In this section we have to take a look at the third method for solving out systems of equations.  For systems of two equations it is possibly a little more complex than the methods we looked at in the previous section.  However, for systems with more equations it is possibly easier than using the method we saw in the previous section.

Before we discussed the method first we need to get some definitions out of the way.

An augmented matrix for system of equations is matrix of numbers wherein each row represents the constants from one equation (both the coefficients & the constant on the other side of the equal sign) and each of the columns represents all the coefficients for a single variable.

Let's consider a look at an example.  Here is the system of equations which we looked at in the previous section.

x - 2 y + 3z = 7

2x + y + z = 4

-3x + 2 y - 2 z = -10

Following is the augmented matrix for this system.

The first row contain all the constants from the first equation  along with the coefficient of the x in the first column, the coefficient of the y into second column, the coefficient of z into the third column and the constant in the final column.  The second row is constants from the second equation along with the similar placement and similarly for the third row. The dashed line show where the equal sign was placed in the original system of equations and is not always involved. This is mostly based on the instructor and/or textbook being used.