Three students buy lunch in the cafeteria. One student buys...

Three students buy lunch in the cafeteria. One student buys 2 hamburgers, 2 orders of fries, and 1 soda for $\$ 9 .$ Another student buys 1 hamburger, 1 order of fries, and 1 soda for $\$ 5 .$ The third student buys 1 hamburger and 1 order of fries for $\$ 5 .$ If possible, find the cost of each item. Interpret the results.

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Key Concepts

Inconsistency in Systems

An inconsistent system is one in which no solution exists that simultaneously satisfies all the equations. Recognizing an inconsistent system is important, as it indicates that the information provided is contradictory or flawed, which can be critical in both theoretical problems and real-world applications.

Elimination and Substitution Methods

The elimination and substitution methods are standard techniques for solving systems of linear equations. Elimination involves combining equations to cancel out one variable, while substitution requires solving one equation for a variable and then substituting that value into the other equations, leading to a solution for the remaining variables.

System of Linear Equations

A system of linear equations involves multiple equations that must be satisfied at the same time by the same set of unknown variables. This concept is fundamental in algebra and is used to model and solve problems where different conditions or transactions relate the unknown quantities.

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