MATHEMATICS II - Deneme Sınavı - 12

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.

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.

Furthermore, one can easily check that

we can express given system as the matrix equation . The determinant of is

By the Cramer’s Rule we get 

Hence the solution is the point (18,-39)

Hence, the inverse matrix of is evaluated as 

Hence we have 

and 

Therefore, we have m-n=2-2=0

Hence the solution is the point (-1,2)

(it does not satisfy the second inequality)

(it does not satisfy the second inequality)

(it does not satisfy the second inequality)

(it satisfies both of the inequalities)

(it does not satisfy the first inequality)

except the choice C. The system of equations corresponding to the system given in C

has the intersection point at 

Hence the minimum value of given constraint set is -15

When we add them we get 3y=9 and y=3. For y=3 by the first equation we get x+6=4 that gives x=-2. Hence the corner point is (-2,3)