Problem 52 - Elementary Algebra

Question

For Problems 43-54, solve each formula for the indicated variable. (Before doing these problems, cover the right-hand column and see how many of these formulas you recognize!) (Objective 2) $$ A=P+\operatorname{Prt} \text { for } t $$

Step-by-Step Solution

Answer

The solution is \( t = \frac{A - P}{Pr} \).

Step 1: Identify the Formula

We start with the provided formula: \( A = P + Prt \), and we need to solve for \( t \). This means we need to isolate \( t \) on one side of the equation.

Step 2: Subtract P from Both Sides

To isolate \( Prt \), subtract \( P \) from both sides of the equation. This gives us: \[ A - P = Prt \].

Step 3: Divide Both Sides by Pr

To solve for \( t \), divide both sides of the equation by \( Pr \): \[ t = \frac{A - P}{Pr} \]. This step isolates \( t \), giving us the final answer.

Key Concepts

Formula Solving

Solving a formula in algebra is like finding the answer to a mystery. When we're given an equation, such as \( A = P + Prt \), each letter stands for something important. In our case, 'A' is the total amount, 'P' might be some principal value, and 'Prt' is a product of three variables. To solve a formula, you simply rearrange these parts to find the answer for a particular letter or variable.

The trick is to ensure that whatever operation you do to one side of the equation, you do the same to the other. This keeps your equation balanced, just like a perfectly leveled seesaw. Solving formulas often involves basic steps, like adding, subtracting, multiplying, or dividing.

• Identify the unknown you're solving for.
• Perform operations to isolate the unknown.
• Ensure to keep the equation balanced.

The goal? To get the variable you're interested in all by itself on one side of the equation. This will give you the solution you're looking for!

Isolating Variables

Isolating variables is one of the most crucial skills in algebra. If we look at our equation \( A = P + Prt \), our mission is to make 't' stand alone. Think of it like untangling a knot; you keep working at it until 't' is free.

Start by looking at what's preventing 't' from being by itself. In \( A = P + Prt \), the 'P' term is in the way, so we subtract 'P' from both sides. This step ensures that whatever happens to one side also happens to the other—maintaining the balance of our equation.

• Identify terms subtracting or adding to the variable.
• Work step by step to remove these terms.
• Use inverse operations to cancel these terms out.

The second obstacle is how 't' is being multiplied by 'Pr'. So we divide both sides by 'Pr'. And just like that, 't' is isolated, showing us its true value!

Mathematical Equations

Mathematical equations are like puzzles. Each piece must fit together to give a complete picture. An equation states that two things are equal, often shown by an "=" sign.

In algebra, these equations can become complex, but they are simply statements of relationship between things. For instance, with our equation \( A = P + Prt \), each letter and operation is a key part of what makes the whole statement true.

• Equations use symbols and numbers to show relationships.
• They can model real-world situations or abstract concepts.
• Equations require operations to be performed to maintain equality.

Different equations demand different approaches for solving them, such as rearranging terms or simplifying expressions. But no matter the complexity, the heart of any equation is the relationship it represents.