5)
The given equations are
- 4x - 2y + 3z = 7
- 3x + 5y - 3z = 13
- 5x + y - z = 11
From equation 3, we have
y = 11 + 5x + z
We would substitute y = 11 + 5x + z into equations 1 and 2. For equation 1, we have
- 4x - 2(11 + 5x + z) + 3z = 7
- 4x - 22 - 10x - 2z + 3z = 7
- 4x - 10x - 2z + 3z = 7 + 22
- 14x + z = 29
For equation 2, we have
- 3x + 5(11 + 5x + z) - 3z = 13
- 3x + 55 + 25x + 5z - 3z = 13
- 3x + 25x + 5z - 3z = 13 - 55
22x + 2z = - 42
Dividing both sides of the equation by 2, we have
11x + z = - 21
z = - 21 - 11x
Substituting z = - 21 - 11x into - 14x + z = 29, we have
- 14x - 21 - 11x = 29
- 14x - 11x = 29 + 21
- 25x = 50
x = 50/- 25
x = - 2
z = - 21 - 11(- 2) = - 21 + 22
z = 1
Substituting x = - 2 and z = 1 into y = 11 + 5x + z, we have
y = 11 + 5(-2) + 1
y = 11 - 10 + 1
y = 2
The solutions are
x = - 2, y = 2, z = 1