# Would this be the most elegant way to approach an elementary system of equations?

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

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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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.