Final answer:
The Smith family bought 4 adult tickets and 3 child tickets to the theater by using a system of equations to solve for the quantity of each ticket type based on the total cost and the number of tickets purchased.
Step-by-step explanation:
To solve this problem, we can use a system of equations. Let 'a' represent the number of adult tickets and 'c' represent the number of child tickets. We are given that adult tickets cost $12 and child tickets cost $9. Also, the family bought 7 tickets in total and spent $75.
The following are our two equations based on the information provided:
- Equation for the total cost: 12a + 9c = 75
- Equation for the total number of tickets: a + c = 7
To solve these equations, we can first solve the second equation for one of the variables and then substitute it into the first equation.
Solve the second equation for 'a':
a = 7 - c
Now, substitute 'a = 7 - c' into the first equation:
12(7 - c) + 9c = 75
84 - 12c + 9c = 75
Merge like terms:
-3c = -9
Divide by -3 to find 'c':
c = 3
Now that we have the number of child tickets, we can find the number of adult tickets by substituting 'c' back into one of the equations. We'll use the equation a + c = 7:
a + 3 = 7
Subtract 3 from both sides to find 'a':
a = 4
So the Smith family bought 4 adult tickets and 3 child tickets.