Question

What is the solution to the following system?
{3x+2y+z=20
{x-4y-z=-10
{2x+y+2z=15

(2, 3, 4)
(4, 3, –2)
(4, 3, 2)
(6, 7, –2)

Answer 1

Solve from the second equation: x = -10 + 4y + z.

Substitute into the first and third:
- First: 3(-10+4y+z) + 2y + z = 20 → 14y + 4z = 50 → 7y + 2z = 25.
- Third: 2(-10+4y+z) + y + 2z = 15 → 9y + 4z = 35.

Subtract: (9y+4z) − (7y+2z) = 35 − 25 → 2y + 2z = 10 → y + z = 5.

Thus z = 5 − y. Plug into 7y + 2z = 25:
7y + 2(5 − y) = 25 → 5y = 15 → y = 3, z = 2, x = -10 + 4·3 + 2 = 4.

Solution: (4, 3, 2).