30S Applied Math Mr. Knight – Killarney School Slide 1 Unit: Linear Programming Lesson 4: The Feasible Region The Feasible Region Learning Outcome B-1 LP-L4 Objectives: To identify the Feasible Region and approximate vertices using Graphmatica.

30S Applied Math Mr. Knight – Killarney School Slide 2 Unit: Linear Programming Lesson 4: The Feasible Region The solution set of a system of inequalities is called the feasible region. Only the feasible region contains all the points that satisfy all the inequalities in the system. This region is a geometric figure created by lines. The points where the lines meet are called the corner points or vertices of the feasible region. The inequalities represent the constraints on the situation. Theory – The Feasible Region

30S Applied Math Mr. Knight – Killarney School Slide 3 Unit: Linear Programming Lesson 4: The Feasible Region Find the feasible region and identify corner points for the system of linear inequalities: x  2y + 4 x + y  -2 4x + y  4 Solution Use Graphmatica to graph each inequality on the same coordinate plane. Change the Grid Range to the values shown. Theory – The Feasible Region and Corner Points

30S Applied Math Mr. Knight – Killarney School Slide 4 Unit: Linear Programming Lesson 4: The Feasible Region The feasible region is the small triangular area (that is highlighted in green) where all three inequalities overlap. If you examine the graph closely, you can get approximate values for the corner points of the feasible region. The values appear to be (0, -2), (2, -4), and (1.3, -1.3). Theory – The Feasible Region and Corner Points

30S Applied Math Mr. Knight – Killarney School Slide 5 Unit: Linear Programming Lesson 4: The Feasible Region Graphmatica allows you to get more accurate values using the Coordinate Cursor. The Coordinate Cursor is a tool that allows you to scroll along a line and read the coordinates value along the line. It is the fourth item from the right end of the button bar. Change the Grid Range to the values as shown Theory – The Feasible Region and Corner Points

30S Applied Math Mr. Knight – Killarney School Slide 6 Unit: Linear Programming Lesson 4: The Feasible Region Click on the Coordinate Cursor. Move the cursor to the top-most vertex point. The coordinates of the vertex are shown at the bottom left corner of the screen. The values are approximately (1.33, -1.33). Move the cursor to each of the other vertex points to find their coordinates more accurately. The values are (0, -2), (2, -4), and (1.33, -1.33). Turn off the Coordinate Cursor by pressing the Esc key on your keyboard. You can continue to "magnify" the feasible region so the coordinate values can be obtained to a greater accuracy. Theory – The Feasible Region and Corner Points

30S Applied Math Mr. Knight – Killarney School Slide 7 Unit: Linear Programming Lesson 4: The Feasible Region Identify the feasible region for the following system: y  3x-5 y  -3x-8 y  0.5x+5 Practice – Example 1

30S Applied Math Mr. Knight – Killarney School Slide 8 Unit: Linear Programming Lesson 4: The Feasible Region Solution The feasible region is the triple-shaded area that is shown highlighted in green. Practice – Example 1

30S Applied Math Mr. Knight – Killarney School Slide 9 Unit: Linear Programming Lesson 4: The Feasible Region Identify the feasible region for the following system and give the coordinates of the corner points. y  x x + y  6 y  1 2x + 3y  18 Practice – Example 2

30S Applied Math Mr. Knight – Killarney School Slide 10 Unit: Linear Programming Lesson 4: The Feasible Region Solution Graph the inequalities on the same graph to get the feasible region. The feasible region is highlighted in green. Using the coordinate cursor, and zooming in on each corner as much as is needed, we find the corner points are (0,6), (3, 3), and (3.6, 3.6). Practice – Example 2