Question 1178756
Let's set up the system of linear equations and solve it using matrices.

**1. Define the Variables:**

* x = amount borrowed at 5%
* y = amount borrowed at 6.5%
* z = amount borrowed at 7.5%

**2. Formulate the Equations:**

* **Total Amount Borrowed:** x + y + z = 2,000,000
* **Total Annual Interest:** 0.05x + 0.065y + 0.075z = 121,250
* **Relationship Between y and z:** y = 4z

**3. Rewrite the Equations in Standard Form:**

1.  x + y + z = 2,000,000
2.  0.05x + 0.065y + 0.075z = 121,250
3.  0x + y - 4z = 0

To simplify the second equation, multiply it by 1000:

1.  x + y + z = 2,000,000
2.  50x + 65y + 75z = 121,250,000
3.  y - 4z = 0

**4. Represent the System as an Augmented Matrix:**

```
[ 1   1   1 | 2000000 ]
[ 50  65  75 | 121250000 ]
[ 0   1  -4 | 0 ]
```

**5. Solve Using Row Operations:**

1.  **Eliminate x from the second row:**
    * R2 = R2 - 50 * R1
    ```
    [ 1   1   1 | 2000000 ]
    [ 0  15  25 | 21250000 ]
    [ 0   1  -4 | 0 ]
    ```

2.  **Swap R2 and R3 to make R2 easier to work with:**
    ```
    [ 1   1   1 | 2000000 ]
    [ 0   1  -4 | 0 ]
    [ 0  15  25 | 21250000 ]
    ```

3.  **Eliminate y from the third row:**
    * R3 = R3 - 15 * R2
    ```
    [ 1   1   1 | 2000000 ]
    [ 0   1  -4 | 0 ]
    [ 0   0  85 | 21250000 ]
    ```

4.  **Solve for z:**
    * 85z = 21,250,000
    * z = 21,250,000 / 85
    * z = 250,000

5.  **Solve for y:**
    * y - 4z = 0
    * y = 4z
    * y = 4 * 250,000
    * y = 1,000,000

6.  **Solve for x:**
    * x + y + z = 2,000,000
    * x = 2,000,000 - y - z
    * x = 2,000,000 - 1,000,000 - 250,000
    * x = 750,000

**6. Answers:**

* Amount borrowed at 5% (x): $750,000
* Amount borrowed at 6.5% (y): $1,000,000
* Amount borrowed at 7.5% (z): $250,000