graphing method to solve linear programming problem.
z=8x+12y
40x+80y=560
6x+8y=72
x=0y=0
Is there a short cut to find the mmaximum without using the graph?
where the fricking answer is?!!
Yes, there is a shortcut to find the maximum (or minimum) value without graphing the linear programming problem. It involves using the concept of the objective function and the constraints.
To solve a linear programming problem using the graphing method, we typically start by graphing the constraints. Then, we find the region of feasibility, which is the intersection of all the constraints. Finally, we evaluate the objective function at each of the corner points of the feasible region to determine the maximum or minimum value.
However, in this case, you have already provided the equations:
Objective function: z = 8x + 12y
Constraint 1: 40x + 80y = 560
Constraint 2: 6x + 8y = 72
To find the maximum value of the objective function without graphing, we can solve the equations using algebraic methods such as substitution or elimination.
First, let's rearrange Constraint 2 to solve for x: x = (72 - 8y) / 6.
Now substitute this value of x into Constraint 1:
40[(72 - 8y) / 6] + 80y = 560
Simplify the equation:
(240 - 32y) + 80y = 560
240 + 48y = 560
48y = 320
y = 320 / 48
y = 6.67 (rounded to two decimal places)
Next, substitute the value of y back into the equation for x:
x = (72 - 8 * 6.67) / 6
x ≈ 1.33 (rounded to two decimal places)
Finally, substitute the values of x and y into the objective function to find the maximum value:
z = 8 * 1.33 + 12 * 6.67
z ≈ 79.28 (rounded to two decimal places)
Therefore, the maximum value of z is approximately 79.28 without using the graph.