Betsy, a recent​ retiree, requires ​$6 comma 000 per year in extra income.  She has ​$50 comma 000 to invest and can invest in​ B-rated bonds paying 17​% per year or in a certificate of deposit​ (CD) paying 7​% per year.  How much money should be invested in each to realize exactly ​$6 comma 000 in interest per​ year?

Let's define two variables to represent the amounts that Betsy will invest in each option:

- Let $ x $ be the amount Betsy invests in the B-rated bonds, which pay 17% per year.
- Let $ y $ be the amount Betsy invests in the CD, which pays 7% per year.

Betsy wants the total interest from both investments to equal $6,000 per year, and she has a total of $50,000 to invest. This gives us two equations:

1. The total investment equation:
   \[
   x + y = 50,000
   \]
   
2. The total interest equation (since she wants $6,000 per year in interest):
   \[
   0.17x + 0.07y = 6,000
   \]

### Solving the System of Equations

1. From the first equation, solve for $ y $:
   \[
   y = 50,000 - x
   \]

2. Substitute this expression for $ y $ into the second equation:
   \[
   0.17x + 0.07(50,000 - x) = 6,000
   \]

3. Simplify and solve for $ x $:
   \[
   0.17x + 0.07 \times 50,000 - 0.07x = 6,000
   \]
   \[
   0.17x + 3,500 - 0.07x = 6,000
   \]
   \[
   0.10x + 3,500 = 6,000
   \]
   \[
   0.10x = 6,000 - 3,500
   \]
   \[
   0.10x = 2,500
   \]
   \[
   x = \frac{2,500}{0.10} = 25,000
   \]

So, Betsy should invest **$25,000** in the B-rated bonds.

4. Now, substitute $ x = 25,000 $ back into the first equation to find $ y $:
   \[
   25,000 + y = 50,000
   \]
   \[
   y = 50,000 - 25,000 = 25,000
   \]

Therefore, Betsy should also invest **$25,000** in the CD.

### Final Answer:
- Betsy should invest **$25,000** in the B-rated bonds and **$25,000** in the CD to generate exactly $6,000 in interest per year.

Would you like further details or have any questions?  
Here are some related questions to deepen your understanding:

1. How would the solution change if the interest requirement was $7,000 per year instead of $6,000?
2. What if the total amount to invest was $60,000 instead of $50,000?
3. How would the solution differ if the interest rates on the B-rated bonds and CD were both lower?
4. Can you graph the system of equations and interpret the intersection point?
5. What are other investment strategies Betsy could consider to meet her needs?

**Tip**: Always check your final answers by substituting them back into the original equations to ensure they satisfy both conditions!

Betsy, a recent retiree, requires $6,000 per year in extra income. She has $50,000 to invest and can invest in B-rated bonds paying 17% per year or in a certificate of deposit (CD) paying 7% per year. How much money should be invested in each to realize exactly $6,000 in interest per year?

Learn how Betsy, a retiree, can invest $50,000 in B-rated bonds and a CD to earn $6,000 per year in interest. This problem involves solving a system of linear equations and explores different investment strategies with 17% and 7% interest rates.

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