Formulate but do not solve the problem.
A theater has a seating capacity of 840 and charges $4 for children (x), $6 for students (y), and $8 for adults (z). At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totaled $5000. How many children attended the show?
= 840
= z
= 5000
To solve this problem, we need to form an equation based on the given information.
Let's assign variables to the number of children, students, and adults attending the show:
- Let x be the number of children.
- Let y be the number of students.
- Let z be the number of adults.
According to the given information:
- The seating capacity of the theater is 840, so the total number of attendees can be represented as x + y + z = 840.
- The price for children is $4, so the total amount earned from children attending the show is 4x.
- The price for students is $6, so the total amount earned from students attending the show is 6y.
- The price for adults is $8, so the total amount earned from adults attending the show is 8z.
- The total amount earned from all attendees is $5000, so the equation can be written as: 4x + 6y + 8z = 5000.
To find the number of children who attended the show, we need to solve the equation system that consists of two equations:
Equation 1: x + y + z = 840
Equation 2: 4x + 6y + 8z = 5000
We can solve this system of equations using various methods such as substitution, elimination, or matrix operations. However, since you only asked to formulate the problem and not solve it, we will leave the solution to you.