MATHEMATICS II

Linear programming is one of the branches of optimization and studies the minimization or maximization of a linear function under linear constraints. The source of linear programming is practical problems, where we are looking for the best way to allocate limited resources. In these problems, the goal may be maximizing profits or minimizing costs, or maximizing output or minimizing waste. The subject of linear programming was developed in the 1940’s in response to the war effort. Since that time, the subject has grown enormously and has found many commercial as well as non-commercial applications. The answer is C.

Let a, b, and c be given real numbers. It is well known that the equation ax + by = c defines a straight line in the plane. The graph of this line can be drawn by two points on it, for example, by the points of intersections with the coordinate axis. The straight-line ax + by = c separates the whole plane into two half-planes, one of the half-planes is represented by ax + by ≤ c,
the other is represented by ax + by ≥ c. Assume that b > 0. Then the half-plane ax + by ≤ c is a half-plane remaining below the graph of the straight line ax + by = c. The half-plane ax + by ≥ c is a half-plane remaining above the graph of the straight line ax + by = c. 

The answer is A.

A solution set of this system is the set of all points (x, y) satisfying all the inequalities, which is the intersection of the corresponding half-planes.

The answer is B.

The answer is C.

For the point (7, 5),

f=11x+9y=11*7+9*5=122. The answer is D.

f=9x+7y,  (1, 3); f=9*1+7*3=30,   (2, 4); f=9*2+7*4=46,   (4, 5); f=9*4+7*5=71,  (1, 8); f=9*1+7*8=65,   (2, 7); f=9*2+7*7=67. The maximum value of the function is 71. The maximum point (4, 5). The answer is B.

For the point (9, 4),

f=4x+13y=4*9+13*4=88. The answer is C.

A linear programming problem includes a linear function in two variables that we want to maximize or minimize. This function is called the goal function. The system of linear inequalities in a linear programming problem is also called the constraint set.

The answer is D

The corner point theorem states that the maximum or minimum of a linear function over the constrained set if it exists will necessarily occur at a corner point of the constraint set.

The answer is B.

In the case n = 2, the linear programming problem consists of the maximization of linear, two variable function f = ax + by under the system of linear inequalities in two variables. The constraint set is the intersection of half-planes. In order to guarantee the boundedness of the constraint set in the sequel, we assume that 
a₁ ≥ 0,…, a_m ≥ 0, b₁ > 0,…, b_m > 0, c₁ > 0,…, c_m > 0.
Under this assumption, the constraint set is bounded and is called a polygonal set with finite number of corner points. The corner points of the constraint set are the intersection points of the straight lines.

The answer is E.

The point (3, 3) satisfies the inequality
2x + 7y + 1 = 2.3 + 7.3 + 1 = 6 + 21 + 1 = 28 ≥ 25

The answer is E.

The point (1,1) satisfies the inequality
5x + 2y = 5.1 + 2.1 = 5+2 = 7 ≤ 10

The answer is A.

For 5x - 6y = -2 and 3x - 2y = 2 Multiply both sides of the second equation by -3 and add to the first equation:

5x - 6y = -2

-9x + 6y = -6 the sum of these equations yields to -4x = -8 which is x = 2

For x = 2, 3x - 2y = 2 becomes  3.2 - 2y = 2 which is 6 - 2y = 2, -2y= -4 which would yield to y=2.

The answer is C.

4x + 5y ≤ 20
2x + 3y ≤ 24
x ≥ 0, y ≥ 0

The system of two equations and their solutions:

4x + 5y ≤ 20 for x = 0, 4.0 + 5y =20 then y = 4, (0,4); for y = 0, 4x + 5.0 = 20 then x = 5, (5,0)

2x + 3y ≤ 24 for x = 0, 2.0 + 3y =24 then y = 8  (0,8); for y = 0, 2x + 3.0 = 24 then x = 12, (12,0)

Multiply the second equation by (-2) and add to the first equation:

-4x - 6y ≤ -48

4x + 5y ≤ 20 therefore -y = -28, y = 28

2x + 84 = 24 this yields to 2x = -60, x = -30, (-30, 28)

The intersection points are (0,4), (5,0), (0,8), (12,0), and (-30, 28). For (0,4) and (5,0) the inequatiy 2x + 3y ≤ 24 is correct, they belong in the constraint set and for (0,8) and (12,0) the inequatiy 4x + 5y ≤ 20 is not correct so they do not belong in the constraint set. The corner points of the constraint set are (0,4), (5,0), and (-30, 28)

The answer is D.