Use graphical methods to solve the linear programming problem.Minimizez = 0.18x + 0.12ysubject to:2x + 6y ≥ 304x + 2y ≥ 20x ≥ 0y ≥ 0

Step 1: Identify the Constraints

The constraints are given by the inequalities:

2x + 6y ≥ 30 4x + 2y ≥ 20 x ≥ 0 y ≥ 0

Step 2: Graph the Constraints

Plot these inequalities on a graph. The feasible region is the area that satisfies all these inequalities.

Step 3: Identify the Objective Function

The objective function is z = 0.18x + 0.12y. This is the function we want to minimize.

Step 4: Find the Optimal Solution

The optimal solution is the point in the feasible region that gives the minimum value of the objective function.

To find this point graphically, draw lines of constant z (also known as isoprofit lines) and move them towards the origin until they just touch the feasible region. The last point at which they touch the feasible region is the optimal solution.

Step 5: Solve

By solving the system of inequalities and finding the optimal solution graphically, you can find the values of x and y that minimize z.