Let's model the cost of a package of comic books with a poster from your store using a linear function in the form \(y = mx + b\), where \(y\) is the total cost, \(x\) is the number of comic books, \(m\) is the cost per comic book, and \(b\) is the initial value (the additional cost, if any).
a. Model with Math
Let's assume:
- The cost per comic book is \(m\) dollars.
- The package has an initial cost of \(b\) dollars (which may be the price of the poster or any fixed cost).
Thus, the linear function representing the cost \(y\) would be: \[ y = mx + b \]
b. Construct Arguments
You mentioned another store that sells a similar package modeled by a linear function with an initial value of $7.99.
This means their cost function can be represented as: \[ y = mx + 7.99 \] However, we do not know the cost per comic book (\(m\)) for that store.
To determine which store has the better deal, we need to compare the effective costs of both packages.
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Your Store's Cost:
- The cost function is \(y = mx + b\).
- As \(x\) increases, so does \(y\), factoring in both the cost of the comics and the initial cost.
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Their Store's Cost:
- The cost function is \(y = mx + 7.99\).
- Similar to yours, the cost increases with more comic books, but starts at $7.99.
Comparison
- If \(b\) (your store's initial cost) is less than $7.99, then your store has a better deal, especially for smaller quantities of comic books.
- If \(b\) is equal to $7.99, then the costs start at the same price, and their deal may become more favorable depending on the value of \(m\).
- If \(b\) is greater than $7.99, then their store has a better deal for smaller quantities, but your store might be better for larger quantities if \(m\) (the cost per additional comic) is less than theirs.
Conclusion
To determine which store has the better deal definitively, you should compare the specific values of \(m\) and \(b\) for both stores. If you provide those values, it will be possible to analyze and make a clear judgment on which store offers a better deal.