Explanation Let the function, p = x + y . Maximize the function under subjected to 2 x + y ≤ 1 , x ≥ 0 , y ≥ 0 Consider the constraints, 2 x + y ≤ 1 x ≥ 0 , y ≥ 0 Step 1: Use slack variables and write the inequalities in equation form. Add the slack variable s . 2 x + y + s = 1 Step 2: Write the equations in an augmented matrix. The equations are; 2 x + y + s = 1 − x − y + p = 0 Write the equations in augmented matrix. x y s p s 2 1 1 0 1 p − 1 − 1 0 1 0 Step 3: Choose the smallest negative number on the left side of the bottom row and pivot the column. Since, the smallest negative number on the left side of the bottom row is − 1 . Thus, pivot the x column. Step 4: Select the pivot which is the smallest of the test ratios a b , where a is the entry in the right most column and b is the corresponding entry in the pivot column. Since, the test ratios are neither 0 nor negative. Thus, the only test ratio is 2 1 = 2 . Thus, the pivot is 2 in column x . x y s p s 2 1 1 0 1 p − 1 − 1 0 1 0 Step 5: Clear the pivot column using the pivot and label the pivot row with the label of the pivot column. Since, the pivot is s in column x and replace s with x . Thus, x is the active variable. Use row operations to clear the pivot column of the augmented matrix. Use the row operations 2 R 2 + R 1 . x y s p x 2 1 1 0 1 p 0 − 1 1 2 1 Step 6: Repeat the above steps till all the entries in the bottom row are non negative

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Chapter 5.3, Problem 52E

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Whether the statement “If there is only 1 constraint in a standard linear programming problem, then you will have to pivot at most once to obtain an optimal solution.” is correct or not.

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Chapter 5.3, Problem 51E

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Whether the statement “If there is only 1 constraint in a standard linear programming problem, then you will have to pivot at most once to obtain an optimal solution.” is correct or not.

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