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Preparing for the General Educational Development (GED) test can be an essential step for many individuals looking to gain a high school equivalency credential. In the realm of mathematics, an understanding of a wide array of topics is necessary, including algebraic expressions and equations - this aligns closely with the type of problem we tackled in the provided exercise.

When studying for the GED, practice with real-world scenarios, such as calculating costs for events, assists in making abstract algebraic concepts more concrete. Such practice problems not only strengthen mathematical skills but also give test-takers a glimpse into how these skills apply to everyday situations.

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as addition, subtraction, multiplication, and division). In the given problem, 400 + 5x represents an algebraic expression where 400 is a constant (the base fee), 5 is a coefficient, and x is the variable (the number of attendees).

Understanding how to construct and interpret these expressions is foundational in algebra. It is crucial to recognize the roles of constants, coefficients, and variables to manipulate and solve problems involving algebraic expressions effectively.

Equations in one variable, such as the one in this problem, are statements that two expressions are equal and they usually involve finding the value of the unknown variable. The equation C = (400 + 5x) / x includes one variable x, which represents the number of legionnaires. Grasping the concept of solving for a single variable is a pivotal part of algebra and helps students understand how to isolate and calculate unknown quantities in various contexts.

Solving these types of equations might involve operations like simplification, addition, subtraction, multiplication, division, or factoring. The exercise showcased an application of dividing an algebraic expression by the variable to find the average cost per attendee, which is a typical example of an equation in one variable.

Problem solving in mathematics is not only about getting the answer but understanding the process of reaching that answer. It involves breaking down a problem into steps as we did in the exercise: identifying the total cost function, calculating the average cost per person, and selecting the correct equation that represents the situation.

To excel at mathematical problem solving, one should practice identifying relevant information, constructing models (like equations), and using logical reasoning to navigate through these models to arrive at a solution. In our provided solution, interpreting the given task and crafting an appropriate equation was key to finding the correct average cost per legionnaire. The process can be further improved by checking if our answer makes sense in the context of the problem, ensuring the solution is both mathematically and logically sound.