Algebra 2 Honors
Section 3.3 Notes Optimization and Linear Programming
Recall: Solving a system of equations
Find the intersection for the graphs
8x + 3y = –5
10x + 6y = –13
Vocabulary
Constraint: Conditions given to variables, often expressed as linear inequalities
Feasible Region: The intersection of the graphs in a system of constraints
Vertex: Any of the points of intersection of the graphs of the constraints that determine a feasible region
Example 1: Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the
maximum and minimum values of the function f(x, y) = 3x – 2y for this region.
x≤5
y≤4
x+y≥2
Example 2: Graph the following system of inequalities. What are the maximum and minimum values of the function
f(x, y) = x + 2y for the feasible region of the graph?
x + 3y ≤ 6
–x – 3y ≤ 9
2y – x ≥ –6
Example 3: A landscaping company has crews who mow lawns and prune shrubbery. The company schedules 1 hour for
mowing jobs and 3 hours for pruning jobs. Each crew is scheduled for no more than 2 pruning jobs per day. Each crew’s
schedule is set up for a maximum of 9 hours per day. On the average, the charge for mowing a lawn is $40 and the charge for
pruning shrubbery is $120. Find a combination of mowing lawns and pruning shrubs that will maximize the income the
company receives per day from one of its crews.
Example 4: Behrens Breakfast Bar Company makes fruit-filled breakfast bars and granola bars. For each case of breakfast
bars, you make $40 profit. For each case of granola bars, you make $55 profit. To make one case of breakfast bars you need 2
machine hours and 5 labor hours. To make one case of granola bars you need 6 machine hours and 4 labor hours. If there are
a maximum of 150 machine hours and a maximum of 155 labor hours, how many of each case should be produced to
maximize profits?
Example 5: A landscaping company has crews who rake leaves and mulch. The company schedules 2 hours for mulching jobs
and 4 hours for raking jobs. Each crew is scheduled for no more than 2 raking jobs per day. Each crew’s schedule is set up for
a maximum of 8 hours per day. On the average, the charge for raking a lawn is $50 and the charge for mulching is $30. What
is a combination of raking leaves and mulching that will maximize the income the company receives per day from one of its
crews?