91Ó°ÊÓ

Isolating variables is a fundamental concept in algebra. It means rearranging an equation so that a particular variable stands alone on one side of the equation. This process allows us to solve for that variable. For instance, when solving for the variable \( t \) in the equation given:
\[ A = P + P r t, \]
we need to get \( t \) by itself.
By subtracting \( P \) from both sides, we isolate the term containing \( t \):
\[ A - P = P r t. \]
The next step involves dividing both sides by \( P r \) to completely isolate \( t \):
\[ t = \frac{A - P}{P r}. \]
This final expression gives us \( t \) in terms of the other variables. Remember, isolating variables often requires inverse operations such as addition to undo subtraction, or division to undo multiplication.

Algebraic equations are mathematical statements that show the equality between two expressions. In this exercise, we looked at the specific equation:
\[ A = P + P r t. \]
Each symbol in the equation represents a variable or a constant. To understand algebraic equations:

• Identify the variables and constants.
  
• Understand the relationships between the terms.
  

For the given equation:

• \( A \) is the total amount.
  
• \( P \) is the principal amount.
  
• \( r \) is the rate.
  
• \( t \) is the time.
  

Solving algebraic equations involves using various techniques like combining like terms, distributing, and performing inverse operations. The goal is to simplify the equation step by step until we find the value of the desired variable.

A step-by-step solution breaks down the process of solving an equation into clear, manageable steps. This approach helps make complex problems more understandable. For the given exercise, we followed these steps:

• Step 1: Identify and understand the equation. We were given \( A = P + P r t \) and needed to solve for \( t \).
  
• Step 2: Isolate the term containing \( t \) by subtracting \( P \) from both sides. This simplifies the equation to \( A - P = P r t \).
  
• Step 3: Solve for \( t \) by dividing both sides by \( P r \). This results in \( t = \frac{A - P}{P r} \).
  

Each of these steps is essential to reach the final solution. By following a detailed step-by-step approach, students can better understand the methods used in algebra and apply them to future problems. Remember to always verify each step and ensure no mistakes are made during the process.