
The simplex method changes constraints (inequalities) to equations in linear programming problems, and then solves the problem by matrix manipulation. The solution set for the altered problem is of higher dimension than the solution set of the original problem, but it is easier to study with matrices. 
Reference : What is a STANDARD MAXIMIZING PROBLEM?
Reference : An example of SIMPLEX METHOD for a standard problem is available.



Write the revised problem as a tableau, with the objective row (= bottom row) consisting of negatives of the coefficients of the objective function z ; z will be maximized. The lower right corner is the value of z when x, y,... are zero ; thus, z usually starts out as zero. See example?  



Simplex method will move the ISM, one column at a time; after each such move, we arrive at (or "hop" to) a new corner point (basic solution) with bigger objective value. Since the solution set has only finitely many corners, this process ultimately yields the biggest value of the objective function. 
An INDICATOR (for standard maximizing problems) is a number in the bottom (objective) row of a tableau, excluding the rightmost number. Thus here, the INDICATOR ROW is the bottom row, but for nonstandard problems the indicator row will be a different row. 
FIRST FIND THE PIVOT: (Rolf, pg. 282)

The pivot column is that column containing the most negative indicator. If no indicator is negative, the tableau is a FINAL TABLEAU : see step 8. 

Form RATIOS (quotients) for each row: divide the rightmost number by the number in the pivot column of that row. 



Apply a pivot operation (Rolf pg 92) to the tableau, including the bottom (objective) row. The pivot column will become a column of a new ISM in the new tableau. Note which column is replaced, and where the new ISM is located ; its columns may not be in the usual order: not to worry. Check out the PIVOT ENGINE to speed up practice. 



If all indicators (in the bottom row) are nonnegative, : your tableau is a FINAL TABLEAU. The basic solution of step 7 is the maximal solution you have been seeking! Note that if you correctly reached this step from a NonStandard Problem, then all rightside numbers above the objective row will also be nonnegative. 

Otherwise, if some indicator remains negative, repeat steps 3 through 9 WITH YOUR NEW TABLEAU. After a finite number of such repetitions (usually 23), simplex method must terminate at step 8. 