Linear Programming-Bellwork
Algebra 2
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)
Linear Programming
Lesson 3-4
Algebra 2
Additional Examples
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).
y > – 2 x + 11
3
3
y < 1 x + 11
4
4
y > 3x – 11
Linear Programming
Lesson 3-4
Algebra 2
Additional Examples
(continued)
Step 3: Evaluate P at each vertex.
Vertex
A(1, 3)
B(5, 4)
C(4, 1)
P = –5x + 4y
P = –5(1) + 4(3) = 7
P = –5(5) + 4(4) = –9
P = –5(4) + 4(1) = –16
When x = 1 and y = 3, P has its maximum value of 7.
(1, 3)
Linear Programming
Lesson 3-4
Algebra 2
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?
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.
No. of Products
No. of Units
Profit
Tables
x
30 < x < 60
150x
Chairs
y
40 < y < 100
65y
Total
x+y
120
150x + 65y
constraint
objective
Linear Programming
Lesson 3-4
Algebra 2
Additional Examples
(continued)
Write: Write the constraints. Write the objective function.
P = 150x + 65y
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)
x > 30
x < 60
y > 40
y < 100
x + y < 120
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.
Linear Programming
Lesson 3-4
Algebra 2
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
Linear Programming
Algebra 2