Another Look at Systems of Equations: Possible Solutions

Question

This is more of a general question than a specific problem. While doing homework, I am being asked to find the solutions for complicated systems of equations, and I was wondering:

Suppose I were to be given 3 equations involving variables a, b, and c. The goal of the problem is to find the number of solutions (they do not necessarily have to be real) for the system of equations. Would it be more logical to use Cramer's Rule to solve the linear system or to use basic algebra? A roadmap of the following example problem or the like would be helpful.

A basic system taken from the book would be:

3a + 4y - z = 4
3a + 2b - 11c = -13
a + 2b + 3c = 7

Answer 1

There are several good ways to solve systems of linear equations, and the best method to use in any given situation is the one that requires the least amount of work. However, that will depend on the particular equations that you’re trying to solve.

Generally linear algebra offer more useful techniques especially if the matrix has some special structure.

In your case, for example, the Rouché-Capelli theorem says that there are infinite solutions with 1 degree of freedom.

Answer 2

I would recommend linear algebra (read up on row reduction). Row-reduction is basically like basic algebra, just formalized and put into a notation that the computer can easily use. Good question though! :) Cramer's Rule is a way, but it uses determinants, which take longer and longer to compute as the matrix (system of equations) gets larger. Don't hesitate to ask questions about row-reduction if you have any! :)