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To understand Kevin’s trip, we first need to grasp the concept of the distance formula. The distance formula relates distance traveled, speed (or rate), and time. This fundamental relationship can be expressed as:\[ \text{Distance} = \text{Rate} \times \text{Time} \]In simple terms, this means the distance Kevin covers can be calculated by multiplying how fast he travels (his rate) with the time he takes. During his trip, Kevin rides a bicycle part of the way and walks the rest. - When riding, his rate is 40 miles per hour.- When walking, his rate is 4 miles per hour.Using these rates, we establish separate equations for the distance bicycled and walked. It's important because it allows us to solve problems by converting word problems into mathematical expressions. With the distance formula, we can explore how Kevin’s entire distance, 186 miles, is split between riding and walking.

Rate and time problems often require identifying the given rates and the relationship between time variables. In Kevin's scenario:- **Rate for bicycling:** 40 miles per hour- **Rate for walking:** 4 miles per hourKevin's total travel time is given as 6 hours. Therefore, the time spent biking plus the time spent walking must equal 6 hours.This is represented as:\[ t_b + t_w = 6 \]In this context:- \( t_b \) is the time spent bicycling- \( t_w \) is the time spent walkingFrom this equation, we learn that every moment not spent biking is spent walking. This equation is central to forming the next steps of solving the problem. By substituting into distance equations, we use this relationship to figure out exactly how long Kevin biked and walked to complete his journey.

To find out how much time Kevin spent on his bicycle, we need to solve for the unknown time variable using simultaneous equations. This problem can initially seem daunting, but breaking it into steps makes solving more manageable.1. **List Equations:** - Total time: \( t_b + t_w = 6 \) - Total distance: \( 40t_b + 4t_w = 186 \)2. **Substitute the Time Equation:** Replace \( t_w \) with \( 6 - t_b \) in the distance equation: \[ 40t_b + 4(6 - t_b) = 186 \]3. **Simplify and Solve:** Simplify the equation to combine like terms: \[ 36t_b + 24 = 186 \] Subtract 24: \[ 36t_b = 162 \] Solve for \( t_b \): \( t_b = \frac{162}{36} = 4.5 \)By following these steps, we determine that Kevin spent 4.5 hours on his bicycle. The process of setting up and manipulating these equations is a crucial algebraic skill that simplifies solving word problems.