3-4 LINEAR PROGRAMMING (p. 135-140)
Linear programming is an extension of solving linear inequalities. It got its name
because it provides a plan or program for solving problems by using linear equations and
inequalities. Basically, you are given constraints (which are the inequalities) and you
graph them.
Linear programming problems are usually looking for maximum or minimum values.
You substitute coordinates of vertices of the region representing the solution to the
constraints into a function to determine the maximum or minimum value.
Linear programming is a technique that identifies the minimum or maximum value of
some quantity. This quantity is modeled with an objective function. Limits on the
variables in the objective function are called constraints or restrictions, which are written
as linear inequalities.
When you graph the linear inequalities, your solution to this system represents the
feasible region.
Vertex Property of Linear Programming
If there is a maximum or a minimum value of the linear objective function, it
occurs at one or more vertices of the feasible region.
Procedure for Solving Linear Programming Problems
1. Determine and graph the constraints or restrictions (the inequalities). You may
want to write each inequality in slope-intercept form. Locate the solution to the
system.
2. Determine the objective function.
3. Find the coordinates of the vertices of the feasible region (the solution of the
system). To do this, use CALC intersect, CALC value, and CALC zero.
4. Evaluate the objective function with the coordinates from each vertex.
Short-cut Note: After finding the coordinates of the first vertex in Step 3, one
could go to the home screen, enter the objective function, and hit enter for the
value of the objective function at that vertex. You can find subsequent values of
the objective function after finding a new vertex without entering the objective
function again. Just hit enter at the home screen after finding the coordinates of a
new vertex.
Example: Find the values of x and y that maximize and minimize P if P = -5x + 4y
2
11

y  - 3 x  3

1
11

y  x 
4
4

 y  3x  11


At (1,3) P =7
At (4,1) P = -16
Examine and discuss Ex. 2 on p. 137. Show how the profit is determined. Put each
inequality in slope-intercept form so that you can use your TI.
500 - 12x

y 
20

600
- 24x
y 

15

x

0

y  0

Do 2 on p. 137.
P = 27x + 15y
Homework p. 138-140: 1,3,5,9,11,13,17,24,25,34,41
11. Let x = no. of spruce
Let y = no. of maple
x  0
y  0


30x  40y  2100
600x  900y  45,000
Use y 
2100  30 x
45,000  600 x
and y 
.
40
900
C = 650x + 300y
Plant 70 spruce and 0 maple.
13. Let x = no. of Type 1 bacteria
Let y = no. of Type 2 bacteria
4x  3y  240

30  x  60
0  y  70

C = 5x + 7y
Coordinates of vertices: (30,70), (30,40), (60,70), (60,0)
60 samples of Type 1 and 0 samples of Type 2
Note: The Linear Programming Technology activity on p. 141 has been incorporated
within 3.4. Students could be assigned a problem on p. 141 for a partner grade if extra
practice is needed.