# How can a system of equations be used to predict media use?

May 26, 2015

Suppose we have collected data about media use, as follows:

Prnt media:
- year 0 (1994) $\to$ 3 hrs per person per day
- year 1 (1995) $\to$ 2.5 hrs per person per day
- ...

Online media:
- year 0 (1994) $\to$ 0.5 hrs per person per day
- year 1 (1995) $\to$ 0.75 hrs per person per day - ...
- ...

We assume a simple, linear model, so that we can easily fill in the yearly data and we already have sufficient information to determine the equations.

Let's denote by $x$ the number of years passed since 1994 and by $y$ the average number of hours per person per day spent on media use .

Our linear equations will have the general form $y = a x + b$

For print media use we have:
$a x + b = 3$ when $x = 0$ $\to$ $b = 3$
and $a x + b = 2.5$ when $x = 1$ $\to$ $a + b = 2.5$ $\to$ $a = - 0.5$
Therefore, our equation for print media use is $y = - 0.5 x + 3$

For online media use we have:
$a x + b = 0.5$ when $x = 0$ $\to$ $b = 0.5$
and $a x + b = 0.75$ when $x = 1$ $\to$ $a + b = 0.75$ $\to$ $a = 0.25$
Therefore, our equation for online media use is $y = 0.25 x + 0.5$

Now, we have a system of two linear equations allowing predictions of media use at various data points. (Of course, this is a fictitious example and an oversimplification meant just to illustrate the general idea. We can build a more realistic model by using exponential instead of linear functions).