Problem 45 - Introductory Algebra for College Students

Question

Exercises \(17-20\) involve using systems of linear equations to compare costs of telephone plans and plans at a discount warehouse. Describe another situation that involves choosing between two options that can be modeled and solved with a linear system.

Step-by-Step Solution

Answer

Two examples of real-life scenarios that can be modeled and solved with linear systems are comparing investment plans and evaluating job offers.

Step 1: Definition of Linear System

In mathematics, a system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables. This kind of system can model situations where different factors vary linearly, meaning they change at a constant rate. This concept is often used to compare costs, salaries, investments and many other entities that can vary linearly.

Step 2: Real-Life Scenario Description 1: Comparing Investment Plans

One instance could be comparing different investment plans. Suppose there are two investment funds available. The first one has a set monthly management fee and a fixed rate of return, while the other has a lower set fee but also offers a lower rate of return. These could be modelled by two linear equations, where one variable is the amount invested, another might be the time, and the equations would represent the total return on each plan.

Step 3: Real-Life Scenario Description 2: Evaluating Job Offers

Another example could be comparing job offers. Suppose one job offer pays a higher salary but comes with higher monthly commuting costs, while another job pays less but comes with lower commuting costs. These could be modelled using two linear equations and solving the system could help determine the minimum salary required to justify taking the job with the higher commute costs.