Math Problem Statement

You have $30,000 to invest. You will invest M dollars in municipal bonds and T dollars in Treasury notes. For security, your financial advisor recommends that you invest twice as much in Treasury notes as you invest in municipal bonds. (a) Write a system of equations that will determine how much to invest in each category. (Suggestion: One equation should show the total amount invested. The second equation should show that twice as much is invested in Treasury notes as in municipal bonds.) M + T = T =

M (b) Solve the system of equations in part (a) to determine how much is invested in each category. municipal bonds $ Treasury notes $

Solution

(a) Writing the System of Equations

Let:

  • MM be the amount invested in municipal bonds (in dollars).
  • TT be the amount invested in Treasury notes (in dollars).

  1. Total Investment Equation: The total amount of money invested is $30,000. M+T=30,000M + T = 30,000

  2. Relationship Between Investments: According to the advisor’s recommendation, twice as much is invested in Treasury notes as in municipal bonds. T=2MT = 2M

System of Equations

M + T = 30,000 \\ T = 2M \end{cases}$$ ### (b) Solving the System of Equations 1. **Substitute** the second equation ($$T = 2M$$) into the first equation: $$M + 2M = 30,000$$ $$3M = 30,000$$ 2. **Solve** for $$M$$: $$M = \frac{30,000}{3} = 10,000$$ 3. **Substitute** $$M = 10,000$$ back into the second equation to solve for $$T$$: $$T = 2M = 2(10,000) = 20,000$$ ### Final Solution - Amount invested in **municipal bonds**: $$M = 10,000$$ dollars. - Amount invested in **Treasury notes**: $$T = 20,000$$ dollars. **Summary:** - **Municipal bonds**: $10,000 - **Treasury notes**: $20,000 Would you like further details or have any questions about the solution? ### Relative Questions 1. What if the total investment amount was $40,000 instead of $30,000? 2. How would the solution change if three times as much was invested in Treasury notes as in municipal bonds? 3. What happens if the total amount is unknown but the relationship between the investments is the same? 4. Can you verify the solution by substituting the values back into the original equations? 5. How would you graph the system of equations to visualize the solution? **Tip:** Always double-check your substitution step when solving systems of equations to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations

Formulas

M + T = 30,000
T = 2M

Theorems

Substitution Method

Suitable Grade Level

Grades 9-12

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