How do you determine the line of possible solutions to an underdetermined system of linear equations?
1 Answer
Trial and error with reasonable factors is really the only possibility.
Explanation:
Mathematically, you must have an independent equation for each variable to be able to calculate a solution. Underdetermined systems may have an infinite number of possible solutions - it all depends on the relationships of those "extra" unknowns.
In application (science) we can usually identify what we need to know in order to generate realistic additional independent equations. For example, I may need to change an experimental parameter, or introduce another "constant" factor to an experiment in order to generate data that can provide the missing equations.
A good example are the Ideal Gas Laws. They involve three possible unknowns. In practice, the parts of the equation were developed separately for two unknowns at a time. Only when those were available did scientists "put them together" into the combined form we use today. What the gas law equation tells us is what kinds of experimental data we need to solve for any particular set of unknowns. If we need an additional set of data to solve a particular problem, we do need to get it before we can solve the equations.
