier to solve.To do this, you need to perform operations that undo any arithmetic surrounding the variable.

• If the variable is being added or subtracted, you'll do the opposite. For instance, if something is added to your variable, you'll subtract that same thing from both sides of the equation.
• If the variable is inside brackets or multiplied by something, usually you'll need to divide or use the distributive property to separate it.

In our example equation \( h = S 2 \pi r - r \), we needed to focus on modifying the equation so that \( S \) stands alone, which means placing all terms with \( S \) on one side. This practice simplifies the path to finding the solution.

Rearranging Equations

Rearranging equations is like organizing your workspace; it helps make the solving process much clearer. This task involves moving terms from one side of the equation to the other, usually to achieve variable isolation. It requires a good understanding of the properties of equality, such as adding, subtracting, multiplying, or dividing both sides by the same number.To rearrange equations successfully:

• Look for terms that are not needed on one side and move them to the other. This means carefully adding or subtracting terms across the equals sign.
• Keep the equation balanced. Whatever you do to one side, you must do to the other.

In the provided step-by-step solution, to solve for \( S \) from \( h = S 2 \pi r - r \), we first rearranged it to \( h + r = S 2 \pi r \). By moving \( -r \) to the other side as \( +r \), we made the equation ready to isolate \( S \). Rearranging is about setting the stage for the actual solving.

Simplifying Expressions

The last essential step in solving an equation is simplifying the expression. Simplification makes your final answer clearer and more understandable. It's crucial to ensure you have done every possible arithmetic operation to express the variable as neatly as possible.In our solved equation \( S = \frac{h + r}{2\pi r} \), the simplification involved two main tasks:

• Combining any like terms and performing all possible arithmetic. Check if anything else can be simplified further, such as factors that can cancel out.
• Writing the answer in the simplest form, which helps avoid any redundancy.

Simplification ensures that the solution is as straightforward as it can be, making it easy to interpret and apply. Don't overlook the importance of checking your work to confirm that everything is in its simplest possible form. Simplification wraps up your work neatly, providing a clean solution to the problem.