Friday, March 2, 2012

Algebra: How to Solve Linear Systems

In our Algebra course, we are currently learning how to solve systems of two linear equations with two variables.  This topic is covered in Chapter 4 of our open Elementary Algebra textbook published by Flat World Knowledge. To find the simultaneous solution of such a system, or point of intersection, we can use one of three methods.

1. The Graphing Method:  Rewrite the two equations in point-slope form ( y = mx + b), graph them on the same set of coordinates and then determine the point of intersection. A video example follows:

2. The Substitution Method:  This completely algebraic method, which requires that we first isolate one of the variables. We then substitute the result into the other equation.  Once we solve for one of the variables, back substitute to find the value of the other variable.  Remember that the answer is an ordered pair (x, y).

3. The Elimination Method:  Usually this is the method of choice.  The idea is to multiply one or both of the equations by appropriate numbers so that one of the variables will eliminate if the equations are added together.  This is sometimes called the "addition method."  Always back substitute to find the value for the other variable and present the solution as an ordered pair.

Each method has its strengths and weaknesses, but whichever method you choose the answer will be the same.  Most of the time, a system will produce one solution, a single point of intersection.  However, sometimes the equations are actually equivalent and in that case there are infinitely many simultaneous solutions.  This describes a dependent system and the algebraic methods will lead to a true statement like 5 = 5 or 0 = 0. Click here for a video example:

It is also important to note that there is not always an answer to some linear systems.  Sometimes the lines are parallel and do not intersect. This describes an inconsistent system and the algebraic methods will lead to a false statement like 0 = 4.  Click here for a video example.

These take time and practice to master.  Try some yourself and you will soon see that solving systems of two linear equations is actually fun.  Also, be sure to look at the many applications found in section 4.4. Hope this helps.

UPDATE:  More Solving Linear Systems videos.  Visually searchable, click on a problem and view it worked out on youTube.