Problem 61 - Introductory Algebra for College Students

Question

The following system models the winning times, \(y,\) in seconds, in the Olympic 500 -meter speed skating event \(x\) years after 1970: $$\left\\{\begin{array}{l}y=-0.19 x+43.7 \\ y=-0.16 x+39.9\end{array}\right.$$ Use the slope of each model to explain why the system has a solution. What does this solution represent?

Step-by-Step Solution

Answer

The system has a solution because the slopes of the two models are different, implying they will intersect at some point. The solution represents the year (2097) when the winning times predicted by the two models are the same, which is approximately 20 seconds.

Step 1: Understand the Role of Slope

The slopes of the given models are -0.19 and -0.16. The numeric value of slope indicates the rate of change in winning time with respect to change in years. Since the slopes of these two lines are different, this means the rate at which the winning times decrease per year is not the same for both models. Therefore, they are not parallel and would intersect at some point.

Step 2: Find the Solution of the System

To find when and at what winning time the two situations are equal, solve the system of equations. Set the two equations equal to each other and solve for \(x\):\n-0.19x + 43.7 = -0.16x + 39.9\nThis simplifies to 0.03x = 3.8\nSolving for \(x\), we get \(x \approx 127\), implying that the two models intersect approximately 127 years after 1970.

Step 3: Interpret the Solution

The solution \(x \approx 127\) represents the number of years after 1970 when the winning times predicted by the two models would be the same. To find the exact year, add 127 to 1970, which gives 2097. The winning time at this point would be calculated by substituting \(x = 127\) into one of the given equations, for example \(y = -0.16(127) + 39.9\) which yields \(y \approx 20\). So, in the year 2097, the winning time would be approximately 20 seconds according to both models.

Key Concepts

Slope

The slope of a line in mathematics is a measure that indicates how steep the line is. It reflects the rate at which one quantity changes with respect to another. In the context of the Olympic 500-meter speed skating event, the slope represents how the winning time changes each year. For example, a slope of -0.19 means that the winning time decreases by 0.19 seconds every year. This is a negative slope, signifying a decrease.

In our given models, the slopes are -0.19 and -0.16, indicating that as years go by, the winning times decrease, but at slightly different rates for each model. This difference in slope confirms that the two lines will eventually intersect because they are not parallel. Parallel lines would imply identical slopes, resulting in no intersection. Thus, the solution emerges at the point where both predictions of winning times align, reflecting the importance of slope in determining the dynamic relationship between variables over time.

Intersection of Lines

The intersection point of two lines is where the two lines cross or meet on a graph. This point represents a solution to a system of equations, as it is the point where both equations balance or are equal. In our example of the speed skating times, the system of equations models how winning times evolve over years since 1970.

By solving these equations, we find that they intersect approximately 127 years after 1970. This translates into the year 2097. At this intersection point, both models predict the same winning time, showing how two different trends can eventually converge. The intersection not only provides a specific point in time but also encapsulates the unique circumstances where the models agree. Understanding intersections is crucial as it shows correlations and reconciliations in changing data points.

Rate of Change

Rate of change is a fundamental concept in understanding how one quantity varies in relation to another. In linear equations, it corresponds to the slope. In our speed skating model, the rate of change indicates how quickly winning times decrease over the years. This concept explains the pace of improvement in the athletes' performances. A higher negative value in rate of change suggests a greater reduction in time across years.

The rates in the models, -0.19 and -0.16, demonstrate that the times are reducing annually but at different speeds. These small yet impactful per-year differences underline the importance of examining rates of change, as they offer deeper insights into trends and dynamics in real-world scenarios such as athletic performance improvements over decades. Understanding this helps predict future outcomes and prepares analysts to adjust models to better align with actual progress.