Explanation:
a. To write a linear function that represents the cost of a package containing any number of comic books, we need to find the slope, m, and the y-intercept, b.
From the given information, we know that:
1 poster + 6 comics = $12.75
1 poster + 13 comics = $19.75
To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1)
Let's use the first equation as (x1, y1) and the second equation as (x2, y2):
m = (19.75 - 12.75) / (13 - 6)
m = 7 / 7
m = 1
Now we can find the y-intercept by plugging in any (x, y) pair into the equation y = mx + b and solving for b. Let's use the first equation:
12.75 = 6(1) + b
b = 6.75
Therefore, the linear function that represents the cost of a package containing any number of comic books is:
y = x + 6.75
b. The other store sells a similar package, modeled by a linear function with initial value $7.99. Let's write this function in the same form as before:
y = mx + b
We know that the initial value, b, is $7.99. To find the slope, we can use another point on the line, such as (x, y) = (13, 19.75):
19.75 = 13m + 7.99
m = 0.998
Therefore, the linear function that represents the cost of a package from the other store is:
y = 0.998x + 7.99
To determine which store has the better deal, we need to compare the prices of packages with the same number of comic books from both stores. Let's say we want to buy a package with 10 comic books.
According to the first store's pricing function:
y = 10 + 6.75
y = $16.75
According to the second store's pricing function:
y = 0.998(10) + 7.99
y = $17.97
Therefore, the first store has the better deal for a package with 10 comic books. However, this may not be true for all numbers of comic books, so we should compare prices for different numbers of comic books to be sure.