Solving A System of Linear Equations

With N Equations and N Unknowns

 

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 following simple examples illustrate 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.

Use GraphFunc online to confirm the result as shown in the following steps:

 

         Begin by going to http://graph.seriesmathstudy.com (you need to wait for the GraphFunc applet to be loaded.)

 

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

 

You will see a popup window is 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

.

Use GraphFunc online to confirm the result as shown in the following steps:

 

         Begin by going to 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.

 

 

Use the GraphFunc online utility to confirm the result

 

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

 

Notice that put a 0 for coefficients that do not exist.

 

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