Super-long algebra question?

Question

Good luck with solving this one! Even I don't know the solution. I only know the answer(but not telling yet).

Neopets Lenny Conundrum Qn 149(www.neopets.com)

8k - 5n = 8r
3u + 2o = 189
5d + 2c = 6m
5L + 4m = 23b
3r - m = 3u
3k - 4i = 2i
3n - 3o = c
o - c = 4d
8i + 9r = 9n
10m + 7d = 3n
4d + 4e = 6b
4y - 6u = 2m
6b + 3c = 14m

L + e + n + n + y + c + o + n + u + n + d + r + u + m = ???

can you include the steps if possible? i know it would be LONG but pleaseeee

really made a fool of myself here...but algebra is soo hard...

Answer 1

OK, I think I found the last error. Here's the correct (I hope, at least) set of values:
b = 20
c = 30
d = 6
e = 24
i = 36
k = 72
L = 80
m = 15
n = 64
o = 54
r = 32
u = 27
y = 48

I didn't actually check the values this time, but I thoroughly checked the calculator for typos. OK, so by this point you're probably wondering how I got my calculator to do all that work. I used an augmented matrix to represent the system of equations and row reduced it with the calculator. There are other ways to do find the solutions, but row reduction is the simplest to understand without any advanced knowledge. This page appears to give a decent explanation:
http://planetmath.org/encyclopedia/GaussianElimination.html
If you don't understand that explanation, you can always search for a better one.

You can solve these beasts by hand, but it's not so easy to do. I just finished a class where we did that, and we never solved anything more than a 4 variable system by hand. And we only had to do it by hand a few times, because calculators were allowed after a certain point! The big systems have the same properties as the little ones, and they're solved the same way. The only difference is the number of calcuations you have to do to solve the system.

Now that you have the values, you should be able to just plug them into the equation and get the result you're looking for! I tried these values in a few equations, and they're right. You may want to check them all, if you want to be sure. Good luck!

Oh, yes . . . one more thing . . . don't be so hard on yourself! This sort of problem is really very simple. It's just so tedious that people give up! Anyone can come up with one of these, it's really very easy to do. Solving one, on the other hand, is a huge pain!

If you can't find any good explanation of row reduction online, email me and I'll try to explain it to you or point you in the right direction. You need to start with a small system (2 or 3 variables tops) when you're learning. Good luck!

Answer 2

574234678 logical guess