CHAPTER
4
Linear Programming with Two Variables
In this chapter, we will study systems of linear inequalities. They are similar to linear systems of equations, but
have inequalitites instead of equalities. We will optimize
(maximize or minimize) a linear function under certain conditions, given in the form of linear inequalities. Such problems are called linear programming problems and the
corresponding mathematical formulation is called a linear
program. In chapter 4, we solve linear programming problems in two variables by graphing. In chapter 5, we will
solve linear programming problems with two or more variables using a matrix method.
First, we will model linear programming problems, but
not solve them.
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HELENE PAYNE, FINITE MATHEMATICS
Exercise 162. An appliance dealer, Andrea Kurtz, wants
to purchase a combined total of no more than 100 refrigerators, and dishwashers for inventory. Refrigerators weigh
200 pound each, and dishwashers weigh 100 pounds each.
Suppose that the dealer is limited to a total of 12, 000
pounds for these two items. If a profit of $35 for each
refrigerator and $20 on each dishwasher is projected, how
many of each should be purchased and sold to make the
largest profit?
(a) Organize the data given into a table.
(b) From the table, formulate the linear programming
problem by writing the objective function and constraints.
CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 177
The inequalities in the above problem,
x + y ≤ 100
200x + 100y ≤ 12000
are called structural constraints. There will always be
some structural constraints in linear programming problems.
The other two inequalities,
x ≥ 0, y ≥ 0
are called non-negativity constraints and they are always added to the structural constraints in linear programming problems.
The profit function,
P = 35x + 20y,
is called the objective function. That is the function we
want to optimize, in this case maximize.
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HELENE PAYNE, FINITE MATHEMATICS
Exercise 163. Write the mathematical formulation of the
linear programming problem, and identify the objective function and all the constraints.
A particular salad contains 4 units of vitamin A, 5 units of
vitamin B complex, and 2 mg of fat per serving. A nutritious soup contains 6 units of vitamin A, 2 units of vitamin
B complex, and 3 mg of fat per serving. If a lunch consisting of these two foods is to have at least 10 units of
vitamin A and at least 10 units of vitamin complex, how
many servings of each should be used to minimize the total
number of milligrams of fat?
CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 179
Exercise 164. Write the mathematical formulation of the
linear programming problem, and identify the objective function and all the constraints.
An investment club has at most $30, 000 to invest in either junk bonds or premium-quality bonds. Each type of
bond is bought in $1, 000 denominations. The junk bonds
have an average yield of 12%, and the premium-quality
bonds yield 7%. The policy if the club is to invest at least
twice the amount of money in premium-quality bonds as in
junk bonds. How should the club invest its money in these
bonds to receive the maximum return on its investment?
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HELENE PAYNE, FINITE MATHEMATICS
4.1. Systems of Linear Inequalities
In linear programming problems in two variables, the structural constraints consist of a system of inequalities. Solutions of a system of inequalities, when graphing, corresponds to a shaded region in the plane. In this section,
we will learn how to graph a system of linear inequalities,
finding which region to shade. In the following section, we
will solve linear programming problems graphically.
Any linear inequality on two variables has one of the
forms
ax + by < c,
ax + by ≤ c,
ax + by > c,
ax + by ≥ c,
where a, b, and c are real numbers, where a and b are not
both zero. Examples of linear inequalities in two variables
are, 3x + 2y ≤ 5, x − 3y > 7, and y ≥ 0. The laws of
inequalities are listed below.
Laws of Inequalities
1. If a < b and c is any number, then a + c < b + c.
2. If a < b and c is any positive number, then ac < bc.
3. If a < b and c is any negative number, then ac > bc.
CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 181
Exercise 165. Solve the inequality 2x − y ≤ 4 for y.
Solutions of a Linear Inequality
The solutions of a linear inequality in two variables consists
of all ordered pairs (x, y) that, when substituted into the
inequality, result in a true statement.
Exercise 166. For the inequality in the above problem,
y ≥ 2x − 4. Check which of the following points are solutions to the inequality.
(a) (3, 3)
(b) (2, 1)
(c) (4, 5)
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HELENE PAYNE, FINITE MATHEMATICS
Exercise 167. Graph the line y = 2x − 4 and the points
from the previous exercise, (3, 3), (2, 1), (4, 5).
Free Multi-Width Graph Paper from http://incompetech.com/graphpaper/multiwidth/
(a) y > 2x − 4 represents points above the line.
(b) y = 2x − 4 represents points on the line.
(c) y < 2x − 4 represents points below the line.
Use this information to shade the area which corresponds
to the solutions of the inequality y ≥ 2x − 4
CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 183
Graphing the Solution of a Linear Inequality
1. Replace the inequality symbol with = to obtain the
equation of the boundary line.
2. Plot the line represented by the boundary line. If the
inequality is ≤ or ≥, plot a solid line. If the inequality is
> or <, plot the line dashed.
3. Select a test point, not on the boundary line and substitute it into the original inequality. If the resulting inequality is true, shade the side of the line where the test
point is located. If the resulting inequality is false, shade
side of the line opposite to where the test point is located.
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HELENE PAYNE, FINITE MATHEMATICS
Exercise 168. Graph the following linear inequalities carefully following the steps listed above.
(a) 3x + y ≤ 9
(b) 3x − 2yFree<Multi-Width
12 Graph Paper from http://incompetech.com/graphpaper/multiwidth/
Free Multi-Width Graph Paper from http://incompetech.com/graphpaper/multiwidth/
CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 185
Exercise 169. Graph the solution of the following system of linear inequalities. List all the corner points of the
shaded area.
x + 2y ≥ 4
x−y ≤ 1
x ≥ 0
y ≥ 0
Free Multi-Width Graph Paper from http://incompetech.com/graphpaper/multiwidth/
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HELENE PAYNE, FINITE MATHEMATICS
The Feasible Region
For any system of linear inequalities, the set of points that
makes all inequalities true is called the feasible region.
