Concept Byte: Linear Programming

For Use With Lesson 3-4

TECHNOLOGY

You can solve linear programming problems using your graphing calculator.

Activity

Find the values of x and y that will maximize the objective function P = 13x + 2y for the constraints below. What is the value of P at this maximum point?

{ − 3 x + 2 y ≤ 8 − 8 x + y ≥ − 48 x ≥ 0 , y ≥ 0 left brace . table with 3 rows and 1 column , row1 column 1 , negative 3 x plus 2 y less than or equal to 8 , row2 column 1 , negative 8 x plus y greater than or equal to negative 48 , row3 column 1 , x greater than or equal to 0 comma y greater than or equal to 0 , end table

• Step 1 Rewrite the first two inequalities to isolate y. Enter the inequalities.

  

• Step 2 Use the VALUE option of CALC to find the upper left vertex. Press 0 enter . begin box , enter , end box , .

  
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• Step 3 Enter the objective function on the home screen. Press enter begin box , enter , end box  for the value of P at the vertex.

  

• Step 4 Use the INTERSECT option of CALC to find the upper right vertex. Go to the home screen and press enter begin box , enter , end box  for the value of P.

  
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• Step 5 Use the ZERO option of CALC to find the lower right vertex. Go to the home screen and press enter begin box , enter , end box  for the value of P. The objective function has a value of 0 when the vertex is at the origin. The maximum value of P is 136.

  

Exercises

Find the values of x and y that maximize or minimize the objective function.

1. { 4 x + 3 y ≥ 30 x + 3 y ≥ 21 x ≥ 0 , y ≥ 0 left brace . table with 3 rows and 1 column , row1 column 1 , 4 x plus 3 y greater than or equal to 30 , row2 column 1 , x plus 3 y greater than or equal to 21 , row3 column 1 , x greater than or equal to 0 comma y greater than or equal to 0 , end table

   Minimum for C = 5x + 8y

2. { 3 x + 5 y ≥ 35 2 x + y ≤ 14 x ≥ 0 , y ≥ 0 left brace . table with 3 rows and 1 column , row1 column 1 , 3 x plus 5 y greater than or equal to 35 , row2 column 1 , 2 x plus y less than or equal to 14 , row3 column 1 , x greater than or equal to 0 comma y greater than or equal to 0 , end table

   Maximum for P = 3x + 2y

3. { x + y ≥ 8 x + 5 y ≥ 20 x ≥ 0 , y ≥ 2 left brace . table with 3 rows and 1 column , row1 column 1 , x plus y greater than or equal to 8 , row2 column 1 , x plus 5 y greater than or equal to 20 , row3 column 1 , x greater than or equal to 0 comma y greater than or equal to 2 , end table

   Minimum for C = 3x + 4y

4. { x + 2 y ≤ 24 3 x + 2 y ≤ 34 3 x + y ≤ 29 x ≥ 0 left brace . table with 4 rows and 1 column , row1 column 1 , x plus 2 y less than or equal to 24 , row2 column 1 , 3 x plus 2 y less than or equal to 34 , row3 column 1 , 3 x plus y less than or equal to 29 , row4 column 1 , x greater than or equal to 0 , end table

   Maximum for P = 2x + 3y

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