The total weekly sales for a newly released portable media...

The total weekly sales for a newly released portable media player $ (PMP) $ increased each week. At the same time, the total weekly sales for another newly released $ PMP $ decreased each week. Models that approximate the total weekly sales $ S $ (in thousands of units) for each $ PMP $ are $ \left\{\begin{array}{l}S = 15x + 50 PMP 1\\S = -20x + 190 PMP 2\end{array}\right. $ where $ x $ represents the number of weeks each $ PMP $ was in stores, with $ x = 0 $ corresponding to the $ PMP $ sales on the day each $ PMP $ was first released in stores. (a) After how many weeks will the sales for the two $ PMPs $ be equal? (b) Use a table to solve the system of equations numerically. Compare your result with that of part (a).

The total weekly sales for a newly released portable media player $ (PMP) $ increased each week. At the same time, the total weekly sales for another newly released $ PMP $ decreased each week. Models that approximate the total weekly sales $ S $ (in thousands of units) for each $ PMP $ are

$ \left\{\begin{array}{l}S = 15x + 50 PMP 1\\S = -20x + 190 PMP 2\end{array}\right. $

where $ x $ represents the number of weeks each $ PMP $ was in stores, with $ x = 0 $ corresponding to the $ PMP $ sales on the day each $ PMP $ was first released in stores.

(a) After how many weeks will the sales for the two $ PMPs $ be equal?

(b) Use a table to solve the system of equations numerically. Compare your result with that of part (a).

Key Concepts

Linear Functions

Linear functions are algebraic expressions of the form y = mx + b, where m is the slope and b is the y-intercept. They represent constant rates of change, which makes them useful for modeling real-world scenarios such as predicting sales over time. In this context, the weekly sales are modeled as linear functions, with each function representing the sales trend of a product over time.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. Solving a system involves finding the values of the variables that satisfy all equations simultaneously. In the sales problem, setting the two linear models equal to each other creates a system whose solution indicates the point in time when the sales numbers for the two products are the same.

Intersection of Lines

The intersection of two lines in a coordinate plane occurs where the two linear equations yield the same output for a given input, meaning they share a common solution. This concept is crucial in this problem because finding the week (x-value) in which the two sales numbers are equal corresponds to finding the intersection point of the two lines representing the sales models.

Numerical Methods Using Tables

Numerical methods, such as using a table of values, provide an alternative approach to solving systems of equations by evaluating the functions at various input values. By constructing a table and comparing the outputs of the two models for different weeks, one can identify the point at which the values converge, thus verifying the solution obtained analytically.

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