This problem involves two motorcyclists traveling in opposite directions, where one is going 7 mph faster than the other. The combined distance they travel in 2 hours is 258 miles. We need to find their individual speeds.

Let's denote the speed of the slower motorcyclist as $x$ mph. The faster motorcyclist is traveling at $x + 7$ mph. Since they are traveling in opposite directions, their combined speeds contribute to the total distance between them.

Step 1: Set up the equation for total distance.

The total distance between them after 2 hours is 258 miles, so: $\text{Distance} = (\text{Speed of the first cyclist}) + (\text{Speed of the second cyclist}) \times 2 \text{ hours}$ $2(x + (x + 7)) = 258$

Step 2: Solve for $x$.

Simplify the equation: $2(2x + 7) = 258$ $4x + 14 = 258$ Subtract 14 from both sides: $4x = 244$ Divide by 4: $x = 61$ So, the speed of the slower motorcyclist is 61 mph.

Step 3: Find the speed of the faster motorcyclist.

The faster motorcyclist is going 7 mph faster: $x + 7 = 61 + 7 = 68 \text{ mph}$

Final Answer:

• The slower motorcyclist is traveling at 61 mph.
• The faster motorcyclist is traveling at 68 mph.

Would you like more details on any part of the solution, or do you have any questions?

Here are 5 relative questions to extend your understanding:

1. How would the solution change if the motorcyclists were traveling for 3 hours instead of 2?
2. What if the faster motorcyclist was traveling 10 mph faster instead of 7 mph faster?
3. Can you create a formula for this type of problem for any time and speed difference?
4. How would the problem change if both motorcyclists were traveling in the same direction?
5. How would you solve this problem if only the distance between them after 2 hours was given without mentioning the speed difference?

Tip: When solving word problems, carefully define variables for unknowns and set up equations that reflect the situation described.