​Betsy, a recent​ retiree, requires ​$5,000 per year in extra income.  She has ​$50,000 to invest and can invest in​ B-rated bonds paying 13​% per year or in a certificate of deposit​ (CD) paying 3​% per year.  How much money should be invested in each to realize exactly ​$5,000 in interest per​ year?
The amount of money invested at 13​% =
The amount of money invested at 3​%=

Let's define the variables:

- Let $ x $ be the amount of money invested in B-rated bonds at 13% interest.
- Let $ y $ be the amount of money invested in the CD at 3% interest.

Betsy has $50,000 to invest in total, so:

\[
x + y = 50,000
\]

Betsy needs $5,000 in interest income per year, which comes from the interest on both investments:

\[
0.13x + 0.03y = 5,000
\]

Now we have a system of two equations:

1. $ x + y = 50,000 $
2. $ 0.13x + 0.03y = 5,000 $

We can solve this system by substitution or elimination. Let's use substitution:

From the first equation, solve for $ y $:

\[
y = 50,000 - x
\]

Substitute this into the second equation:

\[
0.13x + 0.03(50,000 - x) = 5,000
\]

Distribute the 0.03:

\[
0.13x + 1,500 - 0.03x = 5,000
\]

Combine like terms:

\[
0.10x + 1,500 = 5,000
\]

Subtract 1,500 from both sides:

\[
0.10x = 3,500
\]

Divide by 0.10:

\[
x = 35,000
\]

Now, substitute $ x = 35,000 $ back into the equation $ y = 50,000 - x $:

\[
y = 50,000 - 35,000 = 15,000
\]

So, the amounts should be:

- The amount of money invested at 13% = $ \mathbf{35,000} $
- The amount of money invested at 3% = $ \mathbf{15,000} $

Would you like any further details or have any questions? 

Here are five related questions to consider:

1. What would happen if Betsy needed $6,000 per year instead of $5,000?
2. How would the investments change if the CD paid 4% instead of 3%?
3. If Betsy could only invest a total of $40,000, how much would she need to invest at each rate to get $5,000 per year?
4. What if Betsy wanted to minimize her investment in the 3% CD? How much would she need to invest in the bonds to still earn $5,000?
5. How much total interest would Betsy earn in 10 years with these investments?

**Tip:** When solving systems of linear equations, substitution is often useful when one equation can easily be solved for one variable.

Learn how Betsy can invest $50,000 to earn exactly $5,000 per year in extra income using B-rated bonds at 13% and a certificate of deposit (CD) at 3%. Detailed step-by-step solution included.

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