Find the equilibrium quantity and the equilibrium price. In...

Find the equilibrium quantity and the equilibrium price. In the supply and demand equations, $p$ is price (in dollars) and $x$ is quantity (in thousands).A theater charges $\$ 50$ for main floor seats and $\$ 15$ for balecony seats. If all seats are sold, the ticket income is $\$ 27,000$. At one show, $25 \%$ of the main floor seats and $40 \%$ of the Ebalcony seats were sold, and ticket income was $\$ 8,550$. How many scats are on the main floor and how many are in - the balcony?

Key Concepts

Market Equilibrium

This concept refers to the state in which the quantity of a good supplied equals the quantity demanded. It is a fundamental idea in economics, where the market price and quantity are determined by the intersection of the supply and demand curves. In mathematical terms, finding the equilibrium involves setting the supply function equal to the demand function and solving for the unknown variables, commonly the equilibrium price and quantity.

System of Linear Equations

Many economics and real-world problems, such as determining market equilibria or allocating resources based on multiple conditions, can be modeled using systems of linear equations. This key concept involves writing down a set of equations that represent the relationships given by the problem and then solving the system simultaneously to find the values of the unknowns.

Percentage-Based Modeling

This concept involves using percentages to represent parts of a whole in various scenarios. In application problems, percentages are often used to describe proportions of a quantity. For example, when a certain percentage of seats or goods is sold or utilized, it can be converted into an equation that links the percentage to the total number. This technique is particularly useful when translating real-world conditions or constraints into a series of mathematical equations that can be solved.

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