Find the maximum value of the objective function
z = 19x + 7y,
subject to the following constraints. (See Example 2.)
3x + 2y ≤ 18
6x + 2y ≤ 30
  x ≥ 0, y ≥ 0
there are many nice linear programming calculators online. One you can use to check your work is at
http://www.zweigmedia.com/RealWorld/LPGrapher/lpg.html
To find the maximum value of the objective function z = 19x + 7y, subject to the given constraints, we can use the linear programming method.
Step 1: Graph the feasible region:
Start by graphing the inequalities that represent the constraints. Plot the lines represented by the equality part of each constraint, and shade the region that satisfies all the inequalities. In this case, we have two inequalities:
3x + 2y ≤ 18
6x + 2y ≤ 30
Plotting these inequalities and shading the feasible region will give you a set of points that satisfy the constraints.
Step 2: Identify the corner points of the feasible region:
Examine the points where the lines representing the inequalities intersect (corners of the feasible region). These corner points will be the potential solutions to the problem.
Step 3: Evaluate the objective function at each corner point:
Calculate the value of the objective function z = 19x + 7y at each corner point of the feasible region. This will give you a set of values.
Step 4: Determine the maximum value:
Compare the values obtained from evaluating the objective function at each corner point. The maximum value of the objective function will occur at the corner point with the highest value for z.
Therefore, by following these steps, you can find the maximum value of z = 19x + 7y, subject to the given constraints.