Final answer:
To find the complete solution of the linear system or show that it is inconsistent, we need to solve the given system of equations using the method of elimination. The complete solution to the linear system is x = -5/3, y = 2, z = -1/3.
Step-by-step explanation:
To find the complete solution of the linear system or show that it is inconsistent, we need to solve the given system of equations:
- x + y + z = 0
- -x + 2y + 5z = 3
- 3x - y = 6
We will use the method of elimination to solve these equations. First, we will eliminate the x variable:
Multiply equation 1 by 3 and add it to equation 3:
3(x + y + z) + (3x - y) = 0 + 6
Simplifying, we get:
4y + 6z = 6
Next, we will eliminate the x variable again:
Multiply equation 1 by -1 and add it to equation 2:
-(-x + 2y + 5z) + (x + y + z) = -3 + 0
Simplifying, we get:
-y + 3z = -3
Now we have a system of two equations:
4y + 6z = 6
-y + 3z = -3
Multiply equation 2 by 4 and add it to equation 1:
4(-y + 3z) + (4y + 6z) = -12 + 6
Simplifying, we get:
18z = -6
Dividing both sides by 18, we get:
z = -6/18
Simplifying, we get:
z = -1/3
Substitute this value of z back into equation 2:
-y + 3(-1/3) = -3
Simplifying, we get:
-y - 1 = -3
Adding 1 to both sides, we get:
-y = -2
Dividing both sides by -1, we get:
y = 2
Substitute the values of y and z back into equation 1:
x + 2 + (-1/3) = 0
Simplifying, we get:
x + 5/3 = 0
Subtracting 5/3 from both sides, we get:
x = -5/3
Therefore, the complete solution to the linear system is:
x = -5/3, y = 2, z = -1/3