formulate but do not solve the linear programming problem. Tracy has at most $200,000 to invest in stocks,bonds, and money-market funds. She expects anual yields of 15%, 10%, and 8%, respectively, on these investments. If Tracy wants at least $50,000 to be invested in money-market funds and requires that the amount invested in bonds be greater than or equal to the sum of her investments in stocks and money-market funds, determine how much she should invest in each vehicle in order to maximize the returns on her investments.

To formulate the linear programming problem, we first need to define the decision variables, objective function, and the constraints.

Let:
- X be the amount invested in stocks (in dollars).
- Y be the amount invested in bonds (in dollars).
- Z be the amount invested in money-market funds (in dollars).

Now, let's formulate the problem:

Decision Variables:
- X: Amount invested in stocks (in dollars).
- Y: Amount invested in bonds (in dollars).
- Z: Amount invested in money-market funds (in dollars).

Objective Function:
Maximize the total annual yield (returns) on investments:
15% * X + 10% * Y + 8% * Z

Constraints:
1. Total investment amount cannot exceed $200,000:
X + Y + Z ≤ 200,000

2. At least $50,000 should be invested in money-market funds:
Z ≥ 50,000

3. Amount invested in bonds (Y) should be greater than or equal to the sum of investments in stocks (X) and money-market funds (Z):
Y ≥ X + Z

Now you have the linear programming problem formulated. To solve it and find the optimal investment amounts, you can use various optimization techniques like the simplex method, graphical method, or software tools like Excel Solver or MATLAB's linear programming solver.