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Solving for a variable involves finding the value of an unknown in an equation. To achieve this, we need to perform operations that leave the desired variable on one side of the equation by itself. Let's break down the given problem:
In the equation \(A = p + prt\), we are asked to solve for the variable \(r\). This means we need to express \(r\) in terms of the other variables \(A\), \(p\), and \(t\).
We started by looking at every term in the formula that contains \(r\). In this situation, the term involving \(r\) is \(prt\). Our goal is to manipulate the equation so that \(r\) is isolated on one side.
Often, this involves reversing operations that are applied to \(r\). For instance, here, the multiplication of \(r\) by \(p\) and \(t\) might need to be divided. Solving an equation step-by-step ensures that each operation brings us closer to isolating and expressing \(r\) alone.

Rearranging formulas is like putting the pieces of a puzzle in order, allowing us to see how variables interact. In mathematical terms, we change the arrangement of an equation without changing its value. This is essential when solving formulas for a particular variable.
The first step in the example of rearranging the formula \(A = p + prt\) involves getting all terms that aren't related to \(r\) on one side of the equation.
In this case, we move the term \(p\) to the other side by subtracting it from both sides, leading us to \(A - p = prt\). This step is crucial because it helps clear out the clutter and focuses on terms that include the variable of interest.

• Check each side of the equation to ensure all necessary terms have been moved.
• Constant values and coefficients linked directly to the target variable should remain nearby for further calculations.

Rearranging is not just about moving things around; it's about maintaining the balance and understanding the relationships between variables.

Isolating variables is the final step and often involves using inverse operations to wrap up the solving process. After rearranging, the term containing the target variable should be alone on one side of the equation.
Continuing with our example, after subtracting \(p\), we are left with \(A - p = prt\). To isolate \(r\), we need to get it on its own by removing the coefficients \(p\) and \(t\) that are multiplied with it.
This is achieved by performing the inverse operation: dividing both sides of the equation by \(pt\). This results in:\[r = \frac{A - p}{pt}\]

• Divide each term in the equation by the coefficients dividing the variable.
• Confirm that the target variable, \(r\), is isolated with all other variables and constants arranged on the opposite side.

Isolating a variable helps make equations clear and simple, laying out what each variable means clearly. It enables you to see the direct relationship between the variables.