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Chapter 2: Problem 16

Solve for the indicated variable. \(i=p r t\) for \(t\)

Short Answer

Expert verified
The formula solved for 't' is \(t = \frac{i}{pr}\)

Step by step solution

01

Rearrange The Formula

The first step is to rearrange the formula to make 't' the subject. Since 't' is multiplied by 'p' and 'r' in the formula, it would require division operation to isolate 't'. The new formula would be \(t = \frac{i}{pr}\)
02

Verify the Solution

After rearranging the formula, check to make sure 't' is isolated and on one side of the equation. In this case, 't' is indeed isolated and the formula \(t = \frac{i}{pr}\) is the solution.

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Key Concepts

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

Algebraic Formulas
Algebraic formulas are equations that express relationships between different variables, often represented by letters. These formulas are fundamental in mathematics as they provide a concise way to convey how variables interact with one another. In our exercise, the formula \(i = prt\) shows how the interest \(i\) is related to the principal \(p\), rate \(r\), and time \(t\).

Key characteristics of algebraic formulas include:
  • They contain variables (like \(i, p, r, t\) in our exercise) which can vary based on the context or problem.
  • They often involve arithmetic operations such as addition, subtraction, multiplication, and division.
  • They provide a way to solve for unknown variables by rearranging the formula as needed.
Understanding these formulas is crucial as they form the basis for solving equations in both basic and advanced mathematics.
Isolation of Variables
Isolation of variables is a critical step in solving equations. It involves manipulating the equation to get the variable of interest by itself on one side of the equation. This process requires a good understanding of arithmetic operations and maintaining the balance of the equation.

To isolate a variable, you often need to:
  • Perform the same operation on both sides of the equation to maintain equality.
  • Use inverse operations to cancel out other terms. For example, if a variable is multiplied by a number, you divide by that number to isolate it.
  • Systematically move terms from one side of the equation to the other.
In our example \(i = prt\), isolating \(t\) required dividing both sides by \(pr\), thus leading to \(t = \frac{i}{pr}\). This step is essential as it simplifies the equation to clearly express \(t\) in terms of the other variables.
Rearranging Equations
Rearranging equations involves changing the structure of an equation to make solving it easier or to isolate a particular variable. It can be seen as a strategic way to rewrite equations without changing their equality.

Key points about rearranging equations include:
  • Preserving the equality: Whatever operation you apply to one side must be applied to the other.
  • Utilizing arithmetic operations to shift terms around.
  • Understanding the goal: Often, the aim is to express a variable in terms of others, as we did with \(t\) in \(i = prt\).
Rearranging equations is a powerful tool not just in algebra, but in many fields including physics and engineering, where solving for specific variables within complex formulas is often necessary.

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