DO NOW
Determine if the given ordered pair is a
solution of
x+y≥6
x – 2y >10
no
2.
(10, 1)
no
3. (12, 0) yes
4.
(15, 2)
yes
1.
(3, 3)
3.4: Linear
Programming
Linear programming is method of
finding a maximum or minimum value
of a function that satisfies a given set
of conditions called constraints. A
constraint is one of the inequalities in
a linear programming problem. The
solution to the set of constraints can be
graphed as a feasible region.
Example 1:
Maximize the objective function P = 25x + 30y
under the following constraints.
x≥0
y ≥ 1.5
2.5x + 5y ≤ 20
3x + 2y ≤ 12
Example 1:
Step 1 Write the objective function: P= 25x + 30y
Step 2 Use the constraints to graph.
x≥0
y ≥ 1.5
2.5x + 5y ≤ 20
3x + 2y ≤ 12
Vertices:
(0,4)
(0,1.5)
(3,1.5)
(2,3)
Example 1:
Step 3 Evaluate the objective function at the
vertices of the feasible region.
(x, y)
(0, 4)
25x + 30y
25(0) + 30(4)
P($)
120
(0, 1.5)
25(0) + 30(1.5)
45
(2, 3)
(3, 1.5)
25(2) + 30(3)
25(3) + 30(1.5)
140
120
P = 140
The
maximum
value
occurs at
the vertex
(2, 3).
Example 2:
Yum’s Bakery bakes two breads, A and
B. One batch of A uses 5 pounds of
oats and 3 pounds of flour. One batch
of B uses 2 pounds of oats and 3
pounds of flour. The company has 180
pounds of oats and 135 pounds of flour
available. Write the constraints for the
problem and graph the feasible region.
Example 2:
Let x = the number of bread A, and
y = the number of bread B.
Write the constraints:
x≥0
y≥0
The number of batches cannot be negative.
5x + 2y ≤ 180
The combined amount of oats is less than
or equal to 180 pounds.
3x + 3y ≤ 135
The combined amount of flour is less than
or equal to 135 pounds.
Example 2:
Graph the feasible region. The feasible region is a
quadrilateral with vertices at (0, 0), (36, 0), (30, 15),
and (0, 45).
Check A point in the feasible region, such as
(10, 10), satisfies all of the constraints. 
In most linear programming problems, you want to
do more than identify the feasible region. Often you
want to find the best combination of values in order
to minimize or maximize a certain function. This
function is the objective function.
More advanced mathematics can prove that the
maximum or minimum value of the objective
function will always occur at a vertex of the feasible
region.
Example 2: Objective Function
Step 1 Let P = the profit from the bread.
Write the objective function: P = 40x + 30y
Step 2 Recall the constraints and the graph
from Example 1.
Vertices:
x≥0
y≥0
5x + 2y ≤ 180
3x + 3y ≤ 135
(0,0)
(36,0)
(30,15)
(0,45)
Example 2: Results
Step 3 Evaluate the objective function at the vertices
of the feasible region.
(x, y)
(0, 0)
40x + 30y
40(0) + 30(0)
P($)
0
(0, 45)
40(0) + 30(45)
1350
(30, 15)
(36, 0)
40(30) + 30(15)
40(36) + 30(0)
1650
1440
The
maximum
value
occurs at
the vertex
(30, 15).
Yum’s Bakery should make 30 batches of bread A
and 15 batches of bread B to maximize the amount
of profit.
The maximum profit is $1650