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Chapter 1: Problem 89

Solve for the specified variable. (See Examples \(8-9)\) $$ A=P+P r t \text { for } P $$

Short Answer

Expert verified
P = \frac{A}{1 + r t}

Step by step solution

01

Identify the Given Equation

The given equation is: \[ A = P + P r t \].
02

Factor Out the Common Term

Factor out the common term, \(P\), on the right side of the equation: \[ A = P (1 + r t) \].
03

Solve for \(P\)

Isolate \(P\) by dividing both sides of the equation by \((1 + r t)\): \[ P = \frac{A}{1 + r t} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
When solving an equation for a specific variable, our main goal is to rearrange the equation so that the variable of interest is on one side of the equation by itself. This process is known as isolating the variable. Let's look at the equation given in the exercise: \[ A = P + Prt \]Since we want to solve for \(P\), we need to get \(P\) on one side of the equation by itself. To do this, it's often helpful to first simplify the equation if possible. Note that \(P\) appears in two terms on the right side. Simplifying this step will involve factoring.
Factoring
Factoring involves rewriting an expression as a product of simpler terms. In our given equation, we have \(P\) in both terms on the right: \[ P + Prt \]We can factor out the common term \(P\). Factoring \(P\) from both terms gives us: \[ P(1 + rt) \]This factored form is much easier to work with, especially when isolating a variable. By factoring, we can now more clearly see how the term with our variable of interest interacts with other parts of the equation.
Variable Manipulation
Once the variable is factored out, our next step is to isolate it fully. We're aiming to solve for \(P\), so we need to move other factors to the other side of the equation. Starting from: \[ A = P(1 + rt) \]We divide both sides by \((1 + rt)\) to get \(P\) by itself: \[ P = \frac{A}{1 + rt} \]This final step requires an understanding of basic algebraic operations like division and multiplication. By manipulating the variables correctly, we've isolated \(P\) and derived a formula expressing \(P\) in terms of \(A\), \(r\), and \(t\). This method of rearranging equations can be applied to various algebraic problems, making it an essential skill in mathematics.

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