Question

How can Allie determine the number of medium coffees she can buy with the gift card if she does not know the original value of the card?

Answer 1

From the graph in the question, we can get an equation or model which Allie can use to determine the number of medium coffees she can buy with the gift card.

Let the gift card balance on the y-axis of the graph be represented as y

Let the medium cup of coffees on the x-axis of the graph be represented as x.

Now, we have been given two points on the graph which are

(2, 19.70) and (4, 14.40)

We will use these points to determine the slope of the line

This becomes

`\begin{gathered} m=(y_2-y_1)/(x_2-x_1) \\ m=(14.40-19.70)/(4-2) \\ m=(-5.3)/(2) \\ m=-2.65 \end{gathered}`

Now we have the slope, let's get the equation of the line

We will use the formula

`y-y_1=m(x-x_1)`

We have the equation as

`\begin{gathered} y-_{}y_1=m(x-x_1) \\ y-19.70=-2.65(x-2_{}) \\ y-19.70=-2.65x+5.3_{} \\ y=-2.65x+5.3+19.70 \\ y=-2.65x+25 \end{gathered}`

Now, for Allie to know the number of medium coffees she can buy (x), if she does not know the original value of the card, she will use the above equation as model to do this.

Hence if she does not know the original value of the gift card (y), then we put y = 0, and find x from the model. We have

`\begin{gathered} y=-2.65x+25 \\ 0=-2.65x+25 \\ 2.65x=25 \\ x=(25)/(2.65) \\ x=9.43 \end{gathered}`

So, if she does not know the original value of the card, the number of medium cups of coffees she can buy with the gift card cannot be less than 9 cups.

So Allie can determine this using the model

`y=-2.65x+25`