To find a pivot element at anytime when working the simplex method, first, locate the pivot column by selecting the "most negative" element in the bottom (objective) row. This element is the  in Column 3:

Divide each of the elements in the rightmost column by each of the elements in same row and the pivot column. Select the row whose result has the "least positive" quotient:

The pivot row is Row 1. 

The correct choice is that the pivot element is the 5 in Row 1, Column 3.

To find a pivot element at anytime when working the simplex method, first, locate the pivot column by selecting the "most negative" element in the bottom (objective) row. This element is the  in Column 3:

Divide each of the elements in the rightmost column by each of the elements in same row and the pivot column. Select the row whose result has the "least positive" quotient:

The pivot row is Row 1. 

Perform the pivot by first, getting a 1 in that pivot position. Do this by performing the row operation

The pivot will be completed by getting 0's in the other entries in the pivot column by performing the necessary row operations. However, we are only concerned with the upper left entry, so we do not need to complete the pivot to know that the correct response is .

To find a pivot element at anytime when working the simplex method, first, locate the pivot column by selecting the "most negative" element in the bottom (objective) row. This element is the  in Column 3:

Divide each of the elements in the rightmost column by each of the elements in same row and the pivot column. Select the row whose result has the "least positive" quotient:

Perform the pivot by first, getting a 1 in that pivot position. Do this by performing the row operation

The pivot will be completed by getting 0's in the other entries in the pivot column by performing the necessary row operations:

The correct choice is 240.

The student's mistake was that the entries in the right column should be positive, not negative. First, each constraint, other than the nonnegativity statements, should be rewritten as an equality by the introduction of a slack variable:

Each row should include the coefficients of one of these statements, with the number on the right being the actual greatest possible value - not its negative opposite. All other characteristics of the tableau are correct, so the tableau should be (with altered entries in red):

To find a pivot element at anytime when working the simplex method, first, locate the pivot column by selecting the "most negative" element in the bottom (objective) row. At this point, the bottom row only has one negative element, which is in Column 1:

Column 1 is the pivot column.

In each of the non-objective rows, divide the rightmost element by the element in the pivot column. Select the row whose quotient is the "least positive":

The pivot row is Row 2.

The correct choice is that the pivot element is the  in Row 2, Column 1.

The tableau is correct. 

Each row in a matrix operation represents an equation, not an inequality, so each constraint, other than the nonnegativity statements, should be rewritten as an equality by the introduction of a slack variable:

There should be seven columns - one for each variable, slack variables included, and one with the constraint limits.

Also, there should be four rows - one for each constraint, with the coefficients written in the appropriate columns, and one for the objective function (the expression to be maximized), with the additive inverses of the coefficients written in the appropriate columns. The initial tableau should indeed be 

The student's mistake was that three columns are missing. Each row in a matrix operation represents an equation, not an inequality, so each constraint, other than the nonnegativity statements, should be rewritten as an equality by the introduction of a slack variable:

There should be seven columns - one for each variable, slack variables included, and one with the constraint limits. All other characteristics of the tableau are correct, so the tableau should be (with added entries in red):

.

One condition for a matrix to be in reduced row-echelon form is for each leading nonzero entry to be at the right of the entry in the row above it. The leading 1 in the second row is at left of the leading 1 in the first row, as seen below:

This is a violation of that condition. is therefore not a matrix in reduced row-echelon form.

A matrix is in reduced row-echelon form if it meets four conditions.

1) Any rows comprising all zeroes must be below all nonzero rows.

This condition is met, since the only zero row is the bottom row:

2) All leading nonzero entries must be 1's.

3) All other entries in the same column as a leading 1 are zeroes:

These conditions are met, as seen below:

4) Each leading 1 is located to the right of the one above it.

This is vacuously true, since there is only one leading 1.

All four conditions are met; is a matrix in reduced row-echelon form.

One condition for a matrix to be in reduced row-echelon form is for the matrix to have no all-zero rows above any nonzero rows. Row 1 has all zeroes, and Row 2 has a nonzero entry:

This is a violation of that condition. is therefore not a matrix in reduced row-echelon form.