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Chapter 2: Problem 73

Solve each formula for the specified variable. \(d=r t\) for \(t\)

Short Answer

Expert verified
The formula solved for \(t\): \( t = \frac{d}{r} \).

Step by step solution

01

Understand the formula

The given formula is the distance formula: \(d = r t\), where \(d\) is the distance, \(r\) is the rate, and \(t\) is time.
02

Isolate the variable \(t\)

To isolate \(t\), divide both sides of the equation by \(r\). This gives: \( \frac{d}{r} = \frac{r t}{r} \).
03

Simplify the equation

Simplify the right side of the equation. The \(r\) terms on the right side cancel out, leaving: \( \frac{d}{r} = t \).
04

Final solution

The formula solved for \(t\) is: \( t = \frac{d}{r} \).

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Key Concepts

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

Understanding the Distance Formula
In many mathematical problems, we often need to find the distance, rate, or time. The distance formula comes into play here and is given by: \(d = r t\). This formula indicates that distance (\(d\)) equals the rate (\(r\)) multiplied by the time (\(t\)). It's crucial to grasp this basic formula, as it forms the foundation for solving many real-world problems. Whether you鈥檙e calculating how far a car has traveled or figuring out how long a journey will take, this formula is incredibly useful. Notice that each variable represents a key part of the problem:
  • \(d\): Distance traveled
  • \(r\): Rate or speed
  • \(t\): Time taken
Isolating Variables
When solving equations, a common step is isolating the variable you need to solve for. In our example, we need to solve for time (\(t\)). Starting with the distance formula \(d = r t\), our goal is to isolate \(t\). This involves getting \(t\) by itself on one side of the equation. We can do this by dividing both sides of the equation by \(r\). Here鈥檚 how:
  • Original formula: \(d = r t\)
  • Divide both sides by \(r\): \( \frac{d}{r} = \frac{r t}{r} \)
By performing this step, the \(r\) on the right side cancels out, resulting in: \( \frac{d}{r} = t \). Now, \(t\) is isolated, and we鈥檝e solved for time.
Simplifying Equations
Simplifying equations is an essential step to make them easier to understand and solve. After isolating the variable, we often find that terms cancel each other out. In our example, after dividing both sides of the distance formula by \(r\), we are left with \( \frac{d}{r} = t \). Here鈥檚 a breakdown:
  • Start with \( d = r t \).
  • Divide both sides by \(r\): \( \frac{d}{r} = t \).
  • The right side simplifies because \(r\) cancels out.
The final, simplified equation is \( t = \frac{d}{r} \). This simplified form makes it easy to use the equation in various practical applications. Knowing how to simplify makes solving equations faster and more intuitive.

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