Problem 2 - Elementary Algebra

Question

For Problems \(1-10\), solve for the specified variable using the given facts. (Objective 1) $$ \text { Solve } d=r t \quad \text { for } r \text { if } d=486 \text { and } t=9 $$

Step-by-Step Solution

Answer

The value of \(r\) is 54.

Step 1: Understand the formula

The given formula is \( d = rt \), where \(d\) represents distance, \(r\) represents rate or speed, and \(t\) represents time. We need to isolate \(r\) from this equation.

Step 2: Rearrange the formula

To solve for \(r\), we need to rearrange the equation \( d = rt \) to make \(r\) the subject. We do this by dividing both sides by \(t\), which gives us \( r = \frac{d}{t} \).

Step 3: Substitute the given values

Insert the given values \(d = 486\) and \(t = 9\) into the equation \( r = \frac{d}{t} \). This gives us \( r = \frac{486}{9} \).

Step 4: Perform the calculation

Now perform the division: \( \frac{486}{9} = 54 \). Therefore, \(r = 54\).

Key Concepts

Solving Equations

Solving equations involves finding the value of an unknown variable that makes the equation true. In our exercise, we are working with the equation \( d = rt \). Here, the letters represent variables: \(d\) for distance, \(r\) for rate (or speed), and \(t\) for time. Our task is to solve for \(r\), the rate, meaning we need to arrange the equation so we can find its value given certain numbers for the other variables.

Some important steps to solve such problems include:

• Identifying the equation and what each variable represents.
• Understanding which variable you need to find (in this case, \(r\)).
• Using mathematical operations, like division or multiplication, to isolate the unknown variable.

Solving equations is a fundamental skill in algebra because it allows you to figure out unknown quantities needed in real-world scenarios, like determining how fast something is moving.

Distance-Rate-Time Problems

Distance-rate-time problems are a common type of algebraic equation that involves finding one of these three variables—distance, rate (speed), or time. The relationship among them is expressed by the simple formula \( d = rt \). In words, this means, "distance equals rate multiplied by time."This formula is particularly useful in real-life applications such as

• Calculating travel times,
• Determining speeds,
• Estimating distances traveled.

In our example, we are given the distance \(d = 486\) and the time \(t = 9\). Using these values, we can find the rate (\(r\), or how fast something was moving) by rearranging the formula to solve for \(r\). Simply put, this problem allows us to understand how long it takes to travel a certain distance at a fixed speed.

Rearranging Formulas

Rearranging formulas is the process of algebraically manipulating an equation to make a different variable the subject or focus. This is a vital skill in algebra, as it enables you to derive other necessary equations from a given one, making it easier to solve different types of problems.

In our example with the equation \( d = rt \), to solve for \(r\), we need to rearrange the formula by isolating \(r\). This involves:

• Dividing both sides of the equation by \(t\) to get \( r = \frac{d}{t} \).
• Substituting given numerical values into the rearranged equation.

By doing so, we obtain a new equation that helps us directly find the rate when distance and time are known. Mastering the skill of rearranging formulas helps in better understanding and solving a wide range of mathematical problems.