Question

Alyssa was given a gift card for a coffee shop. Each morning, Alyssa uses the card tobuy one cup of coffee. Let A represent the amount money remaining on the card afterbuying x cups of coffee. The table below has select values showing the linearrelationship between x and A. Determine the original amount of money on the giftcard.

Answer 1

We can deduce that the original amount on the gift card correspond to the value of A when x=0, that is no coffee consumed yet.

Then, we have to calculate the function that relates x with A and then calculate A(0).

We know that it is a linear relationship, so we will start by calculating the slope m as:

$m=\frac{A_2-A_1}{x_2-x_1}=\frac{10-15}{4-2}=\frac{-5}{2}=-2.5$

This means that the amount is reduced by 2.5 dollars per coffee.

We then can use one point fo the relationship, like (x,A)=(2,15) and, with the slope m=-2.5, write the slope-point equation form of the line:

$\begin{gathered} A(x)-A_o=m(x-x_0)_{} \\ A(x)-15=-2.5(x-2) \\ A(x)=-2.5\cdot x-2.5(-2)+15 \\ A(x)=-2.5x+5+15 \\ A(x)=-2.5x+20 \end{gathered}$

Now, we can calculate the value of A when x=0:

$A(0)=-2.5(0)+20=20$

The value of A(0) is the same as the y-intercept b=20.

Answer: the original amunt of money in the gift card was $20.