Solve a System of Linear Equations

APL, 1 char

I know it doesn't fit the (revised) requirements, but it's too good not to post:

⌹

Symbol "domino" ⌹ (division ÷ inside a rectangle) performs matrix division, therefore it can solve any system of linear equations. You just have to put it between the constant term vector and the matrix with the other terms:

      8 ¯11 ¯3 ⌹ ⊃(2 1 ¯1)(¯3 ¯1 2)(¯2 1 2)
2 3 ¯1

(if you want to try it on TryApl, ⊃ is ↑)

Javascript (284 181) - Gauss Elimination method

function f(A){l=A.length;d=1;for(i=0;i+1;i+=d){v=A[i][i];for(k=0;k<l+1;k++)A[i][k]/=v;for(j=i+d;A[j];j+=d)for(k=0,w=A[j][i];k<l+1;k++)A[j][k]-=w*A[i][k];if(i==l-d)d=-1,i=l}return A}

Test

f([[2,1,-1,8],[-3,-1,2,-11],[-2,1,2,-3]]);

=> [[1,0,0,2],[0,1,0,3],[-0,-0,1,-1]]

The returned array combine the identity matrix and the solution.

This answer no longer fits the question after the rule change as it uses a matrix function.*

Sage, 32

~matrix(input())*vector(input())

Sample input:

[[2, 1, -1], [-3, -1, 2], [-2, 1, 2]]
[8, -11, -3]

Sample output:

(2, 3, -1)

_{*Arguably, matrix() is a typecast, not a function (running import types; isinstance(matrix, types.FunctionType) gives False). Also, the ~ and * are operators, not functions.}