To solve this problem, we will let $x$, $y$, and $z$ represent the amounts invested in CDs, bonds, and stocks, respectively. We'll establish equations based on the problem's conditions and solve them step by step.

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Step 1: Define the variables

• $x$: amount invested in CDs (earning 5.25% interest).
• $y$: amount invested in bonds (earning 4.2% interest).
• $z$: amount invested in stocks (earning 7.2% interest).

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Step 2: Establish equations based on the problem

1. Total investment: $x + y + z = 210000$

2. Interest income: The annual income from the investments is $11,107.50: $0.0525x + 0.042y + 0.072z = 11107.5$

3. Relationship between bonds and CDs: The investment in bonds is $10,000 more than in CDs: $y = x + 10000$

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Step 3: Substitute $y = x + 10000$ into the equations

Substituting into the total investment equation:

$x + (x + 10000) + z = 210000$ $2x + z = 200000 \quad \text{(Equation 4)}$

Substituting into the interest income equation:

$0.0525x + 0.042(x + 10000) + 0.072z = 11107.5$ Simplify: $0.0525x + 0.042x + 0.042(10000) + 0.072z = 11107.5$ $0.0945x + 420 + 0.072z = 11107.5$ $0.0945x + 0.072z = 10687.5 \quad \text{(Equation 5)}$

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Step 4: Solve the system of equations

We now have two equations:

1. $2x + z = 200000$ (Equation 4)
2. $0.0945x + 0.072z = 10687.5$ (Equation 5)

Solve for $z$ in terms of $x$ using Equation 4:

$z = 200000 - 2x$

Substitute $z = 200000 - 2x$ into Equation 5:

$0.0945x + 0.072(200000 - 2x) = 10687.5$ Expand: $0.0945x + 0.072(200000) - 0.072(2x) = 10687.5$ $0.0945x + 14400 - 0.144x = 10687.5$ Combine like terms: $-0.0495x + 14400 = 10687.5$ $-0.0495x = 10687.5 - 14400$ $-0.0495x = -3712.5$ $x = \frac{-3712.5}{-0.0495} = 75000$

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Step 5: Find $y$ and $z$

Using $x = 75000$:

• $y = x + 10000 = 75000 + 10000 = 85000$
• $z = 200000 - 2x = 200000 - 2(75000) = 50000$

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Final Answer:

• Investment in CDs: $x = 75000$
• Investment in bonds: $y = 85000$
• Investment in stocks: $z = 50000$

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Verification:

1. Total investment: $75000 + 85000 + 50000 = 210000 \quad \text{(Correct)}$

2. Interest income: $0.0525(75000) + 0.042(85000) + 0.072(50000) = 3937.5 + 3570 + 3600 = 11107.5 \quad \text{(Correct)}$

Let me know if you'd like additional explanations!

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Related Questions:

1. How do we derive the equations for systems of linear equations?
2. What strategies exist for solving systems of linear equations with substitution?
3. Can we solve this problem using matrices or determinants?
4. How would the solution change if the interest rates were different?
5. What if the annual income constraint was different—how would that affect the investments?

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Tip:

When solving word problems, break them into smaller parts to derive equations systematically. Always double-check your results for consistency!