Aaden wants to get a subscription to an online library. There are two subscription options, one of which charges a fixed 96 dollar annual fee and the other which charges 3 dollars per book he borrows. Here’s a graph that shows a system of equations for this scenario where
is the number of books Aaden borrows and
is the total cost of the subscription.
The Correct Answer and Explanation is:
To determine which subscription is more cost-effective for Aaden, we analyze the two pricing models:
- Plan A (Fixed Plan): $96 annual fee, regardless of how many books he borrows.
This can be represented by the equation:
C = 96,
where C is the total cost, and it does not depend on the number of books borrowed. - Plan B (Per-Book Plan): No annual fee, but charges $3 for each book borrowed.
This is represented by the equation:
C = 3b,
where C is the total cost, and b is the number of books borrowed.
Solving the System of Equations:
We set the two cost equations equal to find the break-even point — the number of books for which both plans cost the same: 96=3b96 = 3b96=3b
Divide both sides by 3: b=32b = 32b=32
So, at 32 books, both plans cost $96.
Interpreting the Graph:
On the graph:
- The horizontal axis represents the number of books borrowed (b).
- The vertical axis represents the total cost (C).
- The line C = 96 is a horizontal line, showing the fixed cost plan.
- The line C = 3b is a straight line through the origin, increasing with the number of books.
The point where the lines intersect is at (32, 96). This is the break-even point.
Decision:
- If Aaden borrows fewer than 32 books, the $3 per book plan (Plan B) is cheaper.
- If Aaden borrows more than 32 books, the $96 flat fee plan (Plan A) saves money.
Conclusion:
Aaden should consider how many books he plans to borrow in a year. If it’s fewer than 32, he saves money with the per-book plan. If it’s more than 32, the fixed annual subscription is a better choice. This comparison helps make cost-effective decisions using linear equations and graphical interpretation.