When graphing the solution of a system of linear inequalities, we shade the feasible region.
Corner Point of a Feasible Region
For a feasible region in the plane, defined by a system of
linear inequalities, a point of intersection of two boundary lines that is also part of the feasible region is called a
corner point or vertex of the feasible region.
For any system of linear inequalities, the set of points that
makes all inequalities true is called the feasible region.
When graphing the solution of a system of linear inequalities, we shade the feasible region.
CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 187
Exercise 170. Graph the feasible region given by the system of linear inequalities
2x + y ≤ 6
x + 2y ≤ 6
x ≥ 0
y ≥ 0.
List all the corner points of the feasible region.
Free Multi-Width Graph Paper from http://incompetech.com/graphpaper/multiwidth/
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HELENE PAYNE, FINITE MATHEMATICS
4.2. Solving Linear Programming Problems Graphically
In section we will learn how to solve linear programming
problems graphically.
Consider the linear programming problem,
Maximize f = 2x + 3y
subject to x + y ≤ 9
3x + 2y ≤ 24
x + 2y ≤ 16
x ≥ 0; y ≥ 0
Below, we can see what different values of the objective
function correspond to graphically on the feasible region
arising from the structural constraints.
CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 189
f=2
f=25
4
f=1
8
f =1
2
f=6
In the corner point, (0, 8) we have f = 2(0) + 3(8) = 24.
There are many points for which f = 24, for example in the
point, ( 32 , 7), which is in the feasible region. In fact, there is
a whole line for which f = 24, namely the line 2x+3y = 24.
We can also see in the graph that the smaller the values
of f , the lower down in the graph the corresponding line
is located, and the larger the value of f , the higher up the
corresponding line is located. The largest value of the function f on the feasible region is assumed in the corner point
(2, 7), where f = 25. This example suggests that if a linear
objective function, f has a maximum on the feasible region
bounded by the constraints, then such a maximum will occur at a corner point of the feasible region.
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HELENE PAYNE, FINITE MATHEMATICS
Feasible regions can be bounded (i.e. completely enclosed) or unbounded. They can also be convex or not
convex. Here we will only work with convex feasible regions.
unbounded
bounded
bounded
and convex
and convex
and not convex
CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 191
The Fundamental Theorem of Linear Programming
If the feasible region for a linear programming problem is
nonempty and convex, and if the objective function has a
maximum (or minimum) value within that set, then that
maximum (or minimum) will always correspond to at least
one corner point of the region.
For a linear programming problem with a nonempty feasible region, R and an objective function f ,
1. If R is bounded, then f has both a minimum and a
maximum value at some corner point of R.
2. If R is unbounded and if f has a maximum or a minimum value, on R, then that value will occur at a corner
point. However, the nature of f and the shape of R will
determine whether a maximum or a minimum exists.
The graph below shows a function f = 2x − y, which
no maximum value on the unbounded feasible region.
4.3has
SOLVING LINEAR PROGRAMMING PROBLEMS GRAPHICALLY 183
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d Convexity
The Fundamental Theorem
An Unbounded Feosible Region
of Linear Programming
If the feasible region for a linear programming problem is nonempty and convex, and
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HELENE PAYNE, FINITE MATHEMATICS
Exercise 171. Solve the linear programming problem below by following the steps below.
Maximize P = 4x + 4y
subject to x + 3y ≤ 30
2x + y ≤ 20
x ≥ 0; y ≥ 0
(a) Graph the feasible region.
(b) Find the coordinates of the corner points.
(c) Evaluate P at each corner point, and find the extreme value requested.
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CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 193
Exercise 172. Solve the linear programming problem below by following the steps below.
Minimize C = 5x + 3y
subject to x + y ≥ 6
6x + y ≥ 16
x + 6y ≥ 16
x ≥ 0; y ≥ 0
(a) Graph the feasible region.
(b) Find the coordinates of the corner points.
(c) Evaluate C at each corner point, and find the extreme value requested.
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HELENE PAYNE, FINITE MATHEMATICS
Exercise 173. Solve the linear programming problem below by following the steps below.
Minimize C = x − 2y
subject to x ≥ 2
x ≤ 4
y ≥ 1
y ≤ 5
(a) Graph the feasible region.
(b) Find the coordinates of the corner points.
(c) Evaluate C at each corner point, and find the extreme value requested.
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CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 195
4.3. Models Utilizing Linear Programming with Two
Variables
Exercise 174. Two grains, barley and corn, are to be
mixed for animal food. Barley contains 1 unit of fat per
pound, and corn contains 2 units of fat per pound. The total number of units of fat in the mixture are not to exceed
12 units. No more than 6 pounds of barley and no more
than 5 pounds of corn are to be used in the mixture. If
barley and corn each contain 1 unit of protein per pound,
how many pounds of each grain should be used to maximize the number of units of protein in the mixture?
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HELENE PAYNE, FINITE MATHEMATICS
Exercise 175. A sales representative covers territory in
Iowa and Kansas. Her daily travel expenses average $120 in
Iowa and $100 in Kansas. Her company provides an annual
travel allowance of $18, 000. Her company also stipulates
that she must spend at least 50 days in Iowa, and 60 days
in Kansas per year. If sales average $3, 000 per day in Iowa,
and $2, 500 per day in Kansas, how many days should she
spend in each state to maximize sales?
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