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

For my journal entry, I chose the problem of determining the optimal pricing strategy for two different products offered by a local bakery. The challenge was to maximize the bakery's revenue while taking into account the constraints of production costs and consumer demand for each product. I knew that this problem could be solved using a system of linear equations due to several key characteristics that fit the criteria for such a mathematical approach.

First, the problem involves multiple variables, specifically the prices of two products. In this case, we'll denote the price of Product A as \(x\) and the price of Product B as \(y\). Each price point influences revenue differently, and thus requires determination through a linear relationship.

Second, there are constraints based on consumer demand and production capacity that can be modeled linearly. For instance, if the bakery can only produce a limited number of each product due to time or ingredient availability, we can express these limitations as linear equations or inequalities. For example, if the bakery can make no more than 200 units of Product A and 300 units of Product B per day, these constraints can be formulated as \(x \leq 200\) and \(y \leq 300\).

Third, the goal is to determine combinations of the two prices that yield maximum revenue. This can be captured through a revenue equation \(R = P_A \cdot x + P_B \cdot y\), where \(P_A\) and \(P_B\) are the projected sales based on the prices \(x\) and \(y\). The relationship between the prices and the projected sales can be expressed through linear equations, making it possible to create a system for evaluation.

Overall, the specific characteristics of multiple variables (product pricing), linear constraints (production limits), and a linear relationship (revenue generation) confirmed that this real-world problem was suitable for resolution using a system of linear equations. By solving this system, we can make informed decisions that align pricing strategy with the bakery's capacity and market demand, allowing for better revenue management.

I look forward to reading about the problems my classmates chose and how they recognized their suitability for solving through systems of linear equations.