91Ó°ÊÓ

Formula manipulation is a fundamental skill in math and science. It involves changing the form of an equation to isolate a specific variable or to simplify an expression. In this exercise, we start with the equation:

A = p(1 + rt)

The goal is to solve for the variable t. To do this, we need to manipulate the formula step-by-step.
First, we isolate the term containing the variable t. This means rearranging the equation until t is alone on one side. We divide both sides of the equation by the coefficient of the term with t. Here, we divide by p to get:
\[ \frac{A}{p} = 1 + rt \]
Next, we further manipulate the equation to isolate t. We subtract 1 from both sides:
\[ \frac{A}{p} - 1 = rt \]

Finally, we divide by r to completely isolate t:
\[ t = \frac{\frac{A}{p} - 1}{r} \]
Through manipulating the formula, the equation is simplified and t is isolated.

Solving for a variable involves isolating that variable on one side of the equation to find its value. Let's break down the process with the given formula:
Initially, the formula is:
A = p(1 + rt)

We need to find t. Our first task is to get all terms involving t on one side. To do this, divide both sides by p:
\[ \frac{A}{p} = 1 + rt \]
Next, we need to get t by itself. Subtract 1 from both sides:
\[ \frac{A}{p} - 1 = rt \]

The final step is to divide both sides by r, isolating the variable t:
\[ t = \frac{\frac{A}{p} - 1}{r} \]
By following these steps, we have effectively solved for the variable t in the given equation.

Algebraic isolation is the method of rearranging an equation to get a single variable alone on one side. This technique is crucial for solving equations. Take the given example:

A = p(1 + rt)

To isolate t, we follow a systematic process. Start by dividing each side by p:
\[ \frac{A}{p} = 1 + rt \]
This simplifies the equation, bringing us closer to isolating t. Now, subtract 1 from both sides:
\[ \frac{A}{p} - 1 = rt \]

This step removes the constant term, leaving only rt on the right. Finally, divide both sides by r:
\[ t = \frac{\frac{A}{p} - 1}{r} \]

By performing these algebraic operations, we have isolated t and solved the equation. Algebraic isolation allows us to handle more complex problems by breaking them down into simpler steps.