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One of the most important steps when solving an equation is isolating the variable. This means you want to get the variable you are solving for alone on one side of the equation. In the given exercise, we started with the formula: \[d = rt\]To isolate \( r \), we needed to rearrange the equation so that \( r \) is by itself. The key move was to divide both sides of the equation by \( t \), giving us:\[ \frac{d}{t} = r\]This step is crucial because it simplifies the equation while retaining its balance. It's a frequently used method in algebra and helps simplify a wide range of problems. Isolating the variable is all about using inverse operations (like division to counter multiplication) to move everything else to the other side.

Algebraic manipulation involves rearranging and simplifying equations using algebraic rules. In this exercise, the manipulation included:

• Dividing both sides of the equation by the same term
• Simplifying the result

Let's break it down. Starting with:\[d = rt\]We divided both sides by \( t \):\[ \frac{d}{t} = \frac{r t}{t} \]Then, we simplified \( \frac{r t}{t} \) by canceling out \( t \), leading us to:\[ \frac{d}{t} = r \]These steps are examples of algebraic manipulation, where operations like division, multiplication, addition, or subtraction are performed on both sides of the equation equally. It’s important to always perform the same operation on both sides to keep the equation balanced.

The distance formula in algebra is often expressed as \( d = rt \). Here, \( d \) represents distance, \( r \) is the rate (or speed), and \( t \) is time. This formula is fundamental for solving problems related to motion.When asked to solve for rate \( r \), we started with the known formula and isolated the variable \( r \). The formula, after isolation, becomes:\[ r = \frac{d}{t} \]This rearranged equation tells us the rate (or speed) is equal to the distance divided by the time taken. It's a direct and practical application of algebra, commonly used in various real-world scenarios, like calculating travel speed or flow rates in physics and engineering.By understanding and manipulating the distance formula, solving numerous motion-related problems becomes straightforward.