Let the initial amounts invested in the checking, savings, and CD accounts be denoted by:

• $x$ for the checking account,
• $y$ for the savings account,
• $z$ for the CD account.

We are given the following information:

1. The total deposit is $10,000: x + y + z = 10,000 \tag{1}
2. The interest rates are 2%, 4%, and 10% for the checking, savings, and CD accounts, respectively. The total interest earned after 1 year is $560: 0.02x + 0.04y + 0.10z = 560 \tag{2}
3. The total amount invested in the checking and savings accounts combined is $4,000 more than the amount invested in the CD account: x + y = z + 4,000 \tag{3}

Now, let's solve the system of equations.

Step 1: Substitute equation (3) into equation (1)

From equation (3), we know that $x + y = z + 4,000$. Substituting this into equation (1): $z + 4,000 + z = 10,000$ Simplifying: $2z + 4,000 = 10,000$ $2z = 6,000$ $z = 3,000$

Step 2: Substitute $z = 3,000$ into equation (3)

Substitute $z = 3,000$ into equation (3): x + y = 3,000 + 4,000 = 7,000 \tag{4}

Step 3: Substitute $z = 3,000$ into equation (2)

Substitute $z = 3,000$ into equation (2): $0.02x + 0.04y + 0.10(3,000) = 560$ Simplifying: $0.02x + 0.04y + 300 = 560$ 0.02x + 0.04y = 260 \tag{5}

Step 4: Solve the system of equations (4) and (5)

From equation (4), we know $x + y = 7,000$. We can express $x$ as: x = 7,000 - y \tag{6} Substitute equation (6) into equation (5): $0.02(7,000 - y) + 0.04y = 260$ Simplifying: $0.02(7,000) - 0.02y + 0.04y = 260$ $140 - 0.02y + 0.04y = 260$ $140 + 0.02y = 260$ $0.02y = 120$ $y = \frac{120}{0.02} = 6,000$

Step 5: Solve for $x$

Now that we know $y = 6,000$, substitute this into equation (6): $x = 7,000 - 6,000 = 1,000$

Final Answer:

• $x = 1,000$ (initial investment in the checking account),
• $y = 6,000$ (initial investment in the savings account),
• $z = 3,000$ (initial investment in the CD account).

Would you like more details or have any questions?

5 Related Questions:

1. How would the solution change if the interest rates for the accounts were different?
2. What if the total interest earned was different, say $600? How would that affect the investments?
3. How can you express the total interest as a function of the initial investments and interest rates?
4. If you knew the final amounts after 1 year, how could you backtrack to the initial investments?
5. What is the significance of higher interest rates for long-term investment planning?

Tip:

Always try to express variables in terms of each other to reduce the number of unknowns when solving systems of equations.