Final answer:
By solving a system of equations where x represents AirPods and y represents Apple Watches, the answer is 'Yes'. It is possible to purchase 11 AirPods and 3 Apple Watches to get exactly 14 items and spend exactly $2,850.
Step-by-step explanation:
To determine if it's possible to purchase exactly 14 items and spend exactly $2,850, considering AirPods cost $150 and an Apple Watch costs $400, we need to set up an equation that represents the total cost of the items. Let's let the number of AirPods be represented by x and the number of Apple Watches be represented by y.
The corresponding equations would be:
- x + y = 14 (since the student wants to buy a total of 14 items)
- 150x + 400y = 2850 (as the total cost must add up to $2,850)
We can solve this system of equations to see if there is a combination of x (AirPods) and y (Apple Watches) that satisfies both conditions.
Step 1: Solving for y
If we multiply the first equation by 150, we will get:
150x + 150y = 2100
Subtracting this from the second equation gives us:
250y = 750
Dividing both sides by 250 gives us:
y = 3
Step 2: Solving for x
Plug y = 3 into the first equation:
x + 3 = 14
x = 14 - 3 = 11
We've found that buying 11 AirPods and 3 Apple Watches would indeed give us a total of 14 items. The cost for this combination would be:
11 AirPods * $150 = $1650
3 Apple Watches * $400 = $1200
Total = $1650 + $1200 = $2850
Therefore, the answer is 'Yes', it is possible to purchase exactly 14 items and spend exactly $2,850.