A Broadway theater has 500 seats, divided into orchestra,...

A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $\$ 150,$ main seats for $\$ 135,$ and balcony seats for $\$ 110 .$ If all the seats are sold, the gross revenue to the theater is $\$ 64,250 .$ If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $\$ 56,750$. How many of each kind of seat are there?

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Key Concepts

Techniques for Solving Simultaneous Equations

Once a system of equations is set up, various methods such as substitution, elimination, or matrix approaches are used to solve for the unknown variables. These techniques provide a systematic way to isolate and compute variable values that meet all of the given conditions.

System of Linear Equations

This concept involves creating a set of equations where multiple unknown variables are connected by linear relationships. In many problems, especially those dealing with resources or finances, a series of conditions or constraints can be expressed as equations, and the solution requires finding values that satisfy all equations simultaneously.

Translating Word Problems into Mathematical Models

Many practical problems require interpreting a real-world situation and expressing it in mathematical terms. This involves identifying unknown quantities, assigning variables, and setting up equations based on the relationships and constraints described in the problem, such as total counts and total revenue.

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