Solution Possibilities

Because the solution to a system of linear algebraic equationsrepresents the intersection point of those linear algebraic functions, three solution possibilities exist:

1. Unique Solution
   When two equations represent two non-parrallel lines, there exists one and only one intersection point. Therefore, the system has a unique solution.
   Given the following system of linear algebraic equations:
   

2. y = 2x
   y = - 2x + 3

   
   It can easily be determined that the system has a single solution x = 3/4 and y = 3/2, as shown in the figure below. It can also be determined that the determinant of the coefficient matrix representing these set of equations is non-zero.

   

   
   

2. Nonuniqueness of Solution
   When two equations represent the same line there exists an infinite number of common or intersection points. For such a system there are an infinite number of solutions.
   Given the following system of linear algebraic equations:

3. y = 2x + 1
   3y = 6x + 3

   It can be seen that there are an infinite number of common points as shown in the figure below. It can be found that the determinant of the coefficient matrix representing these equations is zero. This means the solution to the set of equations is singular.

   

   
   

3. Nonexistence of Solution
   When two equations represent two parallel lines without a common point, i.e., non-colinear, the lines have no intersection point and accordingly the system has no solution.
   Given the following system of linear algebraic equations:
   y = 3x + 2
   y = 3x

It can be seen that there is no point where the two lines intersect as shown in the figure below. It can be found that the determinant of the coeffiecient matrix representing these equations is zero. This means the solution to the set of equations is singular.

For simple systems containing only 2 or 3 equations it is possible to find the solution either graphically or by solving for one variablein terms of another. However, as equation systems become more complicated other techniques are used to find their solutions. To solve a set of equations with n equations and n unknowns, the process of elimination is very useful. Also related to solving systems of equations is the determinant. Both of these subjects will be covered in this section.