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Using systems of equations to make decisions

Using systems of equations to make decisions

Author: Christopher Danielson

To be able to represent simplified versions of real-world decisions as systems of equations and to consider the meaning of the solutions to these systems in terms of the decision to be made.

This (very large) packet consists of two examples of real-world problems expressed as algebraic systems of equations. The important point is made that the algebra task of "solving the system" is really only the beginning of making a decision in a real-life situation.
In addition to solving the system, we need to consider the other real-life constraints we are operating under. Examples of constraints are considered in both problem situations.

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Many decisions we make in our everyday financial lives can be captured as systems of equations. We might do the computations outside of math class without seeing them as systems of equations, but if we capture those computations and consider them algebraically, we can gain some insight into how to make better financial decisions.

This packet considers two examples using real data: (1) Deciding how to print out your work as a college student, and (2) Buying a car based on gas consumption.

In both cases, we will have to make some simplifying assumptions-that is true whenever we want to do a mathematical analysis. The fewer simplifications we make, the better our mathematical model and the better decisions we can make. But these are introductory-level problems, so we will keep things simple.

Example 1: Printing costs

A student has three main options for printing out her work this semester:

Option 1: Buy a laser printer

Laser printers are expensive. The student has found a good, reliable one for $150. It prints 2000 pages on one toner cartridge and a new cartridge costs $30.

Option 2: Buy an inkjet printer

Inkjet printers are cheap. The student has found a good, reliable one for $30. It prints 500 pages on one ink cartridge and a new cartridge costs $30.

Option 3: Print at school

The college this student attends provides $5.00 in free printing each semester and charges $0.05 per page after that.

What should the student do?

Writing equations for the first two options

This packet describes the process of turning the first two options into equations by the use of the given information, tables and first differences.

Writing the third equation

This video demonstrates the process of finding an equation for the third option.

Looking at the system graphically

This video demonstrates representing the three equations as a system and then considering whether there are solutions to the system of three equations, or to one or more subsystems of two equations at a time.

Example 2: Buying a fuel-efficient car

In our second example, a young man is about to buy a new car. He is interested primarily in fuel consumption and he is especially interested in the cost of gasoline. Presently, the cost of gas is $3.00 per gallon and he knows that he is going to drive long distances on the highway.

He wonders whether it would be more economical-from a fuel-consumption point of view-to buy a Toyota Corolla (cheaper, but not as efficient) or a Toyota Prius (more expensive to buy, but more efficient).

Option 1: Corolla

The Corolla costs $15,060 and gets 35 miles per gallon.

Option 2: Prius

The Prius costs $23,050 and gets 48 miles per gallon.

What should he do?

Writing an equation for the Corolla (and starting for the Prius)

This video demonstrates setting up the equation for the Corolla.

Finishing the equation for the Prius

This video demonstrates finishing writing the equation for the Prius.

Using the system to make a decision

This video demonstrates using the system of equations to decide what to do in the Corolla/Prius situation.


In conclusion, whenever we represent a real-world decision with a system of equations, we need to make some simplifying assumptions. Having done that, we can solve our system.

But the solution itself doesn't tell us what to do. The solution to a system of equations tells us when the two lines cross-when the inputs and outputs are the same for both functions. In the car problem in this packet, the solution tells us how many miles we have to drive each car in order for the gas cost together with the purchase price to be equal.

The system does not make the decision. Instead, it gives us information to use for the decision. The other information we use to make the decision is our set of constraints. Constraints in this case might include a government rebate for buying a fuel-efficient hybrid car, or the number of miles we intend to drive each year, or that we will drive total before giving the car away. Maybe a constraint is that our spouse insists on the most fuel-efficient car, regardless of the cost. Another constraint might be that we expect the cost of gas to increase sharply in the next few years.

Each of these constraints should affect how we interpret the solution to the system we wrote. The constraints help us decide what to do, as does the solution to the system.