The following system models the winning times, y, in...

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?

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?

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Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations with the same variables, representing multiple relationships that must be satisfied simultaneously. In this context, each equation models the winning time as a linear function of time, and the system asks when these two models predict the same outcome.

Slope of a Line

The slope in a linear equation indicates the rate of change of the dependent variable with respect to the independent variable. Here, the slopes -0.19 and -0.16 represent the annual decrease in winning times. Because the slopes are different, the two lines are not parallel, ensuring that they intersect at a unique point.

Intersection of Lines

The intersection point of the two lines is the solution to the system, representing the specific year and corresponding winning time at which both models agree. This point has both a temporal interpretation (the year relative to 1970) and a performance interpretation (the winning time in seconds).

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