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Where do you run into a real world situation involving 3 variables and 3 equations? Can someone think of a specific example from business, etc? I recall taking an operations research course that seemed to involve optimization of 3 variables, but do not recall a single example or theme. Any help is appreciated.

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If you are wondering if a model with 3 variables seems like overkill, you should consider that "real world situations" involve a HUGE number of variables, but they are typically assumed to be constant for the sake of a manageable model. – MPW Mar 31 '14 at 17:57

In spirit of Christmas and new years resolutions, suppose that what we were on an diet and needed to eat precisely $245$ calories, $6$ grams of protein, and $7$ grams of fat for breakfast. Unfortunately, I open my cupboard to see that all I have is three boxes of cereal: Cheerios, Cinnamon Toast Crunch, and Rice Krispies. There nutritional information per serving is as follows:

Cereal                     Calories     Protein     Fat
Cheerios                   120          4           2
Cinnamon Toast Crunch      130          3           5
Rice Krispies              105          1           2

Now, normally, I would dive in and gorge myself on Cinnamon Toast Crunch$^\dagger$ because they're delicious - but, I need to stick to my new years resolution.

First, I denote $c = $ servings of cheerios, $t = $ servings of Cinnamon Toast Crunch, and $r = $ servings of Rice Krispies. Then, I form the following system of $3$ equations in $3$ unknowns:

$$120c + 130t + 105r = 245 \ \text{calories}$$ $$4c + 35t + r = 6 \ \text{grams of protein} $$ $$2c + 5t + 2r = 7 \ \text{grams of fat} $$

Now, I leave it to you to find out if I am stuck with one bland mix of cereal$^1$, whether I will be able to form many mixtures of cereal$^2$, or If I will be forever cursed with the dreaded stomach-tire$^3$.

$^1$The system has a unique solution.

$^2$The system has infinitely many solutions

$^3$The system has no solution

$^\dagger$French Toast Crunch is even better, and baby it's back.

Note: I got the numbers for this (silly) example from here, courtesy of Dr. Dagan Karp.

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Gabriel, thanks, and don't take this the wrong way, but this is a perfect example of a fake contrived textbook problem. In which universe will you ever have this fake constraint of precisely 245 calories, 6 grams of protein, and 7 grams of fat? I am hoping to find a real example that makes sense, not a fake example that is created strictly to justify the math. You know what I mean? – JackOfAll Dec 27 '14 at 16:26

A few examples that may or may not arise in other answers...

  • When you have a set of data of the form $(x_i,y_i,z_i)$ and you want to fit a model of the form


least squares leaves you with three equations in $a$, $b$ and $c$.

  • Any kind of a 'network flow'. For example suppose that you have a roundabout with three exits and the flow in and out of each exit is known then using the principle that the flow into a junction should equal the flow out leaves you with three equations in the unknown flow between exists.

  • When doing a partial fraction expansion --- which occurs in various applications --- you are often left with simultaneous equations.

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Linear programming. look HERE ... in fact the first example (rabbit food) involves three variables.

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