Solving A System of Linear Equations

With N Equations and N Unknowns

A system of linear equations is two or more linear equations that are being solved simultaneously.  The possible results when solving system of linear equations with n equations and n unknowns are one solution, no solution or infinite solutions.   In fact, a system of linear equations involving two or three variables can be solved using techniques learned in elementary algebra.  These techniques are not suitable for system involving larger numbers of variables.  A method called the Gauss-Jordan elimination is used to solve for systems involving larger numbers of variables.  

The followings are some simple examples that are solved by the techniques of elimination using substitution and Gauss-Jordan elimination.  The result of each example is then confirmed by using the GraphFunc utility online.  (You need Java Runtime Environment to run GraphFunc applet in this website.)

Solving System of Linear Equations in Two Variables

Example 1  Solve by elimination using substitution.  Find andthat satisfy the following equations:

Solution

Choose to eliminatefrom equations by adding, and then obtain 

   or

Substituteback into either original equations, say the first equation, and solve foras follows:

Thus, the solution is

   and.

Now we use the GraphFunc to check the result.  See the instructions and solution are shown on the right.

Note that example above can be written in terms of variables x and y, namely

.  Find x and y.

·        Get into the website at  http://graph.seriesmathstudy.com (you need to wait for an applet to be loaded.)

·        Select the Linear Equations item from the Functions drop-down list box, namely .

You will see a popup window displayed in Figure 1.

·        Enter the value 2 in the text field that has label marked as Number of variable.  Then press on the Choose button to setup a mode for the system of linear equations with two linear equations and two unknowns.

·        Enter values of the coefficients of the two equations into the text fields as shown in Figure 1.  Put 0 for the coefficients that do not exist.  Then press the Solve button to get the result.

Figure 1

Solving System of Linear Equations in Three Variables

Example 2   Solve by elimination using substitution..

Solution

·        Look at the coefficients of the variables and choose to eliminatefrom equations (I) and (II) by adding.  Multiply equation (I) by 2 and add to equation (III), we obtain:

·        Multiply equation (IV) by 2 and subtract from equation (V):

  or.

·        Substitutingback to equation (IV):

  .

·        Substitutingandback to either equation (I) or (II) to find:

  .

Thus, the solution is

.

·        Get into the website at  http://graph.seriesmathstudy.com

·        Select Linear Equations from the Functions drop-down list box, namely .

·        Enter Number of variable: 3.  Then click on the Choose button to select a system of linear equation with three equations and three unknowns.

·        Enter the values of the coefficients of the three equations into the text fields as shown in Figure 2.  Then press the Solve button to get the result.

Figure 2

Solving System of Linear Equations in Four Variables

Example 3  Solve by Gauss-Jordan substitution.

Solution

Step 1: Choose leftmost zero column and get a 1 at the top; and a 0 for the rows 1, 2, and 3.

Step 2:

Step 3:

Step 4:

Step 5:

Step 6:

The matrix is in reduced form.

Thus, the solution is.

Follow the same steps as described in Example 1.  Note that the value of number of variables is 4.  The solution is illustrated in Figure 3.

Figure 3