# Service for Solving Linear Programming Problems

and other interesting typical problems
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## Example №10. Solving a Linear Programming Problem Using a Graphical Method. Region of Feasible Solutions is an Empty Set

This solution has been made using the calculator presented on the site.
Problem:
Find the maximum value of the function

F = x1 + x2

subject to the constraints: 3 x1 + 5 x2 ≤ 30 4 x1 - 3 x2 ≤ 12 x1 - 3 x2 ≥ 6

x1 ≥ 0     x2 ≥ 0
Solution:

Points whose coordinates satisfy all the inequalities of the constraint system are called a region of feasible solutions.

It is necessary to solve each inequality of the constraint system to find the region of feasible solutions to this problem. (see step 1 - step 3)

The last two steps are necessary to get the answer.
(see step 4 - step 5)

This is a standard solution plan. If the region of feasible solutions is a point or an empty set then the solution will be shorter.

See the plan for solving this problem in pictures

By the condition of the problem: x1 ≥ 0     x2 ≥ 0.

Now we have the region of feasible solutions shown in the picture. Big pictures
Step №1

Let's solve 1 inequality of the system of constraints.

3 x1 + 5 x2  ≤  30

We need to plot a straight line: 3 x1 + 5 x2 = 30

Let x1 =0 => 5 x2 = 30 => x2 = 6

Let x2 =0 => 3 x1 = 30 => x1 = 10

Two points were found: (0, 6) and (10 ,0)

Now we can plot the straight line (1) through the found two points.

Let's go back to the inequality.

3 x1 + 5 x2  ≤  30

We need to transform the inequality so that only x2 is on the left side.

5 x2  ≤  - 3 x1 + 30

x2  ≤  - 3/5 x1 + 6

The inequality sign is  ≤
Therefore, we must consider points below the straight line (1).

Let's combine this result with the previous picture.
Now we have the region of feasible solutions shown in the picture. Big pictures
Step №2

Let's solve 2 inequality of the system of constraints.

4 x1 - 3 x2  ≤  12

We need to plot a straight line: 4 x1 - 3 x2 = 12

Let x1 =0 => - 3 x2 = 12 => x2 = -4

Let x2 =0 => 4 x1 = 12 => x1 = 3

Two points were found: (0, -4) and (3 ,0)

Now we can plot the straight line (2) through the found two points.

Let's go back to the inequality.

4 x1 - 3 x2  ≤  12

We need to transform the inequality so that only x2 is on the left side.

- 3 x2  ≤  - 4 x1 + 12

x2  ≥  4/3 x1 - 4

The inequality sign is  ≥
Therefore, we must consider points above the straight line (2).

Let's combine this result with the previous picture.
Now we have the region of feasible solutions shown in the picture. Big pictures
Step №3

Let's solve 3 inequality of the system of constraints.

x1 - 3 x2  ≥  6

We need to plot a straight line: x1 - 3 x2 = 6

Let x1 =0 => - 3 x2 = 6 => x2 = -2

Let x2 =0 => x1 = 6

Two points were found: (0, -2) and (6 ,0)

Now we can plot the straight line (3) through the found two points.

Let's go back to the inequality.

x1 - 3 x2  ≥  6

We need to transform the inequality so that only x2 is on the left side.

- 3 x2  ≥  - x1 + 6

x2  ≤  1/3 x1 - 2

The inequality sign is  ≤
Therefore, we must consider points below the straight line (3).

This result does not have common points with the region of possible solutions of the previous step (see picture). Big pictures
Result: