x= -3, y = 1, z = 6
Step-by-step explanation:
-2x + 2y + z = 14 ....equation 1
3x - 2y + z = -5 ....equation 2
-x + y - 2z = -8 ....equation 3
Note: Multiplication of same sign gives positive number. Multiplication of opposite signs give negative number.
Subtract equation 2 from 1:
-2x - 3x +2y -(-2y) + z - z = 14 -(-5)
-5x + 2y + 2y + 0 = 14 + 5
-5x + 4y = 19 ....equation 4
Due to the first equation we derived where we only have x and y. We need to solve the second equation in such a way we get only x and y.
This means we find a way to eliminate z in the second subtraction.
To do this, we multiply 1 by 2. Then we add both equations to eliminate z
-4x + 4y + 2z = 28 ....equation 1
-x + y - 2z = -8 ....equation 3
Add equation 3 and 1:
-4x + (-x) + 4y + y + 2z + (-2z) = -8 + 28
-4x - x + 5y + z + 2z - 2z = 20
-5x + 5y = 20 ....equation 5
-5x + 4y = 19 ....equation 4
-5x + 5y = 20 ....equation 5
subtract equation 4 from 5 (we will be eliminating x):
-5x -(-5x) + 5y - 4y = 20 - 19
-5x + 5x + y = 1
y = 1
we substitue the value of y in any of the equation between 4 and 5:
using equation 4:
-5x + 4(1) = 19
-5x + 4 = 19
-5x = 19-4
-5x = 15
x = 15/-5
x = -3
we substute for both y and x in any of the initial equation to get z:
using equation 2:
3(-3) -2(1) +z = -5
-9 - 2 + z = -5
-11 + z = -5
z = -5 + 11
z = 6