Question 1185918
Here's how to solve this problem using Gaussian elimination:

**1. Define variables:**

*   Let 'x' be the amount borrowed at 8%.
*   Let 'y' be the amount borrowed at 10%.
*   Let 'z' be the amount borrowed at 12%.

**2. Set up the equations:**

*   Equation 1 (Total amount borrowed): x + y + z = 21000
*   Equation 2 (Total interest): 0.08x + 0.10y + 0.12z = 2120
*   Equation 3 (Relationship between amounts): x + y = 2z

**3. Rewrite the equations in standard form:**

*   x + y + z = 21000
*   8x + 10y + 12z = 212000 (Multiply equation 2 by 100 to eliminate decimals)
*   x + y - 2z = 0 (Rewrite equation 3)

**4. Perform Gaussian elimination:**

*   **Eliminate x from the second and third equations:**
    *   Subtract 8 times equation 1 from equation 2:
        2y + 4z = 42000
    *   Subtract equation 1 from equation 3:
        -3z = -21000

*   **Solve for z:**
    z = -21000 / -3 = 7000

*   **Substitute z back into the modified second equation:**
    2y + 4(7000) = 42000
    2y + 28000 = 42000
    2y = 14000
    y = 7000

*   **Substitute y and z back into the first equation:**
    x + 7000 + 7000 = 21000
    x + 14000 = 21000
    x = 7000

**Solution:**

*   Amount borrowed at 8% (x): $7000
*   Amount borrowed at 10% (y): $7000
*   Amount borrowed at 12% (z): $7000