To determine the value of y that maximizes z = x + y subject to the given constraints, we can use a method called linear programming. In this case, we have the following constraints:
1. 2x + 3y ≤ 36
2. x + 2y ≤ 18
3. x ≥ 0, y ≥ 0
To find the optimal value of y that maximizes z, we first need to plot the feasible region formed by the intersection of the inequalities given by the constraints.
After plotting the feasible region, we need to find the corner points of this region where the maximum value of z could occur. These corner points are the intersections of the boundary lines of the inequalities.
Next, we substitute each corner point into the objective function z = x + y and calculate the value of z for each point.
The corner point that gives the highest value of z will be the solution to the maximization problem.
By solving the system of equations formed by the constraints, we can find the corner points and evaluate z at those points to determine the maximum value of z and the corresponding values of x and y that achieve this maximum.
Following these steps will help you determine the value of y that maximizes z under the given constraints.
