Final answer:
Using the balance information after purchasing two and five coffees, the cost of one coffee was found to be $3.36, and the initial balance on the gift card was $16.56. A linear equation was formed, and the total number of coffees that can be purchased was determined to be 4 before the card is depleted.
Step-by-step explanation:
To solve the problem of how many total coffees the student can buy with her gift card and to find the original balance, we'll start by creating a linear equation based on the information provided. We know that after purchasing two coffees, the student's balance is $23.28, and after buying five coffees, the balance is $13.20. We can use these two points to determine the cost of one coffee and the initial balance on the card.
Let's denote the number of coffees as x and the balance left on the gift card as y. We have the points (2, 23.28) and (5, 13.20), which we can use to find the slope of our line (m), representing the cost per coffee. The slope m can be calculated by the change in y over the change in x: m = (13.20 - 23.28) / (5 - 2) = -10.08 / 3 = -$3.36 per coffee.
Now that we have the slope, we can find the y-intercept (b), which represents the initial balance on the gift card. We'll use one of the points to solve for b: 23.28 = 3.36(2) + b. This gives us b = 23.28 - 6.72 = $16.56. So, the equation representing the scenario is y = 16.56 - 3.36x.
Finally, to find out how many coffees can be bought before the funds are depleted, we can set y to 0 (since the balance would be $0) and solve for x: 0 = 16.56 - 3.36x. This gives x = 16.56 / 3.36 = 4.928, which we round down because we can't buy a fraction of a coffee, resulting in a total of 4 coffees the student can purchase before the card runs out of funds.