Okay, there are several ways to do large, simple simultaneous equations w/o a calculator. All are pains in the butt. You can do row reduction, Cramer’s rule, Laplace development, and probably some other stuff too. Here’s how to use Cramer’s rule for matrices w/ n>2:

The determinant “D” for the 2x2 matrix [[a,b],[c,d]] is ad-bc. Simple, right? For the 3x3 matrix [[a,b,c],[d,e,f],[g,h,i]], D is determined using minors and cofactors. To do this, imagine crossing out one row and one column of the 3x3 matrix, leaving a simple 2x2 matrix. Now take the determinant of this 2x2 matrix (which you already know how to do). Now multiply this answer (D1) by the number which got crossed out both vertically *and* horizontally. This number is called the minor. Now multiply by the cofactor. The cofactor is not a number, but either a + or - sign. Start with a + sign in the upper left hand corner, and then just checkerboard the rest of the matrix to see what the cofactor of every other square is. Then repeat this process for an entire row or column, and add 'em all up.

Perhaps an actual example will help here, no?

Take the following matrix:

1 5 2

3 4 9

7 8 6

We can pick any row or column we want to expand upon. Let’s pick the second row. Start with the first number in the second row, 3. Cross out the row and column that it is in. This leaves us with the matrix [[5,2],[8,6]]. The determinant D1 is 14. Thus the first number we get is 3 * 14 * -1=-42. Remember, -1 is the cofactor. Here’s how the cofactors look for the matrix:

Now let’s do this with the second number in the second row, 4. D2= -8. Our second number is -8 * 4 * 1 = -32.

Our third number is -27 * 9 * -1 = 243.

So the determinant is 243±32±42=169. A quick check on Mathematica reveals this to be correct. You could have used any row or column and arrived at the same result. Naturally, to simplify things, you’ll wish to use a row/column that has the most 0’s in it.

Now, for a 4x4 matrix, simply expand the process. Pick a row or column. Cross off a number to get a 3x3 matrix. Find its determinant, multiply by the minor and cofactor, and then continue down the row or column. Keep expanding for a 5x5 and 6x6 matrix. This is already a lot of work, and you’re still just finding the determinant, not using Cramer’s rule yet. Here come’s Cramer’s rule:

The value of “x” in some set of simultaneous equations is found as follows. Take the determinant of the matrix, using the coeffecients next to the variables as the elements (already done). To solve for any single variable, replace its coefs in the matrix with those of the coefs with no variables next to them, and find its determinant (D2). Take D2/D1=answer. Do this six times for all six variables, and you have yourself an answer. I’ll do a simple example with a 3x3 matrix:

3x + 2y + z = 14

x - 4y + 3z = 4

4x + y - z = 2

We find the determinant of:

3 2 1

1 -4 3

4 1 -1

It is 46.

Now, we plug in the numbers with no variables next to them for the x values, giving us the matrix:

14 2 1

4 -4 3

2 1 -1

Take the determinant. It is also 46. 46/46 = 1, so x is 1.

Now do y:

3 14 1

1 4 3

4 2 -1

Now our determinant is 138. 138/46 = 3, so y = 3. You can now solve for z the easy way.

You can usually use Gaussian row reduction to simplify your matrices (and hence your determinant-finding) a little bit, but that’s a bit complicated for high school. It’s definitely too complicated for the SDMB! Good luck.