Systems of Linear Equations: Introduction

What is a system of equations? It is just a (long or short) list of equations in one or more unknowns (also called variables). Why should we be interested in such things? It turns out that many situations in life can be described by systems of equations of various sorts. Often we use these to find conditions that satisfy all the equations. Systems of equations arise in a great variety of areas in life; here is one example:

Example One of the primary functions of air traffic control is to make sure that airplanes don't crash into each other in the air. How do they do this? The path of each airplane is tracked and described by an algebraic equation. Then the equations are compared to see if there are any points at which they intersect. That is, one tries to find a solution for the system of equations that describe the routes of  a set of airplanes - if there is one, that means that the airplanes are on a collision course! The equations that arise may be linear (if a plane is flying in a straight line) or of other types such as quadratic (if a plane is circling the airport, for example).

This example and other situations give rise to possibly very complicated systems of equations, but here we will deal with a particular sort: systems of linear equations. The reason for this name is that this type of equations describes straight lines - in 2-space, 3-space, or higher dimensional space.

What do we mean by a solution to a system of equations? We mean a set of values for the variables so that if we substitute them into the equations, they are all "satisfied". That is, the results are equalities.

Example Consider the system

2x + y = 10 
x - y = 5
 The values x = 5, y = 0 yield a solution for the system, since 2(5 )+ 0 = 10 and 5 - 0 = 5.
The solution to the system is the PAIR of values (x, y) = (5, 0).

Not every system of equations has a solution.

Example Consider the system

2x + y = 10 
2x + y = 20
Clearly this system has no solutions, since whatever values we pick for x and y can satisfy at most one of these equations. In a case like this, we say that the system is inconsistent; go to Inconsistent Systems  to see what this means and what to do in this case.

A third possibility is that a given system has infinitely many solutions! When will this happen? In general, if the system has more unknowns than equations, and if there is a solution, then there will be infinitely many solutions. Alternatively, if it can be transformed into such a system, then it also has infinitely many solutions. We will discuss this transformation in the section on  solutions of systems of linear equations . An example of such a system is:

Example

2x - y + z = 1
For any values one picks for y and z, there is a corresponding value for x which satisfies this equation. Of course there are infinitely many values one could choose for y and z, and so infinitely many solutions to the system.

The algebraic concept of systems of equations and their solutions also has a geometric interpretation. Consider the systems we have just seen; each of their equations describes a line in the (x, y)-plane. If there is a solution, it is the point at which these lines intercept each other; the system is inconsistent when the lines are parallel to each other.

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Now is a great time to get an overview of the concpets of systems of equations and their solutions (or lack thereof); to do this, go through the slide show below:

To learn how to solve systems of equations and to practice solving, go to  Solutions of Sytems of Linear Equations .