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Algebraic manipulation refers to the process of rearranging equations to solve for a desired variable. This requires using various mathematical operations to simplify and transform the equation without changing its original meaning. In the given exercise, we start with the equation \(V = \frac{k}{P}\). By performing algebraic manipulation, we can rearrange this equation to solve for \(P\). By following the steps of multiplication and division, we get the solution \(P = \frac{k}{V}\). Algebric manipulations are essential as they are foundational techniques in algebra to transform complex equations into simpler forms. Remember to always perform the same operation on both sides of the equation to maintain its equality.

Isolating a variable means getting the variable you are solving for alone on one side of the equation. In other words, all other terms should be moved to the opposite side of the equation. For the exercise, we need to solve for \(P\) in the equation \(V = \frac{k}{P}\). We start by isolating \(P\) by multiplying both sides of the equation by \(P\), which results in \(VP = k\). This isolates \(P\) from the denominator, making it easier to re-arrange the equation. The next step involves dividing both sides of the equation by \(V\) to completely isolate \(P\). The final equation we obtain is \(P = \frac{k}{V}\). Always remember that isolating the variable is a critical step for solving any equation.

Inverse operations are operations that undo each other. Addition and subtraction are inverse operations, as are multiplication and division. In the context of solving equations, we often use inverse operations to isolate variables and solve for them. In the problem at hand, we use the inverse operations of multiplication and division. To eliminate the fraction in the equation \(V = \frac{k}{P}\), we multiply by \(P\), using the multiplication operation to cancel out the division by \(P\). This results in the equation \(VP = k\). To further isolate \(P\) on one side, we use the division operation on both sides, which is the inverse of multiplication. This results in \(P = \frac{k}{V}\). Mastering inverse operations is crucial as it allows you to systematically move variables and terms around in an equation to solve for unknowns.