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 200 -seat theater charges $\$ 8$ for adults and $\$ 5$ for children. If all seats were filled and the total ticket income was $\$ 1,435,$ how many adults and how many children were in the audience?

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Key Concepts

Solving Simultaneous Equations

Solving simultaneous equations involves finding the variable values that satisfy all equations in the system. Techniques such as substitution, elimination, or using matrices are standard approaches. This procedure is essential in many applied mathematics problems, including identifying exact figures in economic models or scenario analyses involving multiple unknown quantities.

Market Equilibrium

Market equilibrium occurs when the quantity supplied equals the quantity demanded at a specific price. In this situation, there is no tendency for the price or quantity to change because the market is balanced, meaning that buyers' desires match the sellers' offers.

System of Linear Equations

A system of linear equations consists of two or more equations with common variables. Solving these systems allows us to find the unique values for the variables that satisfy all the equations simultaneously, a method that is particularly useful for determining equilibrium conditions or resolving problems with multiple constraints.

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