Begin the discussion by responding to the following prompt:

Question

Systems of linear equations can be used to solve many, but not all, real-world problems. When you chose the problem for your journal entry, how did you know that this problem could be solved using a system of linear equations? What specific characteristics does your problem have that made it a good choice? List at least three reasons why the problem you described in your journal prompt can be solved using a system of linear equations.

Post a detailed response to the questions about your choice of a real-world problem to solve using a system of linear equations. Then respond to two classmates' posts.

Participate in the discussion to learn more about how to determine if a real-world problem can be solved using systems of linear equations.

Answer 1

In my journal entry, I chose to analyze a problem involving the budgeting of a school event, specifically the planning of a spring fair. Through this scenario, I realized that it could efficiently be modeled and solved using a system of linear equations. Here are three specific characteristics that made this problem a good fit for such an approach:

1. Multiple Variables: The problem involved multiple expenses to consider, such as venue rental, food supplies, and decorations. Each of these expenses can be represented by a different variable. Additionally, I needed to consider different costs associated with the types of activities planned, such as games and entertainment. This complexity naturally lends itself to a system of equations, as each variable can be defined by an equation representing its cost relative to the overall budget.

2. Linear Relationships: The relationships among the different components of the budget were linear. For instance, if I decided to increase the number of food vendors, the total cost would directly increase at a constant rate based on the number of vendors hired. This straightforward relationship allowed me to create linear equations that depict how one change in a variable would impact the total budget. Each decision regarding expenses could be expressed as a linearly dependent equation, making it easier to analyze the interactions among the variables.

3. Constraints: The situation was constrained by a total budget limit set for the fair, meaning that the total costs had to stay within a specific financial boundary. This provides a clear system to work with: the sum of the costs, expressed as equations, must equal this budget limit. By formulating the problem in terms of equations, I could apply methods such as substitution or elimination to find various combinations of expenses that would adhere to this constraint.

Taking these aspects into consideration, I concluded that using a system of linear equations was an appropriate and effective way to solve the problem of budgeting for the spring fair, addressing multiple variables and linear relationships under specific constraints.

I’m looking forward to reading about the problems chosen by my classmates!