This problem involves determining the value of the services exchanged between the three professionals using a system of linear equations. We know the time each professional spent on the others' services, and we also know the financial planner's income ($20,000). The task is to find how much the physician and attorney should be paid.

Step 1: Define Variables

Let:

• $x$ be the physician's income,
• $y$ be the attorney's income,
• $z = 20,000$ (the financial planner's income).

Step 2: Set Up the Equations

Each professional's total income should be equal to the sum of the payments they received based on the time spent on their services.

1. Physician's income $x$:

   • 30% of the physician’s work was on his own care, 40% on the attorney's care, and 30% on the financial planner’s care. Hence: $x = 0.3x + 0.4y + 0.3z$

2. Attorney's income $y$:

   • 50% of the attorney's work was on the physician's care, 20% on her own care, and 30% on the financial planner's care. Hence: $y = 0.5x + 0.2y + 0.3z$

3. Financial planner's income $z = 20,000$:

   • 20% of the financial planner's work was on the physician's care, 20% on the attorney's care, and 60% on his own care. Hence: $z = 0.2x + 0.2y + 0.6z$ But since $z = 20,000$, we can substitute this into the equation.

Step 3: Simplify the Equations

Let’s rearrange and simplify the system of equations.

1. From the first equation: $x = 0.3x + 0.4y + 0.3(20,000)$ $x - 0.3x = 0.4y + 6,000$ $0.7x = 0.4y + 6,000$ $x = \frac{0.4y + 6,000}{0.7}$

2. From the second equation: $y = 0.5x + 0.2y + 0.3(20,000)$ $y - 0.2y = 0.5x + 6,000$ $0.8y = 0.5x + 6,000$ $y = \frac{0.5x + 6,000}{0.8}$

3. From the third equation, substitute $z = 20,000$: $20,000 = 0.2x + 0.2y + 0.6(20,000)$ $20,000 = 0.2x + 0.2y + 12,000$ $0.2x + 0.2y = 8,000$ $x + y = 40,000$

Step 4: Solve the System of Equations

We now have two equations:

1. $x + y = 40,000$
2. $x = \frac{0.4y + 6,000}{0.7}$

Substitute $x = 40,000 - y$ into the second equation: $40,000 - y = \frac{0.4y + 6,000}{0.7}$ Multiply through by 0.7: $0.7(40,000 - y) = 0.4y + 6,000$ $28,000 - 0.7y = 0.4y + 6,000$ Move terms with $y$ to one side: $28,000 - 6,000 = 0.4y + 0.7y$ $22,000 = 1.1y$ $y = \frac{22,000}{1.1} = 20,000$

Substitute $y = 20,000$ into $x + y = 40,000$: $x + 20,000 = 40,000$ $x = 20,000$

Step 5: Conclusion

• The physician should be paid $x = 20,000$.
• The attorney should be paid $y = 20,000$.
• The financial planner already earns $z = 20,000$.

Final Answer:

Each professional should be paid $20,000 for their work.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the payments change if the time spent by the professionals was different?
2. What would happen if one professional charged a different rate per hour of service?
3. How can linear algebra help solve complex systems of professional service exchanges?
4. What would be the result if the financial planner earned more than $20,000?
5. How would additional professionals in the cooperative affect the system of payments?

Tip: Linear systems like this can be solved efficiently using matrix methods such as Gaussian elimination.