How would you describe the method of elimination for solving a system of three linear equations in three unkno?

Let the 3 equations be A, B and C, containing the unknowns x, y and z.

Pick any 2: say A and B.
Re-arranged A so that it reads: z = some expression containing only x and y.
Call this new equation A2.

Repeat for B: z = a different expression containing only x and y.
Call this new equation B2.

So now you can write: z = equation A2 = equation B2.
Thus z is eliminated.
Call this equation N1: it only contains x and y.

Now repeat all of that from the top, but with a different pair: say A and C
You end up with another equation, N2, that also only contains x and y.

N1 and N2 are now a pair of simultaneous equations in two unknowns: x and y.
Solve them for x and y, producing x1 and y1.

Now substitute these values, x1 and y1 in any one of the original 3 equations: A, B or C.
Hence, solve for z, producing z1.

Check your result for z by substituting x1 and y1 in any other one of the other original 3 equations: A, B or C.
You should obtain the same value of z1.

That, in words, is the answer.

An alternative method is to use determinants - but that is difficult to describe here with this very restrictive, non-graphical, text-only format that Yahoo insists on keeping here in a Mathematical Section of all places!

I'm not sure I understand what you are asking?  One approach is to form a 3 x 3 matrix and pose the system of equations in matrix form:

Ax = b

Where entries of A are the coefficients in front of each variable, x is a vector whose entries are the variables to be solved, and b is a vector whose entries are the constants in the system of equations.

From here, several methods can be used to solve for x (i.e. the solutions to the system of equations):

1. Gaussian elimination;
2. Find the inverse of A and set x = inv(A)*b
etc...