Answer:
(1, 1, 3)
Step-by-step explanation:
The initial system of equations is:
-2x + 3y - z = -2
3x + y = 4
-2y + 2z = 4
First, multiply the second equation by -1, so:
3x + y = 4
(-1)3x + (-1)y = (-1)4
- 3x - y = 4
Then, add this equation to the other two:
-2x + 3y - z = -2
- 3x - y = -4
-2y + 2z = 4
-5x + 0 + z = -2
-5x + z = -2
Now, let's solve for y and z as follows:
3x + y = 4
3x + y - 3x = 4 - 3x
y = 4 - 3x
-5x + z = -2
-5x + z + 5x = -2 + 5x
z = -2 + 5x
Finally, replace y = 4 - 3x and z = -2 + 5x in the first equation and solve for x
-2x + 3y - z = -2
-2x + 3(4 - 3x) - (-2 + 5x) = -2
-2x + 3(4) + 3(-3x) + 2 - 5x = -2
-2x + 12 - 9x + 2 - 5x = -2
-16x + 14 = -2
-16x + 14 - 14 = -2 - 14
-16x = -16
-16x/(-16) = -16/(-16)
x = 1
Now, with the value of x, we can calculate y and z as follows
y = 4 - 3x
y = 4 - 3(1)
y = 4 - 3
y = 1
z = -2 + 5x
z = -2 + 5(1)
z = -2 + 5
z = 3
Therefore, the answer in the form (x, y, z) is:
(1, 1, 3)