Write the linear equations representing the costs in both stores and compare them. Is any cost always greater than another?
A
y=x+6.75
B
The first store, see solution.
Practice makes perfect
We are asked to find the linear function that represents the cost y of a package containing any number of comic books x. The desired linear function has to follow a specific format.
y= mx+ b
For an equation in this form, m is the slope and b is the y-intercept. To find the desired linear function we will calculate the slope and the y-intercept. We can do this by using any two points lying on the line that is the graph of the linear function. Let's go through the content of the exercise and get the points we need!
Information From the Exercise
Point (x,y)
1 poster and 6 comics cost $ 12.75.
( 6, 12.75)
1 poster and 13 comics cost $ 19.75.
( 13, 19.75)
Now we can use the obtained points to calculate m. We will start by substituting the points into the Slope Formula.
In our function, a slope of 1 means that if we buy 1 more comic book, we will pay $ 1 more. Now that we know the slope, we can write a partial version of the equation.
y= 1 x+ b
To complete the equation we also need to determine the y-intercept b. Since we know that the given points will satisfy the equation, we can substitute one of them into the equation to solve for b. Let's use ( 6, 12.75).
A y-intercept of 6.75 means that the poster costs $ 6.75. We can now complete the equation.
y= 1x+ 6.75 ⇔ y =x+6.75
We are given that the second store sells a similar package, modeled by a linear function with an initial value of $7.99. We want to determine which store has the better deal. Let's start by recalling the equation of the linear function obtained in Part A.
y =x+6.75
At the second store we have the same value of the slope as at the first store, but the initial value is equal to 7.99. Therefore, we can write the equation of the linear function for the second store.
y =x+7.99
Note that at both stores the price of each comic book is the same, but in the second store the poster costs $ 7.99, which is more than $ 6.75. Since each package contains the poster, it will always cost more in the second store. This means that the first store has the better deal.
