Problem 37 - Algebra 2

Question

Solve each equation or formula for the specified variable. $$ d=r t, \text { for } r $$

Step-by-Step Solution

Answer

The solution for the variable \( r \) is \( r = \frac{d}{t} \).

Step 1: Understand the Given Equation

The equation given is \( d = rt \), where \( d \) represents distance, \( r \) represents rate or speed, and \( t \) represents time. We need to solve this equation for the variable \( r \).

Step 2: Isolate the Variable

To solve the equation \( d = rt \) for \( r \), we need to isolate \( r \) on one side of the equation. This can be done by dividing both sides of the equation by \( t \).

Step 3: Divide Both Sides by t

Divide both sides of the equation by \( t \) to solve for \( r \): \[ r = \frac{d}{t} \]. This isolates \( r \), giving us the formula for rate in terms of distance and time.

Key Concepts

Understanding the Distance-Rate-Time Formula

The distance-rate-time formula is a fundamental equation used extensively in problems involving motion. It helps you understand the relationship between how far an object travels, the speed at which it travels, and the time it takes. In essence, the formula, represented as \( d = rt \), tells us that:

• \( d \) (distance) is the total space covered by an object in motion.
• \( r \) (rate or speed) is how fast the object is moving.
• \( t \) (time) is the duration for which the object has been moving.

This equation is simple yet powerful because knowing any two of these variables allows you to easily find the third. It forms the backbone of understanding motion in physics and everyday scenarios like calculating travel time or speed during a journey.

Isolating Variables in Equations

Isolating a variable means rearranging an equation to make one variable the subject. This technique allows you to solve for that particular variable, crucial when you're given certain known quantities and need to find the unknown.For the equation \( d = rt \), if our aim is to find the rate \( r \), we need to move \( t \) away from \( r \). To do this efficiently:

• Identify which variable you want to isolate. In our case, that's \( r \).
• Rearrange the equation by eliminating the other variables interacting with it.

This involves reversing operations. Since \( r \) is currently multiplied by \( t \), you do the opposite operation (division) to both sides. Isolating variables ensures that you can solve equations consistently across varied contexts, making it a vital skill in algebra and beyond.

Role of Division in Algebra

In algebra, division plays a crucial role in solving equations, especially when isolating variables. It's the process we use to counteract multiplication, effectively undoing it so we can solve equations.Imagine the equation \( d = rt \). Here, \( r \) is multiplied by \( t \). To isolate \( r \), you need to divide both sides of the equation by \( t \):\[ r = \frac{d}{t} \]By dividing, you maintain the equality while separating \( r \) from \( t \). This approach highlights an essential principle of balancing equations: what you do to one side, you must do to the other. Division, therefore, ensures that the solutions remain correct and logically consistent. Mastering division in algebra will empower you to solve a wide array of mathematical problems where similar operations are needed.