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This problem involves solving a system of equations based on simple interest.

We are given the following:
- Phyllis invested $48,000 in total.
- A part of the money earns 5% simple interest, and the other part earns 6%.
- After one year, the total interest earned is $2,650.

Let:
- $ x $ be the amount invested at 5%.
- $ 48000 - x $ be the amount invested at 6%.

We know that:
- The interest from the 5% investment is $ 0.05x $.
- The interest from the 6% investment is $ 0.06(48000 - x) $.

The total interest is given as $2,650, so we can set up the equation:

\[
0.05x + 0.06(48000 - x) = 2650
\]

Now, let's solve this equation step by step.

### Step 1: Distribute the 0.06
\[
0.05x + 0.06(48000) - 0.06x = 2650
\]
\[
0.05x + 2880 - 0.06x = 2650
\]

### Step 2: Combine like terms
\[
(0.05x - 0.06x) + 2880 = 2650
\]
\[
-0.01x + 2880 = 2650
\]

### Step 3: Move the 2880 to the right side
\[
-0.01x = 2650 - 2880
\]
\[
-0.01x = -230
\]

### Step 4: Solve for $ x $
\[
x = \frac{-230}{-0.01} = 23000
\]

So, Phyllis invested $23,000 at 5%.

### Step 5: Find the amount invested at 6%
\[
48000 - 23000 = 25000
\]

Thus, Phyllis invested:
- $23,000 at 5%, and
- $25,000 at 6%.

Would you like more details or have any questions about this solution?

Here are 5 related questions to explore further:
1. What is simple interest, and how does it differ from compound interest?
2. How would the problem change if the interest were compounded annually instead of simple interest?
3. Can you solve the problem using a graph of the two interest rates?
4. What would happen if the total amount invested was different, say $60,000?
5. How do different interest rates affect investment decisions?

**Tip:** To check if your solution is correct, always verify the total interest by plugging the amounts back into the interest formulas.

Phyllis invested 48000 dollars, a portion earning a simple interest rate of 5 percent per year and the rest earning a rate of 6 percent per year. After one year the total interest earned on these investments was 2650 dollars. How much money did she invest at each rate?

Learn how to solve a simple interest problem where Phyllis invested $48,000, splitting her investment between a 5% and 6% interest rate, earning $2,650 after one year. This algebra-based problem involves setting up and solving a system of linear equations.

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