RWM102: Algebra

Two movie rental stores are in competition. Movie House charges an annual membership of $30 and charges $3 per movie rental. Flicks for Cheap charges an annual membership of $15 and charges $3 per movie rental. After how many movie rentals would Movie House become the better option?

It should already be clear to see that Movie House will never become the better option, since its membership is more expensive and it charges the same amount per movie as Flicks for Cheap.

The lines on a graph that describe each option have different $\begin{align*}y-\end{align*}$intercepts - namely 30 for Movie House and 15 for Flicks for Cheap - but the same slope: 3 dollars per movie. This means that the lines are parallel and so the system is inconsistent.

Now let's see how this works algebraically. Once again, we'll call the number of movies you rent $\begin{align*}x\end{align*}$ and the total cost of renting movies for a year $\begin{align*}y\end{align*}$.

The system of equations that describes this problem is:

$\begin{align*}y &= 30 + 3x\!\\ y &= 15 + 3x\end{align*}$

Let's solve this system by substituting the second equation into the first equation:

$\begin{align*}y &= 30 + 3x\!\\ & \Rightarrow 15 + 3x = 30 + 3x \Rightarrow 15 = 30 \qquad \text{This statement is always false.}\!\\ y &= 15 + 3x\end{align*}$

This means that the system is inconsistent.