Maximize z= 5x + 5y

Subject to 3x + 2y < 6
-x + 2y < 4
x,y >0

To solve this linear programming problem and maximize the objective function z = 5x + 5y, subject to the given constraints, we can use the graphical method.

Step 1: Graph the inequality constraints:
First, graph the lines of the constraints as if they were equations.
For the constraint 3x + 2y < 6, rearrange it to the standard form 3x + 2y = 6 by replacing the inequality sign with an equality sign. Plot the line 3x + 2y = 6 on a graph.

For the constraint -x + 2y < 4, rearrange it to the standard form -x + 2y = 4. Plot the line -x + 2y = 4 on the same graph.

Step 2: Shade the feasible region:
Identify the region that satisfies all the given constraints. To do this, shade the region below the line 3x + 2y = 6 (since it has a less-than sign) and below the line -x + 2y = 4.

Step 3: Identify the corner points of the feasible region:
The corner points are the vertices of the shaded region. Find the coordinates of these corner points by solving the systems of equations formed by the lines intersecting at those points.

Step 4: Substitute the corner points into the objective function:
Evaluate the objective function z = 5x + 5y at each corner point to find the maximum value of z within the feasible region.

Step 5: Select the optimal solution:
Choose the corner point that yields the maximum value of z as the optimal solution. This point represents the values of x and y that maximize the objective function z.

That's how you can solve the given linear programming problem graphically to maximize the objective function z = 5x + 5y, subject to the given constraints.