Answer:
1048 student tickets were sold and 1452 adult tickets were sold
Explanation:
We can use a system of equations to find the quantity of both the adult and student tickets. We can allow A to represent the quantity of adult tickets and S to represent the quantity of student tickets.
First equation: (For the context of my explanation, revenue is defined as the product of price of an item and the quantity) We know that the sum of the revenues earned from the adult tickets equals the total revenue as
(price of adult tickets * quantity of adult tickets) + (price of student tickets * quantity of student tickets) = total revenue
Since we know that the adult tickets cost $3, the student tickets cost $2, and the total revenue earned was $6452, our first equation is:
3A + 2S = 6452
Second Equation: We further know that the sum of the quantities of adult and student tickets equals the total amount of tickets sold as
quantity of adult tickets + quantity of student tickets = total amount of tickets sold
Since we know that there were 2500 fans in the gym, our second equation is:
A + S = 2500
Method to solve: We can isolate A in the second equation by subtracting S from both sides. This will allow us to substitute it in the first equation to first solve for S , the quantity of student tickets sold:
Step 1: Isolating S in second equation:
(A + S = 2500) - S
A = -S + 2500
Step 2: Plugging in (substituting) A = -S + 2500 for A in 3A + 2S = 6452:
3(-S + 2500) + 2S = 6452
-3S + 7500 + 2S = 6452
-S + 7500 = 6452
-S = -1048
S = 1048
Now that we know the quantity of student tickets sold was 1048, we can plug in 1048 for S in any of the two equations in our system to solve for A, the quantity of adult tickets sold. Let's use the first equation:
Step 3: Plugging in 1048 for S in 3A + 2S = 6452
3A + 2(1048) = 6452
3A + 2096 = 6452
3A = 4356
A = 1452
Thus, the quantity of adult tickets sold was 1452.
Optional Step 4: We can check that we've found the correct answers by plugging in 1048 for S and 1452 for A in both equations in our system and checking that we get 6452 for the first equation and 2500 for the second equation:
Plugging in 1048 for S and 1452 for A in 3A + 2S = 6452 (i.e., the first equation in our system):
3(1452) + 2(1048) = 6452
4356 + 2096 = 6452
6452
Plugging in 1048 for S and 1452 for a in A + S = 2500 (i.e., the second equation in our system):
1048 + 1452 = 2500
2500 = 2500