A store sells packages of comic books with a poster. a. Model with Math: Write a linear function in the form \(y = mx + b\) that represents the cost, y, of a package containing an... A store sells packages of comic books with a poster. a. Model with Math: 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. Construct Arguments: 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 is asking us to perform the following: Part a involves writing a linear equation in slope-intercept form ( ( y = mx + b ) to represent the cost (( y )) of a package containing any number of comic books (( x )). Part b requires determining which of the two stores offers a better deal. One store's pricing is based on the information needed to determine the slope and ( y )-intercept in part a, while the second store sells a similar package modeled by a linear equation with an initial value ($7.99), and then decide which store has the better deal, explaining your reasoning.
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 has the better deal because its initial value ($6.75) is less than the second store's initial value ($7.99).
Steps to Solve
- Find two points from the given information to define the linear equation
From the image, we can extract two data points:
- 1 poster + 6 comics costs $12.75, which can be written as the point (6, 12.75).
- 1 poster + 13 comics costs $19.75, which can be written as the point (13, 19.75).
- Calculate the slope $m$ using the two points
The slope $m$ is calculated as the change in $y$ divided by the change in $x$: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Using the points (6, 12.75) and (13, 19.75): $$ m = \frac{19.75 - 12.75}{13 - 6} = \frac{7}{7} = 1 $$ So, the slope $m = 1$.
- Calculate the y-intercept $b$ using the slope and one of the points
Using the slope-intercept form $y = mx + b$ and the point (6, 12.75), we can solve for $b$: $$ 12.75 = 1 \times 6 + b $$ $$ 12.75 = 6 + b $$ $$ b = 12.75 - 6 = 6.75 $$ So, the y-intercept $b = 6.75$.
- Write the linear equation for the first store
The linear equation for the first store is: $$ y = 1x + 6.75 $$ $$ y = x + 6.75 $$
- Analyze the second store's pricing model
The second store has an initial value of $7.99. This means the y-intercept $b$ is 7.99. We need to determine the slope (cost per comic book) to compare. The problem indicates that it is a similar package and modeled by a linear function.
- Plug in the slope and y-intercept to write the linear equation for the second store
The value of the slope must be determined using the information provided, which is not available. However, we are given that both stores sell similar package of comic books, we can then assume that the cost per comic book is the same as store 1, meaning the slope $m=1$. Hence, the linear equation for the second store is: $$ y = x + 7.99 $$
- Compare the two stores
To determine which store offers a better deal, we can compare the $y$-intercepts since the slope (cost per comic) is the same ($m=1$). The first store has a $y$-intercept (initial value) of $6.75, while the second store has a $y$-intercept (initial value) of $7.99. Since 6.75 < 7.99, the first store has a lower initial cost.
- Conclusion
The first store offers a better deal.
a. $y = x + 6.75$ b. The first store has the better deal because its initial value ($6.75) is less than the second store's initial value ($7.99).
More Information
The initial value of y-intercept represents the cost of the poster. A linear function is defined by a constant slope and a y-intercept.
Tips
A common mistake is to not correctly identify the two points from the image, or to miscalculate the slope and/or the y-intercept. Also, forgetting to compare the initial values when the cost per comic book is assumed to be the same.
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