A definitive method of solution

The above method is usually described as the Gauss-Jordan elimination method. While it is quite efficient and gives the answer, it has one drawback. We cannot predict the final answer or parts of the final answer without actually running thru the whole algorithm. There is a theoretical alternative which will write down the pattern for the final answer and create formulas for the solution. The problem is that the calculations are usually too long, unless we find a clever short cut. In practice, the short cut is the Gauss-Jordan elimination!

We will describe the theoretical method and learn to use it whenever possible, since it does give a better understanding of the solutions themselves.

For one equation in one variable, this is trivial. Given the equation , the solution is . Of course, we need to observe that this is true for nonzero . If is zero and is not, then there is no solution, while every value of is a solution of the trivial equation when are both zero.

For two equations in two variables, this works as follows. Let the equations be . Define the numbers
, , .

Then the solution is and . This is the so-called Cramer's Rule . Of course, we have to deal with the exceptions as above.

If , then the above solution is valid. If , then we have two cases. If one of the is nonzero, then we have no solution, while if both are zero, then one of the equations is a multiple of another! Thus we are really in the case of one equation in two variables.

For instance, you can check that for the system , we get that
, , , which clearly leads to the correct answer!

This illustrates how the theoretical method progresses. The calculation of the quantities like (also called determinants) is conceptually easy, but can take a long time. However, we will learn enough tricks to minimize the work.