The year is 179 and we are in China. The work of putting together findings from mathematicians in the region over the span of a millennium has just been finalized.
The result is about to become the most important work produced in Chinese in the history of mathematics: The Nine Chapters on the Mathematical Art.
One of the nine chapters in the book is dedicated to the method of solving systems of linear equations.
Already at this time, the Chinese had realized the practical use of formulating and solving problems using this technique, where one example in the book involves comparing the yields of different types of grain.
The procedures of solving a linear system of equations often give rise to variables having negative coefficients, and The Nine Chapters on the Mathematical Art is the earliest known piece of literature dealing with the concept of negative numbers.
An equation with one unknown variable, like , only have one solution. In this case . Simple, right?
If we add a second variable to the equation, the problem becomes less trivial. has infinitely many combinations of values for and as solutions. To obtain definite values, we need two different equations.
Finding unique values for a number of unknowns requires the same number of equations
We use systems to group equations whose variables we want to determine. Systems of linear equations only contain variables multiplied by simple numbers.
A more familiar way to express the equation is as a function of with respect to :
This is the equation of a line, and any point on that line provides a solution to .
With more variables, the equations describe more complicated objects, but the way to solve it is always the same.
A solution is a point on the object, and for a system of equations, we need to find points common to all objects described.
This can be done using the method of elimination which systematically cancels one of the variables at a time until the solution is easily obtained.
If this cannot be done, there are no solutions, but it can also turn out to have an infinite number of solutions.
System of linear equations
An example of a linear equation is the equation for a line or for a plane. The general form is:
The special case where the last constant is equal to zero is called a homogeneous equation. Also note that the variables are of the first degree (lacks potency). Furthermore, there is no product of the variables ().
A collection of several linear equations is called a system of linear equations or a linear system of equations. A solution to the system must then be a solution for all equations. The general form for a system of equations is:
and the general form corresponding to a homogeneous system having 0 as the right-hand side is:
A solution to an equation is a point in , and in particular, one can see that the origin is always a solution to a homogeneous system. In general, exactly one of the following three solution cases applies to each linear system of equations
A unique solution,
infinitely many solutions or
A system that has at least one solution is called a consistent system, while a system that lacks solutions is called an inconsistent system.
A unique solution always means that the solution is a point. What no solutions mean hardly requires any further development, while the case for infinitely many solutions is what is the most interesting. Infinitely many solutions (points) sound chaotic, as if they appear randomly in the space, but they always follow a geometric shape. These infinitely many points either form a line, a plane or a hyperplane.
A good example for maturing the understanding of systems of equations and their solutions (also called solution sets) is the example in the space , where each equation is a plane:
There exists eight possibilities, each of which belongs to one of the three solution cases for the system of equations.