The Gauss-Jordan Method
J. Gauss:
1777-1855
M. Jordan:
1838-1922

GAUSS / JORDAN (G / J) is a device to solve systems of (linear) equations.
Given a system of equations, a solution using G / J follows these steps:

[1]
 Write the given system as an augmented matrix. Examples of this step are below [system] ===> [ A | B ]
[2]
 Use row operations to change A to an Identity Matrix, I:
This step can be done in many ways. A very systematic procedure can be viewed in Prof McFarland's Finite Math Website, but for this Algebra course, you are free to tinker in your own style, perhaps modelling your work on the example below.
[ A | B ] ===> [ I | C ]
[3]
 When [2] is done, re-write the final matrix [ I | C ] as equations.
C will be a (vertical) list of variable values which solve the system, as in the example below

 Note : Professor McFarland names row operations just a bit differently from our text.

 One simple example of Gauss/Jordan appears with our list of row operations.
 Below is a system of equations which we will solve using G/J
 Below is the 1st augmented matrix: Note the location of the red-circled "1" in I3
 Row operations named below change column 1 into what we want
 Next we change "5" in the 2-2 position encircled below. This is where "1" in I3 must be

 Below is the result of "R2 = (1/5)r2". The element in the 2-2 position has become "1" as in I3
 Row operations to change the rest of column 2 are below
 Column 2 below is now what it is in I3. Now alter the "7" encircled in red
 Using the row op "R3 = (-1/ 7)r3" below to change "-7" to "1" as in I3

 Below is the result of "R3 = (-1/ 7)r3". Now we must change the rest of column 3
 Row operations to change the rest of column 3are below
 The result of all row operations is below, with the identity I3 matrix in blue
 Step [3] of G/J
 Re-writing the final matrix as equations gives the solution to the original system