For certain systems of equations, it is obvious what the easiest way to organize and manipulate the equations should be. For instance,

$$y = 10x + 5$$ $$2x + y = 125$$

So you take the first equation and plug it into the second \begin{align} 2x+(10x+5) &= 125\\ 12x+5&=125\\ 12x&=120\\ x &= 10 \end{align}

In this particular example, I would call the "path of least resistance" simply taking $y$ and plugging it into $x$.

However, in my economics problems, the simplifications are not so simply and a direct route is not so obvious. There may be 5-7 variables and 6-8 equations and identifying an order for manipulating the equations becomes a tedious chore.

I was wondering if there are efficient ways to organize equations such that the "path of least" resistance emerges?

"Path of least resistance" is a term I made up to describe the sense that you are manipulating equations in a way that is reducing their complexity rather than increasing it.

Presumably, as you solve a problem, the number of variables in the equations should consistently get less. In my example above, once a substitution has occurred, the equation has in some sense been reduced because now only one variable remains and parameters. It would seem therefore that, as you are solving a problem, there should be qualitative indicators that you are on the right track (e.g. Fewer variables in your equations).

I was wondering if anyone knows of resources that might discuss this kind of issue in a theoretical way and how to approach solving problems of this sort.

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    $\begingroup$ Solving multiple equations in multiple variables can be a whole course just in itself as there are is a range of approaches. Please provide a bit more background about what level of linear algebra you've done before. From your question you've learnt substitution and probably elimination but have you also done matrices or related high level ideas? $\endgroup$ – Ian Miller Mar 17 '16 at 8:29
  • $\begingroup$ Also, the easiest way to solve problems is based on the person's opinion. I don't think there would exist any theory that discusses this. $\endgroup$ – SS_C4 Mar 17 '16 at 8:43

Systems of equations are first sort by the family of math inside: linear/affine equations ? polynomials ? functionals ? (letting apart intro-differentials).

Then they can be classified as full rank (unique / discrete solutions) or not.

They are also classify as being problems within integers or reals number.

Then even in the "simple case" of affine equations there are full textbooks of how to solve Ax=b (A matrix, x,b vectors), depending on the knowledge and structure in the numbers: are a lot of coefficient null ? is the matrix symmetrical ? is the system ill-conditioned (determinant close to zero so most methods will have precision issues). The physics the equation come from can also bring useful structure influencing the choice of the solver.

To choose the appropriate algorithm you could also classify problems and solutions depending on the size (a handful of equations and parameters, or millions ?), depending on the precision required (exact(computers-wise :-) ) or approximative ?), depending on whether you want the full explicit solutions or just know some properties of them, depending if it will be solved once or for many vector b, depending on whether you want to compute on a PC, on a GPU, on a cluster of PC.

So the answer is yes, there are recipes. But none is universal and context-free: they depend on your ingredients, tastes, and diner context :-)

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