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Equation solving involves finding the value of a variable that makes an equation true. In this case, we started with the equation \( m = p \sqrt{\frac{s+t}{s-t}} \). Our goal was to rearrange this equation to solve for the variable \( s \). By squaring both sides in Step 1, we aimed to eliminate the square root. This is a common technique in equation solving that simplifies the equation and makes it easier to handle. Squaring both sides transformed the equation into \( m^2 = p^2 \frac{s+t}{s-t} \), paving the way for further manipulation. Breaking down complex equations into a series of smaller, manageable steps is essential in equation solving. It reduces chances for errors and provides a systematic approach to reach the solution. Let's look at how breaking down a complex problem into simpler steps helps facilitate the solving process. - **Start with simplifying**: Look for any operations, like square roots or fractions, to simplify. - **Isolate the target variable**: Always aim to get the variable for which you need a solution on one side of the equation. - **Keep the equation balanced**: Any operation performed on one side must be done to the other.

Variable isolation is a fundamental skill in algebra that involves rearranging an equation to get the variable of interest alone on one side. In the original exercise, we needed to isolate \( s \). After eliminating the square root and dealing with fractional terms, isolation becomes crucial. This began after we got the equation in the form \( m^2 (s-t) = p^2 (s+t) \) in Step 2. To isolate \( s \), we moved all terms involving \( s \) to one side of the equation. Once there, we factored \( s \) out of the expression. Factoring is a powerful tool in algebra allowing us to pull a common factor from an expression, simplifying further steps of isolation. Ultimately, we obtained \( s(m^2 - p^2) = p^2 t + m^2 t \). - **Identify terms with the variable**: Move terms involving the variable of interest to one side of the equation. - **Factor whenever possible**: This step drastically simplifies equations, especially when solving for specific variables. - **Use inverse operations**: To isolate a variable, effectively undo operations like addition/subtraction or multiplication/division.

Mathematical manipulation refers to the strategic use of mathematical operations to rearrange equations. Each step utilized in the original problem enlists a form of manipulation. After ensuring the equation \( m^2 = p^2 \frac{s+t}{s-t} \) was free of square roots and fractions, manipulation allowed us to balance and simplify the equation further. Throughout this process, operations such as cross-multiplying, expanding, and factoring played critical roles. Cross-multiplication removed fractions, transforming the equation into \( m^2 (s-t) = p^2 (s+t) \). Expansion allowed us to express terms in a simpler form, which we later simplified and factored to isolate \( s \). - **Cross-multiplication**: A technique to eliminate fractions by equating each cross-product of terms.- **Distribution and expansion**: Used to simplify complex expressions into manageable terms. - **By factoring**: We separate shared components within an expression, assisting in solving equations.These steps illustrate how creating detailed actions using mathematical manipulation supports solving equations efficiently by reducing complexity.