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In the document here, I have solved problem $58.2$ on page $636$. My solution is below:

Solution to the system of equations

Now, I hope that this is not a duplicate (and I suspect it isn't, since this seems to be a slightly obscure pdf). My question is: Is there a more intelligent way to do this? The problem itself states, "This may look long and boring: Look for a shortcut." My interpretation of that was to notice how the equations relate to each other. That is, they form a diagonal of $0$s if you arrange the terms into columns.

Based on this, I observed the fact that, for any of the two equations, a subtraction results in the $0$ of the first being filled and in the $0$ of the second being filled, leaving $0$s everywhere else. (I realize that's not really mathematical language, but you can see what I mean when I subtracted $E_{4}$ from $E_{2}$.)

After I realized that, I just did the two subtractions that seemed most interesting. Then I realized, "Oh hey, if you add those, you get an equation that you can use to make the originals only 2 variable!" Thus, giving me the solution after the trivial work.

I'm hoping this isn't too elementary for math.stackexchange, but feel free to close it if so.

P.S. Maybe I am just being perfectionistic in this case, as the actual amount of work required in this method wasn't that much. But, I'd love to see any other approaches if they exist. Oh, and, I've always kinda disliked systems of equations, but this one seemed interesting.

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1 Answer 1

up vote 7 down vote accepted

If you add the four original equations, you get $3x+3y+3u+3v=-9$, so that $x+y+u+v=-3$. Now, subtract any one of the original equations from this one will give the value of a particular variable.

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Wow, I didn't see that at all. Nice job. I'm going to accept this answer if no one provides a better one. :) –  000 Jan 16 '12 at 1:52
Hi again. I accepted your answer. As a slight addendum, is there any particular set of methods you employ when attacking systems of equations? For example, (note: this is just a specific example) do you prefer Gaussian elimination over (sometimes tedious) substitution? Are there any methods that you employ that might not be that common? (This is a bit of a hard question, given the vast area to answer it, so I am more than happy with almost any constructive input you provide. :)) –  000 Jan 16 '12 at 20:56
@user22144: What I try depends a good bit on what the equations look like. For this particular problem, the uniformity and sort of symmetry of the given equations (each involved all but 1 of the variables and all with coefficient 1) suggested to me that a trick like adding the equations together might be helpful. If equation(s) are given in a form that is ready for substitution, I might give that a try. If the numbers would make elimination particularly annoying, I might try substitution or look for a trick... Otherwise, it's pretty much just elimination. –  Isaac Jan 16 '12 at 21:02
Thank you. I see that we're not too different in our thinking: I essentially bruteforce it whenever nothing elegant arises after a good 3-5 minutes. I think I should probably try to be more patient and think things through rather than just start plowing away immediately, however. That might've be why I didn't immediately realize how cute this system was. –  000 Jan 16 '12 at 21:08
@user22144: A lot of it is just experience—I've seen problems of exactly this sort on dozens of contests and written them into numerous contests, so it's a pattern that I recognize immediately. –  Isaac Jan 16 '12 at 21:10

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