Consider the following linear program:

Max 6x₁ + 7x₂

s.t.
	x1		6
	2x1 + 3x2		19
	x1 + x2		8
	x1, x2		0

The graphical solution to the problem is shown below. From the graph, we see that the optimal solution occurs at

x₁ = 5, x₂ = 3

and obtains a maximum value of 51.

On the coordinate plane the horizontal axis is labeled x₁ and the vertical axis is labeled x₂. There are 4 lines on the graph.

The line labeled x₁ + x₂ = 8 enters the window at x₂ = 8 on the positive x₂-axis, goes down and right, passes through the point (5, 3) crossing the line labeled 2 x₁ + 3 x₂ = 19, passes through the point (6, 2) crossing the vertical line labeled x₁ = 6, and exits the window at x₁ = 8 on the positive x₁-axis.

The vertical line labeled x₁ = 6 enters at the point (6, 0).

The line labeled 2 x₁ + 3 x₂ = 19 enters the window at approximately x₂ = 6.3 on the positive x₂-axis, goes down and right, passes through the point (5, 3) crossing the line labeled x₁ + x₂ = 8, passes through the approximate point (6, 2.3) crossing the vertical line labeled x₁ = 6, and exits the window at x₁ = 9.5 on the positive x₁-axis.

The line labeled Max 6 x₁ + 7 x₂ enters the window at x₂ = 6 on the positive x₂-axis, goes down and right, passes through the approximate point (5, 1.7), passes through the approximate point (6, 0.9) crossing the vertical line labeled x₁ = 6, and exits the window at x₁ = 7 on the positive x₁-axis.

Two arrows, pointing up and right, are drawn perpendicularly extending from the line labeled "Max 6 x₁ + 7 x₂ = 42."

The point (5, 3) is labeled "Optimal Solution (x₁ = 5, x₂ = 3) max = 51."

(a)

Calculate the range of optimality for the objective function coefficient for

x₁.

(Round your answers to two decimal places.)

to

Calculate the range of optimality for the objective function coefficient for

x₂.

(Round your answers to two decimal places.)

to

(b)

Calculate the dual value for the first constraint.

Calculate the dual value for the second constraint.

Calculate the dual value for the third constraint.