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!! :[
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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.