How Can You Model Data With A System of Equations?

In December of 2014, Sony released the movie The Interview online after threats to theaters cancelled the debut in theaters. As originally reported in Wall Street Journal, the sales figures reported in January contained an interesting math problem appropriate for algebra students.

The following January, Sony reported sales of $31 million from the sales and rentals of The Interview. They sold the movies online for $15 and rented through various sites for $6. If there were 4.3 million transactions, how many of the transaction were sales of the movie and how many of the transactions were rentals?

If you recognize that this problem may be modeled by a system of linear equations, you are dead on. Let’s start by defining two variables to what we are looking for:

S: number of sales transactions for The Interview in millions

R: number of rental transactions for The Interview in millions

These variables also relate to the total sales as well as the total number of transactions. Since there were a total of 4.3 million transactions,

S + R = 4.3

Each sale of the movie yields $15 in sales and each rental results in $6 in sales. Thus the total sales yields

15S + 6R = 31

Putting these equations together gives the system of linear equations,

S + R = 4.3
15S + 6R = 31

There are two strategies for solving this system. In this post we’ll look at the simpler strategy.

Strategy 1: Substitution or Elimination Method

Either methods allows us to solve for the variables. Since the first equation is easy to solve for a variable, we’ll show how to solve the system that way. If we solve the first equation for S, we get

S = 4.3 – R

When we put this into the second equation,

15(4.3 – R) + 6R = 31

Simplify to give the value for R:

64.5 – 15R + 6R = 31
-9R = -33.5
R ≈ 3.72

or 3.72 million rental transactions. Putting this into the equation solved for S give

S = 4.3 – 3.72 = 0.58

or 0.58 million sale transactions.