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# Applications of Linear Systems

## Word problems using two equations and two unknowns

Estimated10 minsto complete
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Progress
Practice Applications of Linear Systems
Progress
Estimated10 minsto complete
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Point of Intersection

Learning Goal

By the end of this lesson you will be able to determine the point of intersection in context and state when to choose each scenario.

What if you were playing a game in which you collected houses and hotels. Three houses and one hotel are worth $1750. One house and three hotels are worth$3250. How could you find the value of each house and each hotel? After completing this Concept, you'll be able to solve real-world applications like this one that involve linear systems.

### Guidance

In this section we will use Real Life Applications of Linear Relations to determine under what conditions we should choose each scenario.

#### Example A

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. Determine under which conditions each option would be a better choice. Solution 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*}. flat fee rental fee total membership45 \begin{align*}2x\end{align*} \begin{align*}y = 45 + 2x\end{align*}
no membership 0 \begin{align*}3.50x\end{align*} \begin{align*}y = 3.5x\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 pay2 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 two equations are: \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. We can use the graph to find the Point of Intersection. The Point of Intersection occurs where the two lines cross each other. Looking at the graph you can see at 30 movies per year both options would cost105.

We need to report both numbers.

Now we need to decide when each option would be a better choice.

BEFORE the point of intersection:

If you are going to rent less than 30 movies than being a Non-Member would cost less and therefore be the better choice.

AFTER the point of intersection:

If you are going to rent more than 30 movies than being a Member would cost less and therefore be the better choice.

AT the point of intersection: