Independent systems contain equations that have only one solution that satisfies all of the equations. The solution is a pair of values for the two variables given in all equations. Graphically, this pair of values appears as the single point where the equations intersect. The values are represented as a pair of x and y coordinates on a graph. For example "y = x + 5" and "y = 7-x" are both linear equations that have "x = 1" and "y = 6" as a common solution. In other words, when graphing the two lines, "y = x + 5" and "y = 7-x," they intersect at the point (1,6).
Systems of linear equations that are inconsistent do not have a solution common to both equations. Graphically, this means that the equations in the system do not intersect, but instead run parallel to one another. For instance, "y = 6" and "y = -1" are two lines that are both parallel to the x-axis (horizontal axis), but do not ever intersect each other. In this case, there is no pair of numbers that satisfy both equations at the same time.
Systems of equations that are dependent have equations that are actually identical, but written in a different form. This means that the equations have all answers common to one another because they are really the same equation. Graphically, linear systems that are dependent are lines that lie on top of one another with all points or solutions in common. For example, "y = x + 5" and "x= y --5" are really the same equation and so would appear as the same line on a graph. The two equations would have the same points in common.
Systems of linear equations with a large number of variables can still be categorized into three types, primarily -- independent, inconsistent and dependent. When solving a large number of equations, there are various techniques, such as determinants and Gauss Jordan Row Reduction, in the branch of linear algebra that can assist in finding solutions efficiently. Using a computer program, such as MatLab, can also be helpful in determining the type of system involved and what solutions exist, if any.