Problem 39 - Intermediate Algebra

Question

Solve each formula for the specified variable. See Example 5. $$ d=r t \quad \text { for } t $$

Step-by-Step Solution

Answer

\( t = \frac{d}{r} \)

Step 1: Understand the Formula

The formula we have is the distance formula, which states: \( d = rt \). In this formula, \( d \) is the distance, \( r \) is the rate (or speed), and \( t \) is the time. We need to solve this formula for \( t \).

Step 2: Isolate the Variable

To solve for \( t \), we want to isolate \( t \) on one side of the equation. Currently, \( t \) is being multiplied by \( r \). We can isolate \( t \) by dividing both sides of the equation by \( r \).

Step 3: Divide Both Sides by r

We divide both sides of the equation \( d = rt \) by \( r \). This gives us:\[ \frac{d}{r} = \frac{rt}{r} \]On the right side, \( r \) cancels out, leaving:\[ \frac{d}{r} = t \]

Step 4: Rewrite the Solution

Now that we have isolated \( t \), we can rewrite the equation as:\[ t = \frac{d}{r} \]

Key Concepts

Distance Formula

The distance formula is a fundamental concept in algebra. It is expressed as:

• \( d = rt \)

where \( d \) is the distance, \( r \) is the rate or speed, and \( t \) is the time taken. This formula is widely used in various fields to calculate how far something has traveled over a certain period at a specific speed.
To visualize this, imagine driving a car. If you know how fast you're going (rate) and how long you drive (time), you can use this formula to figure out how far you've gone (distance).
Understanding this relationship helps in planning travel, predicting arrival times, and even in scientific calculations such as studying the movement of objects in physics. It all boils down to the basic multiplication of rate and time to find distance. Remember that you can rearrange this formula to solve for any one of the variables, provided you have the other two.

Solving Equations

Solving equations involves finding the value of the unknown variable. In the case of the distance formula, our equation is \( d = rt \). To solve for a particular variable, we rearrange the equation in terms of that variable.
When solving equations:

• Identify the variable you need to solve for. In our example, it's \( t \).
• Perform operations to isolate this variable on one side of the equation.

This process might involve addition, subtraction, multiplication, or division, depending on the structure of the equation.
Solving equations is a critical skill in algebra as it helps in understanding relationships between different variables. It enables predicting unknown values, verifying results, and applying this knowledge in real-world scenarios.

Isolate the Variable

To isolate the variable means to get the unknown variable alone on one side of the equation. For the equation \( d = rt \), isolating \( t \) involves rearranging the equation to express \( t \) in terms of \( d \) and \( r \).
Here’s how you do it:

• Start with the equation: \( d = rt \).
• To isolate \( t \), notice that it is multiplied by \( r \). To "undo" this multiplication, divide both sides of the equation by \( r \).
• This gives you: \( \frac{d}{r} = \frac{rt}{r} \).
• The \( r \) on the right side cancels out, leaving you with \( \frac{d}{r} = t \).
• Finally, arrange it as: \( t = \frac{d}{r} \).

By isolating the variable, you have successfully rewritten the equation to solve for \( t \), making it easier to determine the time if you know the distance and rate. Isolating variables is a key technique used in algebra to simplify and solve equations effectively.