A store sells packages of comic books with a poster. a. Write a linear function in the form y = mx + b that represents the cost, y, of a package containing any number of comic boo... A store sells packages of comic books with a poster. a. Write a linear function in the form y = mx + b that represents the cost, y, of a package containing any number of comic books, x. b. Suppose another store sells a similar package, modeled by a linear function with initial value $7.99. Which store has the better deal? Explain.
Understand the Problem
The problem presents a scenario involving two stores selling packages of comic books with a poster. Part (a) requires creating a linear function to model the cost of the package based on the number of comic books. Part (b) introduces another store with a different pricing model and asks which store offers the better deal, requiring an explanation.
Answer
a. $y = x + 6.75$ b. The first store has the better deal.
Answer for screen readers
a. $y = x + 6.75$ b. The first store (COMIX) has the better deal because the poster's cost ($6.75) is lower than the second store's initial value ($7.99).
Steps to Solve
- Find the slope $m$ for the first store
The store sells 6 comic books with a poster for $12.75, and 13 comic books with a poster for $19.75. We can define two points $(x_1, y_1) = (6, 12.75)$ and $(x_2, y_2) = (13, 19.75)$. The slope $m$ can be calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{19.75 - 12.75}{13 - 6} = \frac{7}{7} = 1$
- Find the y-intercept $b$ for the first store
Now that we have the slope $m = 1$, we can use one of the points to find the y-intercept $b$. We can use the point (6, 12.75) and the equation $y = mx + b$. $12.75 = 1 * 6 + b$ $12.75 = 6 + b$ $b = 12.75 - 6 = 6.75$
- Write the linear function for the first store
The linear function for the first store is $y = 1x + 6.75$ or $y = x + 6.75$.
- Analyze the second store's offer
The second store's package is modeled by a linear function with an initial value of $7.99. This means when $x = 0$ (no comic books), the cost $y = 7.99$. Essentially, the poster costs $7.99. The equation is thus $y = mx + 7.99$. Without more information we can't solve this store's price per book. We must assume that the 'initial value' means that the cost of the poster is $7.99. Let us find the prices of each poster to see which is a better deal.
- Compare the cost of the poster at the first store to that one in the second store
In the first store, the equation $y = x + 6.75$ represents the total cost. If we buy zero comic books ($x = 0$), $y = 0 + 6.75 = 6.75$. This means the poster alone costs $6.75 at the first store.
The poster alone at the second store costs $7.99
- Determine which store has the better deal
Since the poster costs $6.75 at the first store and $7.99 at the second store, the first store has a better deal because the initial cost (poster cost) is lower. Note that each additional comic book at the first store cost $1 each.
a. $y = x + 6.75$ b. The first store (COMIX) has the better deal because the poster's cost ($6.75) is lower than the second store's initial value ($7.99).
More Information
The linear function represents the cost of the package where the slope (1) is the cost of each comic book, and the y-intercept (6.75) is the cost of the poster.
Tips
- Forgetting to calculate the slope correctly. Ensure the correct values are subtracted in the right order.
- Incorrectly interpreting the y-intercept in the context of the problem.
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