Solve the linear programming problem by the method of corners.
Minimize C = 3x + 7y
subject to 4x + y ≥ 39
2x + y ≥ 29
x + 3y ≥ 42
x ≥ 0, y ≥
To solve the linear programming problem using the method of corners, follow these steps:
1. Plot the feasible region:
- Start by graphing the equations of the constraints on the x-y plane.
- To do this, convert each constraint into an equation and then plot the corresponding line.
- Shade the region that satisfies all the constraints.
2. Identify the corner points:
- The corner points are the vertices of the shaded feasible region.
- They represent the possible solutions to the problem.
3. Evaluate the objective function at each corner point:
- Substitute the x and y values of each corner point into the objective function (C = 3x + 7y) and calculate the corresponding value.
4. Determine the minimum value:
- Among the values obtained in step 3, choose the minimum value of the objective function.
- The corresponding values of x and y will give the optimal solution to the linear programming problem.
Let's solve the problem step by step:
1. Plot the feasible region:
- Convert the constraints into equations:
4x + y = 39
2x + y = 29
x + 3y = 42
- Graph these equations on the x-y plane.
2. Identify the corner points:
- Identify the vertices of the shaded feasible region.
3. Evaluate the objective function at each corner point:
- Substitute the x and y values of each corner point into the objective function C = 3x + 7y.
4. Determine the minimum value:
- Choose the minimum value of the objective function obtained in step 3.
- The corresponding x and y values will give the optimal solution to the linear programming problem.
To solve the linear programming problem using the method of corners, we will follow these steps:
Step 1: Graph the constraints
- Graph the lines representing each constraint on the coordinate plane.
- Shade the region of the graph that satisfies all the constraints.
In our case, we have the following constraints:
- 4x + y ≥ 39
- 2x + y ≥ 29
- x + 3y ≥ 42
- x ≥ 0
- y ≥ 0
Step 2: Find the corners of the feasible region
- The feasible region is the shaded area on the graph.
- Identify the vertices or corners of this region by finding the points where the lines intersect or touch.
Step 3: Evaluate the objective function at each corner
- Substitute the x and y values of each corner point into the objective function (C = 3x + 7y).
- Calculate the value of the objective function at each corner point.
Step 4: Determine the optimal solution
- For a minimization problem, select the corner point with the smallest objective function value.
- This corner point represents the optimal solution to the linear programming problem.
By following these steps, we will be able to solve the given linear programming problem using the method of corners.