Here is a summary of the simplex method for standard linear programming problems.

Step1: Using slack variables, convert the LP problem to a system of linear equations.
Step 2: Set up the initial tableau.
Step 3: Select the pivot column.

    The rule for the selecting a pivot column is this: Look at all the numbers in the bottom row, excluding the Answer column. From these, choose the negative number with the largest magnitude. Its column is the pivot column. (If there are two candidates, choose either one.) If all the numbers in the bottom row are zero or positive, then you are done, and the basic solution is the optimal solution.
Step 4: Select the pivot in the pivot column.

    1) The pivot must always be a positive number. (This rules out zeros and negative numbers.)
    2) For each positive entry b in the pivot column, compute the ratio a/b, where a is the number in the rightmost column in that row. We call this a test ratio.
    3) Of these ratios, choose the smallest one. The corresponding number b is the pivot.
Step 5: Use the pivot to clear the pivot column in the normal manner. This gives the next tableau.

    (For quick instructions on how to pivot, press here.)
Step 6: Repeat Steps 3-5 until there are no more negative numbers in the bottom row (with the possible exception of the Answer column).

    The solution for the LP problem is then the basic solution associated with the final tableau.