Problem 30 - Elementary Algebra Within Reach

Question

Solve for the specified variable. Solve for \(r: A=P+P r t\)

Step-by-Step Solution

Answer

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

Step 1: Subtract P from Both Sides

Subtract \(P\) from each side of the equation in order to isolate terms with \(r\): \(A - P = P r t\).

Step 2: Isolate r

To isolate \(r\), we divide both sides of the equation by \(P t\): \(r = \frac{A - P}{P t}\).

Key Concepts

Understanding Algebraic Equations

An algebraic equation is a statement of equality between two expressions formed by variables and constants. In the example provided, we have the equation A = P + Prt. This formula is commonly seen in financial mathematics, where A represents the total amount accrued through some principal P, with a rate r, over a time t.

To solve an algebraic equation, we perform operations to simplify and isolate the variables. Here, the goal is to find the value of r, thus making r the subject of our formula. Solving such equations often requires familiarity with the basic principles of algebra like addition, subtraction, multiplication, and division, as well as understanding how to manipulate equations to keep them balanced.

How to Isolate Variables

Isolating a variable means rearranging an algebraic equation so that the variable we're solving for is by itself on one side of the equation. In other words, we want to make the variable the subject. In the exercise, we start with A = P + Prt and want to solve for r. To do this, we use inverse operations, performing the same operation on both sides to maintain the equation's balance.

The first inverse operation used here is subtraction; by subtracting P from both sides, we group all terms involving r on one side: A - P = Prt. Next, we divide both sides by Pt, leaving r on one side of the equation: r = (A - P) / (Pt). Mastering the steps to isolate variables is crucial and often involves these simple operations to obtain the desired variable.

Algebra Problem Solving

Algebra problem solving often refers to the process of finding unknown variables within expressions and equations. The key is to understand and apply mathematical laws and properties consistently. To approach algebraic problems, one should:

• Identify what needs to be solved for and carefully read the equation to understand the relationship between the variables.
• Apply algebraic operations in a logical sequence to avoid errors.
• Check your answer by plugging it back into the original equation to see if it holds true.

Using these strategies in the given exercise helps us to accurately determine the value of r by subtracting P from both sides and then dividing by Pt. This process of careful step-by-step manipulation, while maintaining the balance of the equation, exemplifies algebra problem solving. The ability to solve such problems is fundamental to understanding more complex mathematical concepts.