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Understanding the distance-time relationship is essential for solving problems involving motion. In this exercise, we are dealing with three cars moving at different speeds. We know that distance, speed, and time are related by the formula:

• Distance = Speed × Time

When the first two cars begin their journey, the distances they travel can be calculated by their respective speeds multiplied by the time spent traveling. By considering the third car starts 0.5 hours after the first two and reaches the second car first, we need to consider how their relative speeds affect the distances each car covers.
By examining the time it takes for the third car to catch up, we can use the formula to find out how far ahead each car was initially. Calculating these initial distances helps simplify the problem and focus on how the third car manages to close the gaps based on its speed.

Linear equations can be used to solve many motion-related problems, such as calculating unknown speeds or distances. In our problem, we need to find the speed of the third car, which can be treated as the variable in our equation. This involves setting up a time equation:

• Time to reach the second car = \( \frac{5}{x-40} \)
• Time to reach the first car = \( \frac{25}{x-50} \)

Given these times, we know:\[ \frac{25}{x-50} = \frac{5}{x-40} + 1.5 \]This equation represents the relationship between the times it takes for the third car to reach the first two cars. By solving this equation, we can find the value of \( x \), which is the speed of the third car. It’s a process of substituting values and solving step-by-step to isolate \( x \). This process involves typical algebraic techniques like cross-multiplication and simplifying expressions.

Effective problem-solving strategies are key to reaching a solution in a structured manner. For motion problems like this, follow these strategies:

• Read the problem carefully to understand the relationships between distances, speeds, and times.
• Identify what information is given and what needs to be found. Here, we need the speed of the third car.
• Set up equations based on these identified relationships. Use known formulas to express the distances and times in linear equations.
• Substitute known values into the equation and simplify them to isolate the variable of interest.
• Solve step-by-step. Break down the problem into manageable sections and tackle each logically.
• Verify the solution by checking if the result satisfies the original conditions of the problem.

By following these organized strategies, you not only find the solution but also deepen your understanding of the concepts involved, making future problems easier to tackle.