SOLUTION: Gonzalez manufacturing borrowed $21000. , part of the money was borrowed at 8%, part 10% and part 12%. annual interest was $2120 and the total at 8%, and 10% was twice the amount
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Question 1185918: Gonzalez manufacturing borrowed $21000. , part of the money was borrowed at 8%, part 10% and part 12%. annual interest was $2120 and the total at 8%, and 10% was twice the amount borrowed at 12%. find amount borrowed at each rate using the Gaussian elimination or Gauss-Jordan elimination. Need steps please Answer by CPhill(1959) (Show Source):
You can put this solution on YOUR website! 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
