When analyzing solutions in the context of real-world problems, it is crucial to ensure that the solutions are not only mathematically correct but also make sense within the scenario being modelled. In this particular system of equations where x represents the time spent riding a bike and y represents the distance traveled, we must interpret the variables with these real-world constraints in mind.
The solution to this system of equations is given as (-1, 7). Let's analyze the solution:
- x = -1 represents the time spent riding a bike. In the real world, time cannot be negative. Time is a quantity that, when measured from a particular event or start point, is always non-negative—it either starts at zero and increases, or you can think of it as the difference between two time points, which also must be non-negative. Therefore, having -1 as a measure of time does not make sense for a real-world scenario. We cannot ride a bike for negative one hour; we can only ride for zero or positive amounts of time.
- y = 7 represents the distance traveled. In real-world terms, this is a plausible value because we can indeed travel 7 units of distance (miles, kilometers, etc.). Distance, like time, is non-negative in the real world; you cannot travel a negative distance.
Given that the solution includes a negative time, this solution is not physically plausible. Despite being a valid mathematical solution to the equations, it cannot represent a real-life situation where someone rides a bike.
The importance of analyzing solutions in their real-world context becomes apparent here. Mathematically obtained solutions need to be vetted against the practical scenarios they are supposed to represent. In modeling, verifying, or problem-solving within real-world contexts, we must reject solutions that do not make sense, such as negative values for quantities that can only be positive or zero. As a math teacher, emphasizing this to students is fundamental in teaching them not just how to solve equations, but also how to apply mathematical thinking to real-life problems. In cases such as this one, we must go back and review our calculations or constraints to find a solution that is both mathematically correct and contextually plausible.
