RWM102: Algebra

The movie rental store CineStar offers customers two choices. Customers can pay a yearly membership of $45 and then rent each movie for $2 or they can choose not to pay the membership fee and rent each movie for $3.50. How many movies would you have to rent before the membership becomes the cheaper option?

Let's translate this problem into algebra. Since there are two different options to consider, we can write two different equations and form a system.

The choices are "membership" and "no membership". 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 flat fee is the dollar amount you pay per year and the rental fee is the dollar amount you pay when you rent a movie. For the membership option the rental fee is $\begin{align*}2x\end{align*}$, since you would pay $2 for each movie you rented; for the no membership option the rental fee is $\begin{align*}3.50x\end{align*}$, since you would pay $3.50 for each movie you rented.

Our system of equations is:

$\begin{align*}y = 45 + 2x\!\\ y = 3.50x\end{align*}$

Here's a graph of the system:

Now we need to find the exact intersection point. Since each equation is already solved for $\begin{align*}y\end{align*}$, we can easily solve the system with substitution. Substitute the second equation into the first one:

$\begin{align*}y = 45 + 2x\!\\ {\;} \qquad \qquad \qquad \ \Rightarrow 3.50x = 45 + 2x \Rightarrow 1.50x = 45 \Rightarrow x = 30 \ \text{movies}\!\\ y = 3.50x\end{align*}$

You would have to rent 30 movies per year before the membership becomes the better option.

This example shows a real situation where a consistent system of equations is useful in finding a solution. Remember that for a consistent system, the lines that make up the system intersect at single point. In other words, the lines are not parallel or the slopes are different.

In this case, the slopes of the lines represent the price of a rental per movie. The lines cross because the price of rental per movie is different for the two options in the problem

Now let's look at a situation where the system is inconsistent. From the previous explanation, we can conclude that the lines will not intersect if the slopes are the same (and the $\begin{align*}y-\end{align*}$intercept is different). Let's change the previous problem so that this is the case.