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Linear Programming
Lesson summary:
In this activity, students will use inequalities to determine the feasible region and
conclude the optimal solution given certain constraints of a linear programming problem.
Key Words:
Inequality, Feasible Region, Optimal Solution, Constraint
Background Knowledge:
This activity will have the students exploring inequalities to determine the optimal
solution to maximize a profit for a company. Students should have a general understanding of
how to work with inequalities. They should also have a basic knowledge of how to use the TINspire CAS handheld components such as Calculator, Function Tables, and Graphs &
Geometry.
Materials:
TI-Nspire CAS handheld
Worksheet
Computer
Suggested Procedure:
Students will be put into groups of two or three. The students will then open the file on
their computer that will guide them through the activities. There are questions that will provide a
place to record their observations. Remind the students that they are looking to maximize profit.
Discuss and review how to use different applications on the TI-Nspire CAS handheld, such as
Calculator, Function Tables, and Graphs & Geometry. Instruct the groups to open the activity
document, and have the teams complete the activity.
Standards: Patterns, Functions and Algebra Standard
7. Use symbolic algebra (equations and inequalities), graphs and tables to represent
situations and solve problems
9. Solve linear equations and inequalities graphically, symbolically and using technology
Benchmark/ Grade Level Indicator:
F. Solve and graph linear equations and inequalities.
Assessment: Check students progress in class on the TI-Nspire CAS handheld. Have the
students submit their completed activities.
***Note: The TI-Nspire CAS calculator at this time is a new piece of technology for many students. In the future students will be
more knowledgeable with the TI-Nspire. Throughout the lesson, comment boxes explaining how to complete procedures are
included. In the future, once students are comfortable with the TI-Nspire, the comment boxes may be removed.
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Activity 1: Inequalities. In this activity, you will learn to graph inequalities.
Goal: The students will discover how to graph inequalities, and how to determine the algebraic
representation of a half-plane
Students’ Names: _____________________________________
1. Start by creating a new document. Choose Graphs & Geometry.
2. Graph the inequality: y < 3x
In f1(x), delete the = sign. Then, type in the portion “< 3x”. Then hit enter.
3. What do you notice about the displayed graph, i.e., how is the boundary? How is the
region to the right?
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It should look like the following:
4. Let’s try to characterize the points on the shaded region called a half-plane. What does
the shaded region represent?
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5. Let’s check to see if you are correct. Create a point P that is in the shaded region and
display its coordinates. Do the coordinates of P satisfy the given inequality?
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6. Grab and move point P on the shaded region. Do the coordinates of P keep satisfying the
inequality at every position?
________________________________________________________________________
7. Now, pick a point Q that is located on the line and display its coordinates. Do the
coordinates of Q satisfy the given inequality? What equation do the coordinates seem to
satisfy?
________________________________________________________________________
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8. Grab and move point Q on the line. Do the coordinates of Q keep satisfying the equation
at every position?
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9. Now, pick a point R that is located on the non-shaded region (also called a half-plane)
and display its coordinates. Do the coordinates of R satisfy the given inequality? What
inequality do the coordinates seem to satisfy?
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10. Grab and move point R on the non-shaded region. Do the coordinates of R keep satisfying
the inequality at every position?
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11. In general we know that the graph of a linear equation in two variables is a line. The line
divides the plane in three regions: two half-planes and the line itself. What algebraic
expressions have you discovered represent each half-plane?
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12. The shaded region representing the solution to a linear inequality is known as the feasible
region. In your own words, define feasible region.
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Next, graph the following equation in a new page. y ≤ -2x + 10
1. Start by creating a new page.
2. Graph the inequality y ≤ -2x + 10
In f1(x), delete the “=” sign. To get the “≤” sign, hit ctrl, < . Fill in the rest of the
equation and hit enter.
3. What do you notice about this graph?
________________________________________________________________________
________________________________________________________________________
It should look like the following:
4. What is the feasible region?
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________________________________________________________________________
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5. Let’s check to see if you are correct. Create a point P that is not in the shaded region and
display its coordinates. Do the coordinates satisfy the given inequality?
________________________________________________________________________
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6. Now, pick a point Q that is located on the line and a point R that is located in the shaded
region and display their coordinates. Do the coordinates of both points satisfy the given
inequality?
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7. Comparing the resulting graphs on the two inequalities studied, how does the inclusion of
the equal sign impacts the graph of the inequality?
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Extension:
1. Create a new Graphs & Geometry page.
2. Graph the inequalities: y < 3x and y ≤ -2x + 10
3. What is different about the shading regions of this graph? What is the feasible region of
this set of inequalities? How can you confirm your answer?
________________________________________________________________________
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It should look like the following:
4. Consider the points: (-7, 2), (2, -5), (2, 6), and (12, 5). Based on the graph alone,
determine if the points satisfy the set of inequalities. Explain your reasoning.
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________________________________________________________________________
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5. Check to see if your predictions were correct using the coordinates of each point and the
two inequalities given. In the following space, show all work.
6. If your prediction(s) were incorrect, what is the reason(s) for them being incorrect?
________________________________________________________________________
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7. Describe what you have learned about inequalities in your own words.
