2 x + 2 y = 8and subtract it from the second equation, like this: The result is one equation in the one unknown, y. The other unknown, x, has been eliminated. Solving this equation yields y = 0.4.
{ x = 3.6, y = 0.4 }.(Note that we could have instead found x without backsubstitution if we had subtracted 3 times the first equation from the second equation, since this eliminates y.)
The Elementary Row Operations (E.R.O.’s) are:

Algorithm for Gaussian Elimination We transform one column at a time into row echelon (or Gauss) form. The column presently being transformed is called the pivot column. We proceed systematically, letting the pivot column be the first column, then the second column, etc. until the last column before the vertical line of the augmented matrix. For each pivot column, we do the following two steps before moving on to the next pivot column:

{ x = 7, y = 5, z = 3 }.
Algorithm for GaussJordan Elimination We transform one column at a time into reduced row echelon (or GaussJordan) form. The column presently being transformed is called the pivot column. We proceed systematically, letting the pivot column be the first column, then the second column, etc. until the last column before the vertical line of the augmented matrix. For each pivot column, we do the following two steps before moving on to the next pivot column:

{x = 49, y = −18, z = 8}.
In general, an augmented matrix which has been put into row echelon form and which contains one or more rows of zeros at the bottom of the matrix indicates a redundant system of equations.
In general, an augmented matrix which has been put into row echelon form and which contains one or more bottom rows consisting of all zeros to the left of the vertical line and a nonzero number to the right indicates an inconsistent system of equations with no solution.
Algebra Coach Exercises 