Marie invested K12000 in two different accounts at simple interest. The first one at 3 % for two years and the second one at 5.5% for 3 years. Both accounts earned K850 at their accumulated interest after their terms has matured. How much did Marie invested in each amount? 

To solve this problem, let's denote the amount Marie invested in the first account as \( x \) and the amount she invested in the second account as \( y \). We are given the following information:

1. The total investment: 
   \[
   x + y = 12000
   \]
   
2. The interest earned from the first account:
   \[
   \text{Interest}_1 = x \times 0.03 \times 2 = 0.06x
   \]
   
3. The interest earned from the second account:
   \[
   \text{Interest}_2 = y \times 0.055 \times 3 = 0.165y
   \]
   
4. The total interest earned from both accounts:
   \[
   0.06x + 0.165y = 850
   \]

Now, we have two equations:
\[
x + y = 12000
\]
\[
0.06x + 0.165y = 850
\]

### Step 1: Solve the system of equations
We can solve these two equations simultaneously to find the values of \( x \) and \( y \).

From the first equation, express \( y \) in terms of \( x \):
\[
y = 12000 - x
\]

### Step 2: Substitute \( y \) into the second equation
Substitute \( y = 12000 - x \) into the second equation:
\[
0.06x + 0.165(12000 - x) = 850
\]

### Step 3: Expand and simplify
Expand the second equation:
\[
0.06x + 1980 - 0.165x = 850
\]
Combine like terms:
\[
-0.105x + 1980 = 850
\]
Subtract 1980 from both sides:
\[
-0.105x = 850 - 1980 = -1130
\]

### Step 4: Solve for \( x \)
\[
x = \frac{-1130}{-0.105} \approx 10761.90
\]

So, \( x \approx K10761.90 \).

### Step 5: Find \( y \)
Substitute \( x = 10761.90 \) back into the equation \( y = 12000 - x \):
\[
y = 12000 - 10761.90 \approx K1238.10
\]

### Final Answer:
Marie invested approximately **K10761.90** in the first account and **K1238.10** in the second account.

---

Would you like more details on this calculation or have any other questions?

### Related Questions:
1. How much interest would Marie have earned if both investments were made at 3%?
2. What would be the total interest if the entire K12000 was invested in the second account?
3. How would the interest change if the second account had a 6% interest rate instead?
4. How does the duration of the investment affect the total interest earned?
5. What is simple interest, and how does it differ from compound interest?
6. Can you calculate the total amount in each account after the interest is added?
7. What is the formula for simple interest, and how is it applied?
8. How would the investment amounts change if the total interest earned was K900 instead of K850?

### Tip:
Always double-check your substitution and simplification steps when solving systems of equations, as small errors can lead to incorrect results.

Learn how to solve a complex simple interest problem involving two accounts with different interest rates and durations. This problem covers systems of equations and financial mathematics concepts. Suitable for high school students.

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