________________________________________________________________________
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Activity 2: Determine the optimal solution.
Goal: Given a set of inequalities (called constrains) and a function (called the function),
the students will determine the maximum and minimum (if they exist) values of the function on
the feasible region.
Definition: A constraint for a linear programming problem is an inequality that takes
part in defining the feasible region.
Definition.
1. Create a new Graphs & Geometry page.
2. Graph the following set of inequalities:
x≥0
y≥0
y ≤ -2x + 10
This is how your window should look: (Note: We cannot graph x ≥ 0 (why?), so we
know the feasible region will be to the right of the y-axis)
What polygon do the feasible region form in this case?
3. Look at the feasible region. Suppose we pick any two points on the perimeter of the
region, if we connect them with a line, will the line be totally contained in the feasible
region? If not, give an example of two such points.
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Definition: A region is said to be convex if any two points of the region can be
joined by a line segment totally contained in the region.
4. Is the feasible region convex? ____________________________________________
Using these constraints, we want to maximize the function f(x,y) = 2x + 3y.
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5. Determine the coordinates of the vertices of the feasible region.
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To determine the coordinates, you must create points. Press menu, 6: Points &
Lines, 3: Intersection Point(s). Then, click the x-axis and y-axis. After you create
this point, choose the y-axis and the inequality y ≤ -2x + 10. Finally, choose the
inequality and the x-axis. Next, you may need to get the labels for these points. Press
menu, 1: Actions, 6: Coordinates and Equations. Then, click on the points.
Your graph should look like the following:
6. Next, we want to find the optimal solution for the equation f(x,y) = 2x + 3y.
Definition: An optimal solution is a solution to an optimization problem that
maximizes (or minimizes) the corresponding function.
Evaluate the equation using the points we found in step 5. Show all of your work in
the space provided below.
7. Next, we must check some of the interior points of the feasible region. Choose five
interior points and evaluate the equation in the space provided below.
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What do you notice about the values of these points in relation to the values of the
coordinates that make up the feasible region?
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8. Generalize your findings from step 7.
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9. It turns out that your generalization from step 8 is an actual theorem. It states:
The point (x0, y0) of a convex region that produces the maximum value of a quantity
k = ax + by, where a and b are positive, will be at a vertex of the region.
10. Based on this theorem, what point maximizes the equation k = 2x + 3y under the
specified constraints? What is the value of the maximum?
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Extension:
1. Suppose we have the constraints:
x≥0
y≥0
y < -2x + 10
2. Graph these inequalities in a new page. It should look like the following:
Using these constraints, we want to maximize the function f(x,y) = 2x + 3y.
3. Is this region convex? Explain your reasoning.
_____________________________________________________________________
_____________________________________________________________________
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4. Are you able to determine the maximum of f(x,y) = 2x + 3y under these constraints?
If yes, give the coordinates of the point and the maximum value. If no, explain the
reason(s) why.
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Activity 3: A Linear Programming Example
Goal: The students will be able to use their recent findings to determine the optimal solution
for the following linear programming example.
Problem: A gold processor has two sources of gold ore, source A and source B. In order to
keep his plant running, at least three tons of ore must be processed each day. Ore from source A
costs $20 per ton to process, and ore from source B costs $10 per ton to process. Costs must be
kept to less than $80 per day. Moreover, Federal Regulations require that the amount of ore from
source B cannot exceed twice the amount of ore from source A. If ore from source A yields 2 oz.
of gold per ton, and ore from source B yields 3 oz. of gold per ton, how many tons of ore from
both sources must be processed each day to maximize the amount of gold extracted subject to the
above constraints?
1. Start by creating a new Graphs & Geometry page.
2. Next, we must define our variables in the problem above.
Let x = ____________________________________________________________
Let y = ____________________________________________________________
3. Determine the objective of this linear programming problem and give its equation.
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4. Determine the equations of the constraints:
Processing: ______________________________________________________________
Costs: __________________________________________________________________
Federal Regulations: ______________________________________________________
5. We also have the implied constraints of x ≥ 0 and y ≥ 0. What is the reason for this?
________________________________________________________________________
________________________________________________________________________
6. Graph the constraints. Your graph should look like the following:
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7. Give the coordinates of the feasible region.
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8. Look at the feasible region. Does it appear to be convex? Justify your answer.
________________________________________________________________________
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9. Based on your answer is 8, is there a theorem that may be helpful in solving this linear
programming problem? State the theorem.
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10. Next, evaluate the equation for the amount of gold yielded using the coordinates of the
feasible region. Show all of your work in the space provided.
11. Based on your results, determine the optimal solution. (Give the point and the maximum
amount of gold yielded) Interpret your results.
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Extension:
Using the same format for this activity, complete the following optimization problem:
A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres. However, he has
only $1200 to spend and each acre of wheat costs $200 to plant and each acre of rye costs $100
to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to
plant an acre of wheat and 2 hours to plant an acre of rye. If the profit is $500 per acre of wheat
and $300 per acre of rye how many acres of each should be planted to maximize profits?
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