1. Linear Programming-Bellwork
Multiple Choice: What values of x and y minimize P for the objective function P = 3x + 2y?
A. (2,0) B. ( 2,6) C. (4,3) D. (4,4)

2. Find the values of x and y that maximize P if P = –5x + 4y.
Linear Programming
Lesson 3-4
Additional Examples
y – x +
y x +
y 3x – 11 2 3 1 4 11
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Find the values of x and y that maximize
P if P = –5x + 4y.
Step 1: Graph the constraints.
Step 2: Find the coordinates for each vertex.
The solution is (1, 3), so A is at (1, 3).
The solution is (5, 4), so B is at (5, 4).
The solution is (4, 1), so C is at (4, 1).

3. Step 3: Evaluate P at each vertex.
Linear Programming
Lesson 3-4
Additional Examples
(continued)
Step 3: Evaluate P at each vertex.
Vertex P = –5x + 4y
A(1, 3) P = –5(1) + 4(3) = 7
B(5, 4) P = –5(5) + 4(4) = –9
C(4, 1) P = –5(4) + 4(1) = –16
When x = 1 and y = 3, P has its maximum value of 7.
(1, 3)

4. Define: Let x = number of tables made in a day.
Linear Programming
Lesson 3-4
Additional Examples
A furniture manufacturer can make from 30 to 60 tables a day and from 40 to 100 chairs a day. It can make at most 120 units in one day. The profit on a table is $150, and the profit on a chair is $65. How many tables and chairs should they make per day to maximize profit? How much is the maximum profit?
Tables Chairs Total
No. of Products x y x + y
No. of Units x y
Profit 150x 65y 150x + 65y
Define: Let x = number of tables made in a day.
Let y = number of chairs made in a day.
Let P = total profit.
Relate:  Organize the information in a table.
constraint
objective
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5. Write: Write the constraints. Write the objective function. > <
Linear Programming
Lesson 3-4
Additional Examples
(continued)
x 30
x 60
y 40 y
x + y
>
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Write: Write the constraints. Write the objective function.
>
<
P = 150x + 65y
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Step 1: Graph the
constraints.
Step 2: Find the
coordinates of each vertex.
Vertex A(30, 90)
B(60, 60)
C(60, 40)
D(30, 40)
Step 3: Evaluate P
at each vertex.
P = 150x + 65y
P = 150(30) + 65(90) = 10,350
P = 150(60) + 65(60) = 12,900
P = 150(60) + 65(40) = 11,600
P = 150(30) + 65(40) = 7100
The furniture manufacturer can maximize their profit by making
60 tables and 60 chairs. The maximum profit is $12,900.

6. Find the values of x and y that maximize P if P = 3x + 2y. x 2 x 6 y 1
Linear Programming
Lesson 3-4
Find the values of x and y that maximize
P if P = 3x + 2y.
x 2
x 6
y 1
y 5
x + y 8
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>
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7. Linear Programming
Homework
pg 142
1, 5, 7, 10