systems of linear equations with three or more variables

[Bob The Great]

Does any one have any good links on systems of linear equations with three or more variables? I was trying to help a friend on them, and I was stumped.

The prob is...

x-2y=7
y+z=1
x+y=4

Any help would be really appreciated!

[TheBurningCat]

I think x= 5 and y= -1

Let me check

[jkrohn]

A very easy way is to use matracies to solve those, especially since that is overdetermined (3 equations, 2 variables).

Sorry no links for it though
When a system is overdetermined there are 2 options,
1) it has a solution (as in teh one above)
In that case you set up teh matrix

A = | 1  -2  0|
    | 1   0  1|
    | 1   1  0|
and
b = |7|
    |1|
    |4|

and just solve for Ax=b where x is a vector belonging to R3

The second option is that there is no real solution so a Least squares solution must be found.

This can be found by the formula. (use the same A and b from above)

(At)A*x = (At)b where (At) is A transposed

This will leave you with a projected vector onto your space closest to a "real solution".

Both of these can be solved use Guassian reduction.

Hope I helped and didn't just confuse!
Jkrohn

[TheBurningCat]

OK, i did ut in my head and then wrote it down..might be wrong...

Never done this before, just pure logic used.

BAH cant find a host!

[zskillz]

this looked sort of helpful Simultaneous Linear Equations

good luck
-Z

btw - if I'm not mistaken, you can solve any of those sort of equations using the simple substitution method. although jkrohn's method should also work.. it's a bit much for this level of math. you can solve these using a TI-85 style calculator (at least I positive for that model and anything higher than it)

[Bob The Great]

ok so

x=5
y=-1
z=2

cool! Thanks

[zskillz]

hehe.. oops - jkrohn.. just notices something. that system is not overdetermined. you mistaked the 'z' for a '2'. which means you used the wrong matrix as well. (been a while since I've taken math like that.... can't remember all of the little tricks)

-Z

[jkrohn]

Well simple logic can solve that one, but he made no inidication of the math level and how good of an example that problem was

A ti-85 isn't gonna solve an over determined set of equations, thats for sure

Jkrohn

[zskillz]

this looked like helpful information for the method that jkrohn was using coefficient matrices

good luck!

-Z

[zskillz]

jkrohn - no harm intended... sorry if that came off wrong!

-Z