Types of Linear Systems

Date: 10/13/97 at 16:30:20
From: John 
Subject: Types of linear systems

In my math class Al/Geo/Sat 2, I was told to find out what is meant 
when a type of linear system is said to be a consistent or an
inconsistent system.

I have looked up this question in a few math books and have asked 3 
or 4 teachers without any success. Can you help?

Thanks a lot, 
  John

Date: 10/13/97 at 17:32:42
From: Doctor Scott
Subject: Re: Types of linear systems

Hi John!

Since you are talking about linear systems, it might be helpful to 
think about the possibilities when you graph the equations that make 
up the system. 

If the system consists of 2 equations, there are 3 possibilities 
when you graph the equations:

(1) The lines may cross at ONE point. This point is usually called 
    the solution of the system.

(2) The lines may be coincident (lie on on another). There are  
    infinitely many solutions to the system. In fact, any point that 
    lies on the line is a solution.

(3) The lines may be parallel. There are NO solutions to the system.

When the system leads to (3), it is usually called INCONSISTENT.  
There is no solution that is common to the two equations.  Note that 
we can tell if a system is inconsistent without actually graphing it 
by looking at the slopes of the lines and their y-intercepts. If the 
slopes are the same but the y-intercepts are different, the lines are 
parallel and not the same line.

When (2) occurs, the system is said to be DEPENDENT.

I would guess that (1) would be called CONSISTENT.

-Doctor Scott,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/