Actually Applicable Application Problems and Brainteasers/Linear Systems, Underdetermined by One Constraint

Overview

We find it convenient for systems with three variables to have three equations or inequalities, of four variables to have four, and so on, because this gives us unique solutions. However, sometimes situations are more flexible. These situations are called "underdetermined" and this method is a way of dealing with that.

If the constraints are based on a real situation, this is as Actually Applicable as any other strategy in the linear systems/linear programming family.

General Method

1. As you would usually do, set up an equation or inequality for each constraint, put their coefficients and constants into an augmented matrix, and apply RREF to that augmented coefficient matrix.
   • Unlike the ideal situation, this RREF will probably have two columns that do not resemble an identity matrix. The second is the constants for the right hand side of the equation, as usual, but the first is the (probably mostly nonzero) coefficients that apply to your "free variable."
2. Extract equations from the rows of the RREF matrix.
3. Assign any value that makes sense to your free variable (this is why it is called free), and use the equations you extracted in the previous step to determine the values of the other variables.
   • If you assign the value 0, the other variables' values can be read directly from the constants column as usual.
   • You may need to try more than one value depending on the problem.
   • If there are a finite number of possible values for the free variable, you can try each and give all possible solutions.
   • A spreadsheet can be helpful for evaluating the other variables' values, especially if there are many options you want to try for your free variable.

Problems

Zoo Animals

A zoo wants to buy a total of 11 animals. Their options are elephants, giraffes, and rhinos. Each elephant eats 200kg food per day, each giraffe eats 75kg, and each rhino eats 50kg. The zoo's budget can provide 950kg of food per day to these animals. How many of each type should the zoo buy?

This problem was found in the wild on Facebook. Its original source is unknown.

In the interest of honesty I'll admit that this particular problem is a bit of a brainteaser, since it seems improbable that elephants, giraffes, and rhinos would eat the exact same foods in the same proportions, such that the zoo's budget for them could be accurately expressed in terms of total weight of the food rather than in money. If anyone can point me toward the needed information I'd be happy to develop a better problem of this kind!

Make Your Own Problem

For a situation, write down some options you need to choose between, and count them.

Find one less linear relationship between those options than the number of options.

Apply this method and make a decision.