Solve the following LP problem graphically
MAX 2X + 7Y
Subject to:
5X + 9Y >= 90
9X + 8Y <= 144
Y <= 8
X, Y >= 0
To solve the given linear programming (LP) problem graphically, we need to follow these steps:
Step 1: Convert the inequalities into equations
We need to convert the given constraints into equations to plot them on the graph. Here are the converted equations:
Equation 1: 5X + 9Y = 90 (converted from 5X + 9Y >= 90)
Equation 2: 9X + 8Y = 144 (converted from 9X + 8Y <= 144)
Equation 3: Y = 8 (converted from Y <= 8)
Step 2: Plotting the equations on a graph
Now, we can plot these equations on a two-dimensional graph. The X-axis represents the X variable, and the Y-axis represents the Y variable.
Let's start with Equation 1: 5X + 9Y = 90
To plot this equation, we need to find two points on this line. We can assign arbitrary values to either X or Y and calculate the corresponding values for the other variable.
When X = 0, 9Y = 90, so Y = 10. Thus, we have the point (0, 10).
When Y = 0, 5X = 90, so X = 18. Thus, we have the point (18, 0).
Plot these two points and draw the line passing through them.
Next, Equation 2: 9X + 8Y = 144
Similarly, find two points on this line:
When X = 0, 8Y = 144, so Y = 18. We have the point (0, 18).
When Y = 0, 9X = 144, so X = 16. We have the point (16, 0).
Plot these two points and draw the line passing through them.
Finally, Equation 3: Y = 8
This equation represents a horizontal line parallel to the X-axis at Y = 8. Draw a horizontal line passing through the point (0, 8).
Step 3: Determine the feasible region
The feasible region is the area of the graph that satisfies all the given constraints. Shade the region on the graph that lies below the line of Equation 1 (5X + 8Y = 90), above the line of Equation 2 (9X + 8Y = 144), and below the horizontal line of Equation 3 (Y = 8).
Step 4: Find the optimal solution
Now, we need to find the highest point in the feasible region that maximizes the objective function 2X + 7Y. This point represents the optimal solution.
To find the optimal solution graphically, we can draw the objective function line 2X + 7Y = C, where C is a constant.
Start by selecting an arbitrary value for C and solving for X and Y. Repeat this process by adjusting the value of C to find the highest value for the line that intersects the feasible region.
The highest value of the objective function occurs at the corner point(s) of the feasible region where the objective function line intersects.
Note: Without specific values for X and Y, it is not possible to provide the exact graphical solution. Hence, follow the above steps to find the solution by plotting the given information on a graph.