minimize z= 5x + 9y

subject to
6x + y >= 18
2x + 3y >= 30
x<= 12
x,y>= 0

find the corner points, ans subject the objective function to the corner points, finding the minimum value of z giving the constraints.

i know the first and second points but i couldn't figure out the third one and the minimal value. please help me!! :[

Submit your question at tutoteddy and get the answer.

No Answer

To find the corner points, we need to solve the system of linear inequalities. Let's go step by step:

1. Plot the equations on a graph:

Rewrite the inequalities in the form of equations to find the border lines.

Equation 1: 6x + y = 18 (I)
Equation 2: 2x + 3y = 30 (II)

Now plot these two lines on a graph.

2. Find the intersection points:

To find the intersection points, we need to solve the equations (I) and (II) simultaneously:

Solve the equations (I) and (II) by elimination or substitution method:

Multiply equation (I) by 2 and equation (II) by 6 to eliminate the x term:

12x + 2y = 36 (III)
12x + 18y = 180 (IV)

Now subtract equation (III) from equation (IV):

16y = 144
y = 9

Substitute the value of y back into equation (I) or (II):

6x + 9 = 18
6x = 9
x = 3

So the first intersection point is (3, 9).

Observe that the value of x = 0 makes the equation (I) true, so we have another intersection point at (0, 18).

3. Check the boundary lines:

Now, we need to check the region bounded by the constraints. Let's check the values of x and y at the corners of the feasible region:

- Point A: (0, 0)
- Point B: (0, 18)
- Point C: (12, 0)
- Point D: (3, 9)

4. Calculate the value of the objective function (z = 5x + 9y):

For each corner point, plug in the x and y values into the objective function to find the corresponding z value:

- For Point A: z = 5(0) + 9(0) = 0
- For Point B: z = 5(0) + 9(18) = 162
- For Point C: z = 5(12) + 9(0) = 60
- For Point D: z = 5(3) + 9(9) = 84

5. Find the optimal solution:

Among the values of z at the corner points, the minimum value of z is 0, which occurs at point A (0, 0).

Therefore, the optimal solution occurs at (0, 0) with z = 0.