Adult and children's tickets for a certain show sell for 8...

Adult and children's tickets for a certain show sell for $\$ 8$ each. A total of 1000 tickets are sold, with total sales of $\$ 8000 .$ Is it possible to figure out exactly how many of each type of ticket were sold? Why or why not?

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Key Concepts

Redundancy of Information

Sometimes, the conditions provided in a problem are not independent of each other. When the information given leads to equations that are effectively the same, one encounters redundancy. This means that the system does not offer enough distinct constraints to determine the unique values of the unknowns.

Underdetermined Systems

An underdetermined system occurs when there are fewer independent equations than unknowns, resulting in infinitely many solutions. In such cases, there isn’t enough information to pin down a unique solution because the constraints do not sufficiently restrict all variables.

System of Linear Equations

This concept involves setting up equations to represent relationships between quantities based on given conditions, such as totals and sums. In many problems, each condition (like total quantity and total revenue) creates an equation, and the solution comes from finding the intersection of these equations.

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